Encyclopedia of Crystallographic Prototypes

Space Group Notation

Space Group Properties and Notation

This table lists the space group notations used throughout the Encyclopedia. In general, we use the first space group orientation listed in the International Tables for Crystallography. There are several exceptions:

  • Some centrosymmetric space groups, e.g. $Fm\overline{3}m$ #227, have two settings, reflecting a choice of origin. We always take the second setting, which places the origin of the real-space coordinate system at an inversion site. This is denoted as setting 2.
  • For rhombohedral unit cells we always use the label for the full hexagonal unit cell (H) rather than the primitive rhombohedral cell (R), which follows our choice to describe the unit cell in terms of the hexagonal lattice parameters $a$ and $c$, rather than the rhombohedral lattice parameters $a$ and $\alpha$. (Setting H)
  • Monoclinic cells are always represented using the “unique axis $b$, ” i.e. the $b$ axis is perpendicular to both the $a$ and the $c$ axes, and those two axes are not orthogonal to one another. (Setting b)

Over the year multiple methods have been devised to label space groups. We include some of the most common.

  • The space group number, as listed in the International Tables
  • Hermann-Mauguin: (Hermann, 1928) and ( Mauguin, 1931), as updated by (de Wolff, 1992)
  • Hall: (Hall, 1981)
  • International: a compact form of the Hermann-Mauguin notation used in the International Tables of Crystallography (Brock, 2016)
  • Schoenflies: (Schoenflies, 1891) and (Schoenflies, 1923). Unlike the previous three notations, the label here remains the same regardless of the orientation of the crystal.

A more comprehensive list of all space group orientations and settings can be found at

Class classifies each space group by its chirality and the presence or lack of an inversion site:

  • Centrosymmetric space groups are achiral, and have an inversion site. Because of this, any structure appearing in one of these space groups is achiral.
  • Achiral space groups (not the best name) are achiral, lack an inversion site, and include mirror operations so that they do not support chiral structures.
  • Sohncke II space groups are themselves chiral. They are paired, with a space group with a screw axis of the form $n_{m}$ having an enantiomorphic twin with a $n_{n-m}$ axis.
  • Sohncke III space groups are themselves achiral, but have no mirror or mirror-like operations and so support chiral structures.

The Other column contains other information about the groups:

  • Polar space groups have at least one axis where the origin is not fixed by symmetry. This automatically excludes centrosymmetric and cubic space groups.
  • Space groups with a Screw Axis can be rotated along an axis and translated along the same axis without changing the crystal structure. Centrosymmetric, chiral, achiral, polar and cubic space groups may have screw axes. Monoclinic space groups can have zero, one, two, or three screw axes, while higher symmetry space groups are limited to one.
Number Setting Hermann-Mauguin Hall International Schoenflies Class Other
Triclinic Space Groups
1 P 1 P 1 $P1$ $C_{s}^{2}$ Sohncke III Polar
2 P -1 -P 1 $P\overline{1}$ $C_{i}^{1}$ Centrosymmetric
Monoclinic Space Groups
3 b P 1 2 1 P 2y $P2$ $C_{2}^{1}$ Sohncke III Polar
4 b P 1 21 1 P 2yb $P2_{1}$ $C_{2}^{2}$ Sohncke III   Polar   & Screw Axis
5 b1 C 1 2 1 C 2y $C2$ $C_{2}^{3}$ Sohncke III Polar
6 b P 1 m 1 P -2y $Pm$ $C_{s}^{1}$ Achiral Polar
7 b1 P 1 c 1 P -2yc $Pc$ $C_{s}^{2}$ Achiral Polar
8 b1 C 1 m 1 C -2y $Cm$ $C_{s}^{3}$ Achiral Polar
9 b1 C 1 c 1 C -2yc $Cc$ $C_{s}^{4}$ Achiral Polar
10 b P 1 2/m 1 -P 2y $P2/m$ $C_{2h}^{1}$ Centrosymmetric
11 b P 1 21/m 1 -P 2yb $P2_{1}/m$ $C_{2h}^{2} $ Centrosymmetric Screw Axis
12 b1 C 1 2/m 1 -C 2y $C2/m$ $C_{2h}^{3}$ Centrosymmetric
13 b1 P 1 2/c 1 -P 2yc $P2/c$ $C_{2h}^{4}$ Centrosymmetric
14 b1 P 1 21/c 1 -P 2ybc $P2_{1}/c$ $C_{2h}^{5}$ Centrosymmetric Screw Axis
15 b1 C 1 2/c 1 -C 2yc $C2/c$ $C_{2h}^{6}$ Centrosymmetric
Orthorhombic Space Groups
16 P 2 2 2 P 2 2 $P222$ $D_{2}^{1}$ Sohncke III
17 P 2 2 21 P 2c 2 $P222_{1}$ $D_{2}^{2}$ Sohncke III Screw Axis
18 P 21 21 2 P 2 2ab $P2_{1}2_{1}2$ $D_{2}^{3}$ Sohncke III Screw Axis
19 P 21 21 21 P 2ac 2ab $P2_{1}2_{1}2_{1}$ $D_{2}^{4}$ Sohncke III Screw Axis
20 C 2 2 21 C 2c 2 $C222_{1}$ $D_{2}^{5}$ Sohncke III Screw Axis
21 C 2 2 2 C 2 2 $C222$ $D_{2}^{6}$ Sohncke III
22 F 2 2 2 F 2 2 $F222$ $D_{2}^{7}$ Sohncke III
23 I 2 2 2 I 2 2 $I222$ $D_{2}^{8}$ Sohncke III
24 I 21 21 21 I 2b 2c $I2_{1}2_{1}2_{1}$ $D_{2}^{9}$ Sohncke III Screw Axis
25 P m m 2 P 2 -2 $Pmm2$ $C_{2v}^{1}$ Achiral Polar
26 P m c 21 P 2c -2 $Pmc2_{1}$ $C_{2v}^{2}$ Achiral  Polar   & Screw Axis
27 P c c 2 P 2 -2c $Pcc2$ $C_{2v}^{3}$ Achiral Polar
28 P m a 2 P 2 -2a $Pma2$ $C_{2v}^{4}$ Achiral Polar
29 P c a 21 P 2c -2ac $Pca2_{1}$ $C_{2v}^{5}$ Achiral   Polar   & Screw Axis
30 P n c 2 P 2 -2bc $Pnc2$ $C_{2v}^{6}$ Achiral Polar
31 P m n 21 P 2ac -2 $Pmn2_{1}$ $C_{2v}^{7}$ Achiral   Polar   & Screw Axis
32 P b a 2 P 2 -2ab $Pba2$ $C_{2v}^{8}$ Achiral Polar
33 P n a 21 P 2c -2n $Pna2_{1}$ $C_{2v}^{9}$ Achiral   Polar   & Screw Axis
34 P n n 2 P 2 -2n $Pnn2$ $C_{2v}^{10}$ Achiral Polar
35 C m m 2 C 2 -2 $Cmm2$ $C_{2v}^{11}$ Achiral Polar
36 C m c 21 C 2c -2 $Cmc2_{1}$ $C_{2v}^{12}$ Achiral   Polar   & Screw Axis
37 C c c 2 C 2 -2c $Ccc2$ $C_{2v}^{13}$ Achiral Polar
38 A m m 2 A 2 -2 $Amm2$ $C_{2v}^{14}$ AchiralPolar
39 A e m 2 A 2 -2c $Aem2$ $C_{2v}^{15}$ Achiral Polar
40 A m a 2 A 2 -2a $Ama2$ $C_{2v}^{16}$ Achiral Polar
41 A e a 2 A 2 -2ac $Aea2$ $C_{2v}^{17}$ Achiral Polar
42 F m m 2 F 2 -2 $Fmm2$ $C_{2v}^{18}$ Achiral Polar
43 F d d 2 F 2 -2d $Fdd2$ $C_{2v}^{19}$ Achiral Polar
44 I m m 2 I 2 -2 $Imm2$ $C_{2v}^{20}$ Achiral Polar
45 I b a 2 I 2 -2c $Iba2$ $C_{2v}^{21}$ Achiral Polar
46 I m a 2 I 2 -2a $Ima2$ $C_{2v}^{22}$ Achiral Polar
47 P 2/m 2/m 2/m -P 2 2 $Pmmm$ $D_{2h}^{1}$ Centrosymmetric
48 2 P 2/n 2/n 2/n:2 -P 2ab 2bc $Pnnn$ $D_{2h}^{2}$ Centrosymmetric
49 P 2/c 2/c 2/m -P 2 2c $Pccm$ $D_{2h}^{3}$ Centrosymmetric
50 2 P 2/b 2/a 2/n:2 -P 2ab 2b $Pban$ $D_{2h}^{4}$ Centrosymmetric
51 P 21/m 2/m 2/a -P 2a 2a $Pmma$ $D_{2h}^{5}$ Centrosymmetric Screw Axis
52 P 2/n 21/n 2/a -P 2a 2bc $Pnna$ $D_{2h}^{6i}$ Centrosymmetric Screw Axis
53 P 2/m 2/n 21/a -P 2ac 2 $Pmna$ $D_{2h}^{7}$ Centrosymmetric Screw Axis
54 P 21/c 2/c 2/a -P 2a 2ac $Pcca$ $D_{2h}^{8}$ Centrosymmetric Screw Axis
55 P 21/b 21/a 2/m -P 2 2ab $Pbam$ $D_{2h}^{9}$ Centrosymmetric Screw Axis
56 P 21/c 21/c 2/n -P 2ab 2ac $Pccn$ $D_{2h}^{10}$ Centrosymmetric Screw Axis
57 P 2/b 21/c 21/m -P 2c 2b $Pbcm$ $D_{2h}^{11}$ Centrosymmetric Screw Axis
58 P 21/n 21/n 2/m -P 2 2n $Pnnm$ $D_{2h}^{12}$ Centrosymmetric Screw Axis
59 2 P 21/m 21/m 2/n:2 -P 2ab 2a $Pmmn$ $D_{2h}^{13}$ Centrosymmetric Screw Axis
60 P 21/b 2/c 21/n -P 2n 2ab $Pbcn$ $D_{2h}^{14}$ Centrosymmetric Screw Axis
61 P 21/b 21/c 21/a -P 2ac 2ab $Pbca$ $D_{2h}^{15}$ Centrosymmetric Screw Axis
62 P 21/n 21/m 21/a -P 2ac 2n $Pnma$ $D_{2h}^{16}$ Centrosymmetric Screw Axis
63 C 2/m 2/c 21/m -C 2c 2 $Cmcm$ $D_{2h}^{17}$ Centrosymmetric Screw Axis
64 C 2/m 2/c 21/a -C 2bc 2 $Cmce$ $D_{2h}^{18}$ Centrosymmetric Screw Axis
65 C 2/m 2/m 2/m -C 2 2 $Cmmm$ $D_{2h}^{19}$ Centrosymmetric Screw Axis
66 C 2/c 2/c 2/m -C 2 2c $Cccm$ $D_{2h}^{20}$ Centrosymmetric
67 C 2/m 2/m 2/a -C 2b 2 $Cmme$ $D_{2h}^{21}$ Centrosymmetric
68 2 C 2/c 2/c 2/a:2 -C 2b 2bc $Ccca$ $D_{2h}^{22}$ Centrosymmetric
69 F 2/m 2/m 2/m -F 2 2 $Fmmm$ $D_{2h}^{23}$ Centrosymmetric
70 2 F 2/d 2/d 2/d:2 -F 2uv 2vw $Fddd$ $D_{2h}^{24}$ Centrosymmetric
71 I 2/m 2/m 2/m -I 2 2 $Immm$ $D_{2h}^{25}$ Centrosymmetric
72 I 2/b 2/a 2/m -I 2 2c $Ibam$ $D_{2h}^{26}$ Centrosymmetric
73 I 2/b 2/c 2/a -I 2b 2c $Ibca$ $D_{2h}^{27}$ Centrosymmetric
74 I 2/m 2/m 2/a -I 2b 2 $Imma$ $D_{2h}^{28}$ Centrosymmetric
Tetragonal Space Groups
75 P 4 P 4 $P4$ $C_{4}^{1}$ Sohncke III Polar
76 P 41 P 4w $P4_{1}$ $C_{4}^{2}$ Sohncke II   Polar   & Screw Axis
77 P 42 P 4c $P4_{2}$ $C_{4}^{3}$ Sohncke III   Polar   & Screw Axis
78 P 43 P 4cw $P4_{3}$ $C_{4}^{4}$ Sohncke II   Polar   & Screw Axis
79 I 4 I 4 $I4$ $C_{4}^{5}$ Sohncke III Polar
80 I 41 I 4bw $I4_{1}$ $C_{4}^{6}$ Sohncke III   Polar   & Screw Axis
81 P -4 P -4 $P\overline{4}$ $S_{4}^{1}$ Achiral
82 I -4 I -4 $I\overline{4}$ $S_{4}^{2}$ Achiral
83 P 4/m -P 4 $P4/m$ $C_{4h}^{1}$ Centrosymmetric
84 P 42/m -P 4c $P4_{2}/m$ $C_{4h}^{2}$ Centrosymmetric Screw Axis
85 2 P 4/n:2 -P 4a $P4/n$ $C_{4h}^{3}$ Centrosymmetric
86 2 P 42/n:2 -P 4bc $P4_{2}/n$ $C_{4h}^{4}$ Centrosymmetric Screw Axis
87 I 4/m -I 4 $I4/m$ $C_{4h}^{5}$ Centrosymmetric
88 2 I 41/a:2 -I 4ad $I4_{1}/a$ $C_{4h}^{6}$ Centrosymmetric Screw Axis
89 P 4 2 2 P 4 2 $P422$ $D_{4}^{1}$ Sohncke III
90 P 4 21 2 P 4ab 2ab $P42_{1}2$ $D_{4}^{2}$ Sohncke III Screw Axis
91 P 41 2 2 P 4w 2c $P4_{1}22$ $D_{4}^{3}$ Sohncke II Screw Axis
92 P 41 21 2 P 4abw 2nw $P4_{1}2_{1}2$ $D_{4}^{4}$ Sohncke II Screw Axis
93 P 42 2 2 P 4c 2 $P4_{2}22$ $D_{4}^{5}$ Sohncke III Screw Axis
94 P 42 21 2 P 4n 2n $P4_{2}2_{1}2$ $D_{4}^{6}$ Sohncke III Screw Axis
95 P 43 2 2 P 4cw 2c $P4_{3}22$ $D_{4}^{7}$ Sohncke II Screw Axis
96 P 43 21 2 P 4nw 2abw $P4_{3}2_{1}2$ $D_{4}^{8}$ Sohncke II Screw Axis
97 I 4 2 2 I 4 2 $I422$ $D_{4}^{9}$ Sohncke III
98 I 41 2 2 I 4bw 2bw $I4_{1}22$ $D_{4}^{10}$ Sohncke III Screw Axis
99 P 4 m m P 4 -2 $P4mm$ $C_{4v}^{1}$ Achiral Polar
100 P 4 b m P 4 -2ab $P4bm$ $C_{4v}^{2}$ Achiral Polar
101 P 42 c m P 4c -2c $P4_{2}cm$ $C_{4v}^{3}$ Achiral   Polar   & Screw Axis
102 P 42 n m P 4n -2n $P4_{2}nm$ $C_{4v}^{4}$ Achiral   Polar   & Screw Axis
103 P 4 c c P 4 -2c $P4cc$ $C_{4v}^{5}$ Achiral Polar
104 P 4 n c P 4 -2n $P4nc$ $C_{4v}^{6}$ Achiral Polar
105 P 42 m c P 4c -2 $P4_{2}mc$ $C_{4v}^{7}$ Achiral   Polar   & Screw Axis
106 P 42 b c P 4c -2ab $P4_{2}bc$ $C_{4v}^{8}$ Achiral   Polar   & Screw Axis
107 I 4 m m I 4 -2 $I4mm$ $C_{4v}^{9}$ Achiral Polar
108 I 4 c m I 4 -2c $I4cm$ $C_{4v}^{10} $ Achiral Polar
109 I 41 m d I 4bw -2 $I4_{1}md$ $C_{4v}^{11}$ Achiral   Polar   & Screw Axis
110 I 41 c d I 4bw -2c $I4_{1}cd$ $C_{4v}^{12}$ Achiral   Polar   & Screw Axis
111 P -4 2 m P -4 2 $P\overline{4}2m$ $D_{2d}^{1}$ Achiral
112 P -4 2 c P -4 2c $P\overline{4}2c$ $D_{2d}^{2}$ Achiral
113 P -4 21 m P -4 2ab $P\overline{4}2_{1}m$ $D_{2d}^{3}$ Achiral Screw Axis
114 P -4 21 c P -4 2n $P\overline{4}2_{1}c$ $D_{2d}^{4}$ Achiral Screw Axis
115 P -4 m 2 P -4 -2 $P\overline{4}m2$ $D_{2d}^{5}$ Achiral
116 P -4 c 2 P -4 -2c $P\overline{4}c2$ $D_{2d}^{6}$ Achiral
117 P -4 b 2 P -4 -2ab $P\overline{4}b2$ $D_{2d}^{7}$ Achiral
118 P -4 n 2 P -4 -2n $P\overline{4}n2$ $D_{2d}^{8}$ Achiral
119 I -4 m 2 I -4 -2 $I\overline{4}m2$ $D_{2d}^{9}$ Achiral
120 I -4 c 2 I -4 -2c $I\overline{4}c2$ $D_{2d}^{10}$ Achiral
121 I -4 2 m I -4 2 $I\overline{4}2m$ $D_{2d}^{11}$ Achiral
122 I -4 2 d I -4 2bw $I\overline{4}2d$ $D_{2d}^{12}$ Achiral
123 P 4/m 2/m 2/m -P 4 2 $P4/mmm$ $D_{4h}^{1}$ Centrosymmetric
124 P 4/m 2/c 2/c -P 4 2c $P4/mcc$ $D_{4h}^{2}$ Centrosymmetric
125 2 P 4/n 2/b 2/m:2 -P 4a 2b $P4/nbm$ $D_{4h}^{3}$ Centrosymmetric
126 2 P 4/n 2/n 2/c:2 -P 4a 2bc $P4/nnc$ $D_{4h}^{4}$ Centrosymmetric
127 P 4/m 21/b 2/m -P 4 2ab $P4/mbm$ $D_{4h}^{5}$ Centrosymmetric Screw Axis
128 P 4/m 21/n 2/c -P 4 2n $P4/mnc$ $D_{4h}^{6}$ Centrosymmetric Screw Axis
129 2 P 4/n 21/m 2/m:2 -P 4a 2a $P4/nmm$ $D_{4h}^{7}$ Centrosymmetric Screw Axis
130 2 P 4/n 21/c 2/c:2 -P 4a 2ac $P4/ncc$ $D_{4h}^{8}$ Centrosymmetric Screw Axis
131 P 42/m 2/m 2/c -P 4c 2 $P4_{2}/mmc$ $D_{4h}^{9}$ Centrosymmetric Screw Axis
132 P 42/m 2/c 2/m -P 4c 2c $P4_{2}/mcm$ $D_{4h}^{10}$ Centrosymmetric Screw Axis
133 2 P 42/n 2/b 2/c:2 -P 4ac 2b $P4_{2}/nbc$ $D_{4h}^{11}$ Centrosymmetric Screw Axis
134 2 P 42/n 2/n 2/m:2 -P 4ac 2bc $P4_{2}/nnm$ $D_{4h}^{12}$ Centrosymmetric Screw Axis
135 P 42/m 21/b 2/c -P 4c 2ab $P4_{2}/mbc$ $D_{4h}^{13}$ Centrosymmetric Screw Axis
136 P 42/m 21/n 2/m -P 4n 2n $P4_{2}/mnm$ $D_{4h}^{14}$ Centrosymmetric Screw Axis
137 2 P 42/n 21/m 2/c:2 -P 4ac 2a $P4_{2}/nmc$ $D_{4h}^{15}$ Centrosymmetric Screw Axis
138 2 P 42/n 21/c 2/m:2 -P 4ac 2ac $P4_{2}/ncm$ $D_{4h}^{16}$ Centrosymmetric Screw Axis
139 I 4/m 2/m 2/m -I 4 2 $I4/mmm$ $D_{4h}^{17}$ Centrosymmetric
140 I 4/m 2/c 2/m -I 4 2c $I4/mcm$ $D_{4h}^{18}$ Centrosymmetric
141 2 I 41/a 2/m 2/d:2 -I 4bd 2 $I4_{1}/amd$ $D_{4h}^{19}$ Centrosymmetric Screw Axis
142 2 I 41/a 2/c 2/d:2 -I 4bd 2c $I4_{1}/acd$ $D_{4h}^{20}$ Centrosymmetric Screw Axis
Trigonal Space Groups
143 P 3 P 3 $P3$ $C_{3}^{1}$ Sohncke III Polar
144 P 31 P 31 $P3_{1}$ $C_{3}^{2}$ Sohncke II   Polar   & Screw Axis
145 P 32 P 32 $P3_{2}$ $C_{3}^{3}$ Sohncke II   Polar   & Screw Axis
146 H R 3:H R 3 $R3$ $C_{3}^{4}$ Sohncke III Polar
147 P -3 -P 3 $P\overline{3}$ $C_{3i}^{1}$ Centrosymmetric
148 H R -3:H -R 3 $R\overline{3}$ $C_{3i}^{2}$ Centrosymmetric
149 P 3 1 2 P 3 2 $P312$ $D_{3}^{1}$ Sohncke III
150 P 3 2 1 P 3 2'' $P321$ $D_{3}^{2}$ Sohncke III
151 P 31 1 2 P 31 2c (0 0 1) $P3_{1}12$ $D_{3}^{3}$ Sohncke II Screw Axis
152 P 31 2 1 P 31 2'' $P3_{1}21$ $D_{3}^{4}$ Sohncke II Screw Axis
153 P 32 1 2 P 32 2c (0 0 -1) $P3_{2}12$ $D_{3}^{5}$ Sohncke II Screw Axis
154 P 32 2 1 P 32 2'' $P3_{2}21$ $D_{3}^{6}$ Sohncke II Screw Axis
155 H R 32:H R 3 2'' $R32$ $D_{3}^{7}$ Sohncke III Screw Axis
156 P 3 m 1 P 3 -2'' $P3m1$ $C_{3v}^{1} $ Achiral Polar
157 P 3 1 m P 3 -2 $P31m$ $C_{3v}^{2}$ Achiral Polar
158 P 3 c 1 P 3 -2''c $P3c1$ $C_{3v}^{3}$ Achiral Polar
159 P 3 1 c P 3 -2c $P31c$ $C_{3v}^{4} $ Achiral Polar
160 H R 3 m:H R 3 -2'' $R3m$ $C_{3v}^{5}$ Achiral Polar
161 H R 3 c:H R 3 -2''c $R3c$ $C_{3v}^{6}$ Achiral Polar
162 P -3 1 2/m -P 3 2 $P\overline{3}1m$ $D_{3d}^{1}$ Centrosymmetric
163 P -3 1 2/c -P 3 2c $P\overline{3}1c$ $D_{3d}^{2}$ Centrosymmetric
164 P -3 2/m 1 -P 3 2'' $P\overline{3}m1$ $D_{3d}^{3}$ Centrosymmetric
165 P -3 2/c 1 -P 3 2''c $P\overline{3}c1$ $D_{3d}^{4}$ Centrosymmetric
166 H R -3 2/m:H -R 3 2'' $R\overline{3}m$ $D_{3d}^{5}$ Centrosymmetric
167 H R -3 2/c:H -R 3 2''c $R\overline{3}c$ $D_{3d}^{6}$ Centrosymmetric
Hexagonal Space Groups
168 P 6 P 6 $P6$ $C_{6}^{1}$ Sohncke III Polar
169 P 61 P 61 $P6_{1}$ $C_{6}^{2}$ Sohncke II   Polar   & Screw Axis
170 P 65 P 65 $P6_{5}$ $C_{6}^{3}$ Sohncke II   Polar   & Screw Axis
171 P 62 P 62 $P6_{2}$ $C_{6}^{4}$ Sohncke II   Polar   & Screw Axis
172 P 64 P 64 $P6_{4}$ $C_{6}^{5}$ Sohncke II   Polar   & Screw Axis
173 P 63 P 6c $P6_{3}$ $C_{6}^{6}$ Sohncke III   Polar   & Screw Axis
174 P -6 P -6 $P\overline{6}$ $C_{3h}^{1}$ Achiral
175 P 6/m -P 6 $P6/m$ $C_{6h}^{1}$ Centrosymmetric
176 P 63/m -P 6c $P6_{3}/m$ $C_{6h}^{2}$ Centrosymmetric Screw Axis
177 P 6 2 2 P 6 2 $P622$ $D_{6}^{1}$ Sohncke III
178 P 61 2 2 P 61 2 (0 0 -1) $P6_{1}22$ $D_{6}^{2}$ Sohncke II Screw Axis
179 P 65 2 2 P 65 2 (0 0 1) $P6_{5}22$ $D_{6}^{3}$ Sohncke II Screw Axis
180 P 62 2 2 P 62 2c (0 0 1) $P6_{2}22$ $D_{6}^{4}$ Sohncke II Screw Axis
181 P 64 2 2 P 64 2c (0 0 -1) $P6_{4}22$ $D_{6}^{5}$ Sohncke II Screw Axis
182 P 63 2 2 P 6c 2c $P6_{3}22$ $D_{6}^{6}$ Sohncke III Screw Axis
183 P 6 m m P 6 -2 $P6mm$ $C_{6v}^{1}$ Achiral Polar
184 P 6 c c P 6 -2c $P6cc$ $C_{6v}^{2}$ Achiral Polar
185 P 63 c m P 6c -2 $P6_{3}cm$ $C_{6v}^{3}$ Achiral   Polar   & Screw Axis
186 P 63 m c P 6c -2c $P6_{3}mc$ $C_{6v}^{4}$ Achiral   Polar   & Screw Axis
187 P -6 m 2 P -6 2 $P\overline{6}m2$ $D_{3h}^{1}$ Achiral
188 P -6 c 2 P -6c 2 $P\overline{6}c2$ $D_{3h}^{2}$ Achiral
189 P -6 2 m P -6 -2 $P\overline{6}2m$ $D_{3h}^{3}$ Achiral
190 P -6 2 c P -6c -2c $P\overline{6}2c$ $D_{3h}^{4}$ Achiral
191 P 6/m 2/m 2/m -P 6 2 $P6/mmm$ $D_{6h}^{1}$ Centrosymmetric
192 P 6/m 2/c 2/c -P 6 2c $P6/mcc$ $D_{6h}^{2}$ Centrosymmetric
193 P 63/m 2/c 2/m -P 6c 2 $P6_{3}/mcm$ $D_{6h}^{3}$ Centrosymmetric Screw Axis
194 P 63/m 2/m 2/c -P 6c 2c $P6_{3}/mmc$ $D_{6h}^{4}$ Centrosymmetric Screw Axis
Cubic Space Groups
195 P 2 3 P 2 2 3 $P23$ $T_{}^{1}$ Sohncke III
196 F 2 3 F 2 2 3 $F23$ $T_{}^{2}$ Sohncke III
197 I 2 3 I 2 2 3 $I23$ $T_{}^{3}$ Sohncke III
198 P 21 3 P 2ac 2ab 3 $P2_{1}3$ $T_{}^{4}$ Sohncke III Screw Axis
199 I 21 3 I 2b 2c 3 $I2_{1}3$ $T_{}^{5}$ Sohncke III Screw Axis
200 P 2/m -3 -P 2 2 3 $Pm\overline{3}$ $T_{h}^{1}$ Centrosymmetric
201 2 P 2/n -3:2 -P 2ab 2bc 3 $Pn\overline{3}$ $T_{h}^{2}$ Centrosymmetric
202 F 2/m -3 -F 2 2 3 $Fm\overline{3}$ $T_{h}^{3}$ Centrosymmetric
203 2 F 2/d -3:2 -F 2uv 2vw 3 $Fd\overline{3}$ $T_{h}^{4}$ Centrosymmetric
204 I 2/m -3 -I 2 2 3 $Im\overline{3}$ $T_{h}^{5}$ Centrosymmetric
205 P 21/a -3 -P 2ac 2ab 3 $Pa\overline{3}$ $T_{h}^{6}$ Centrosymmetric Screw Axis
206 I 21/a -3 -I 2b 2c 3 $Ia\overline{3}$ $T_{h}^{7}$ Centrosymmetric Screw Axis
207 P 4 3 2 P 4 2 3 $P432$ $O^{1}$ Sohncke III
208 P 42 3 2 P 4n 2 3 $P4_{2}32$ $O^{2}$ Sohncke III Screw Axis
209 F 4 3 2 F 4 2 3 $F432$ $O^{3}$ Sohncke III
210 F 41 3 2 F 4d 2 3 $F4_{1}32$ $O^{4}$ Sohncke III Screw Axis
211 I 4 3 2 I 4 2 3 $I432$ $O^{5}$ Sohncke III
212 P 43 3 2 P 4acd 2ab 3 $P4_{3}32$ $O^{6}$ Sohncke II Screw Axis
213 P 41 3 2 P 4bd 2ab 3 $P4_{1}32$ $O^{7}$ Sohncke II Screw Axis
214 I 41 3 2 I 4bd 2c 3 $I4_{1}32$ $O^{8}$ Sohncke III Screw Axis
215 P -4 3 m P -4 2 3 $P\overline{4}3m$ $T_{d}^{1}$ Achiral
216 F -4 3 m F -4 2 3 $F\overline{4}3m$ $T_{d}^{2}$ Achiral
217 I -4 3 m I -4 2 3 $I\overline{4}3m$ $T_{d}^{3}$ Achiral
218 P -4 3 n P -4n 2 3 $P\overline{4}3n$ $T_{d}^{4}$ Achiral
219 F -4 3 c F -4c 2 3 $F\overline{4}3c$ $T_{d}^{5}$ Achiral
220 I -4 3 d I -4bd 2c 3 $I\overline{4}3d$ $T_{d}^{6}$ Achiral
221 P 4/m -3 2/m -P 4 2 3 $Pm\overline{3}m$ $O_{h}^{1}$ Centrosymmetric
222 2 P 4/n -3 2/n:2 -P 4a 2bc 3 $Pn\overline{3}n$ $O_{h}^{2}$ Centrosymmetric
223 P 42/m -3 2/n -P 4n 2 3 $Pm\overline{3}n$ $O_{h}^{3}$ Centrosymmetric Screw Axis
224 2 P 42/n -3 2/m:2 -P 4bc 2bc 3 $Pn\overline{3}m$ $O_{h}^{4}$ Centrosymmetric Screw Axis
225 F 4/m -3 2/m -F 4 2 3 $Fm\overline{3}m$ $O_{h}^{5}$ Centrosymmetric
226 F 4/m -3 2/c -F 4c 2 3 $Fm\overline{3}c$ $O_{h}^{6}$ Centrosymmetric
227 2 F 41/d -3 2/m:2 -F 4vw 2vw 3 $Fd\overline{3}m$ $O_{h}^{7}$ Centrosymmetric Screw Axis
228 2 F 41/d -3 2/c:2 -F 4cvw 2vw 3 $Fd\overline{3}c$ $O_{h}^{8}$ Centrosymmetric Screw Axis
229 I 4/m -3 2/m -I 4 2 3 $Im\overline{3}m$ $O_{h}^{9}$ Centrosymmetric
230 I 41/a -3 2/d -I 4bd 2c 3 $Ia\overline{3}d$ $O_{h}^{10}$ Centrosymmetric Screw Axis