Involutional symmetry C_{s}, (*) [ ] = 
Cyclic symmetry C_{nv}, (*nn) [n] = 
Dihedral symmetry D_{nh}, (*n22) [n,2] =  
Polyhedral group, [n,3], (*n32)  

Tetrahedral symmetry T_{d}, (*332) [3,3] = 
Octahedral symmetry O_{h}, (*432) [4,3] = 
Icosahedral symmetry I_{h}, (*532) [5,3] = 
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.
This article lists the groups by Schoenflies notation, Coxeter notation,^{[1]} orbifold notation,^{[2]} and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion.^{[3]}
Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.^{[4]}
Involutional symmetry
There are four involutional groups: no symmetry (C_{1}), reflection symmetry (C_{s}), 2fold rotational symmetry (C_{2}), and central point symmetry (C_{i}).
Intl  Geo ^{[5]} 
Orb.  Schön.  Con.  Cox.  Ord.  Fund. domain 

1  1  11  C_{1}  C_{1}  ][ [ ]^{+} 
1  
2  2  22  D_{1} = C_{2} 
D_{2} = C_{2} 
[2]^{+}  2  
1  22  ×  C_{i} = S_{2} 
CC_{2}  [2^{+},2^{+}]  2  
2 = m 
1  *  C_{s} = C_{1v} = C_{1h} 
±C_{1} = CD_{2} 
[ ]  2 
Cyclic symmetry
There are four infinite cyclic symmetry families, with n = 2 or higher. (n may be 1 as a special case as no symmetry)
Intl  Geo 
Orb.  Schön.  Con.  Cox.  Ord.  Fund. domain 

4  42  2×  S_{4}  CC_{4}  [2^{+},4^{+}]  4  
2/m  22  2*  C_{2h} = D_{1d} 
±C_{2} = ±D_{2} 
[2,2^{+}] [2^{+},2] 
4 
Intl  Geo 
Orb.  Schön.  Con.  Cox.  Ord.  Fund. domain 

2 3 4 5 6 n 
2 3 4 5 6 n 
22 33 44 55 66 nn 
C_{2} C_{3} C_{4} C_{5} C_{6} C_{n} 
C_{2} C_{3} C_{4} C_{5} C_{6} C_{n} 
[2]^{+} [3]^{+} [4]^{+} [5]^{+} [6]^{+} [n]^{+} 
2 3 4 5 6 n 

2mm 3m 4mm 5m 6mm nm (n is odd) nmm (n is even) 
2 3 4 5 6 n 
*22 *33 *44 *55 *66 *nn 
C_{2v} C_{3v} C_{4v} C_{5v} C_{6v} C_{nv} 
CD_{4} CD_{6} CD_{8} CD_{10} CD_{12} CD_{2n} 
[2] [3] [4] [5] [6] [n] 
4 6 8 10 12 2n 

3 8 5 12  
62 82 10.2 12.2 2n.2 
3× 4× 5× 6× n× 
S_{6} S_{8} S_{10} S_{12} S_{2n} 
±C_{3} CC_{8} ±C_{5} CC_{12} CC_{2n} / ±C_{n} 
[2^{+},6^{+}] [2^{+},8^{+}] [2^{+},10^{+}] [2^{+},12^{+}] [2^{+},2n^{+}] 
6 8 10 12 2n 

3/m=6 4/m 5/m=10 6/m n/m 
32 42 52 62 n2 
3* 4* 5* 6* n* 
C_{3h} C_{4h} C_{5h} C_{6h} C_{nh} 
CC_{6} ±C_{4} CC_{10} ±C_{6} ±C_{n} / CC_{2n} 
[2,3^{+}] [2,4^{+}] [2,5^{+}] [2,6^{+}] [2,n^{+}] 
6 8 10 12 2n 
Dihedral symmetry
There are three infinite dihedral symmetry families, with n = 2 or higher (n may be 1 as a special case).
Intl  Geo 
Orb.  Schön.  Con.  Cox.  Ord.  Fund. domain 

222  2.2  222  D_{2}  D_{4}  [2,2]^{+}  4  
42m  42  2*2  D_{2d}  DD_{8}  [2^{+},4]  8  
mmm  22  *222  D_{2h}  ±D_{4}  [2,2]  8 
Intl  Geo 
Orb.  Schön.  Con.  Cox.  Ord.  Fund. domain 

32 422 52 622 
3.2 4.2 5.2 6.2 n.2 
223 224 225 226 22n 
D_{3} D_{4} D_{5} D_{6} D_{n} 
D_{6} D_{8} D_{10} D_{12} D_{2n} 
[2,3]^{+} [2,4]^{+} [2,5]^{+} [2,6]^{+} [2,n]^{+} 
6 8 10 12 2n 

3m 82m 5m 12.2m 
62 82 10.2 12.2 n2 
2*3 2*4 2*5 2*6 2*n 
D_{3d} D_{4d} D_{5d} D_{6d} D_{nd} 
±D_{6} DD_{16} ±D_{10} DD_{24} DD_{4n} / ±D_{2n} 
[2^{+},6] [2^{+},8] [2^{+},10] [2^{+},12] [2^{+},2n] 
12 16 20 24 4n 

6m2 4/mmm 10m2 6/mmm 
32 42 52 62 n2 
*223 *224 *225 *226 *22n 
D_{3h} D_{4h} D_{5h} D_{6h} D_{nh} 
DD_{12} ±D_{8} DD_{20} ±D_{12} ±D_{2n} / DD_{4n} 
[2,3] [2,4] [2,5] [2,6] [2,n] 
12 16 20 24 4n 
Polyhedral symmetry
There are three types of polyhedral symmetry: tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the trianglefaced regular polyhedra with these symmetries.
Intl  Geo 
Orb.  Schön.  Con.  Cox.  Ord.  Fund. domain 

23  3.3  332  T  T  [3,3]^{+} = [4,3^{+}]^{+} 
12  
m3  43  3*2  T_{h}  ±T  [4,3^{+}]  24  
43m  33  *332  T_{d}  TO  [3,3] = [1^{+},4,3] 
24 
Intl  Geo  Orb.  Schön.  Con.  Cox.  Ord.  Fund. domain 

432  4.3  432  O  O  [4,3]^{+} = [[3,3]]^{+} 
24  
m3m  43  *432  O_{h}  ±O  [4,3] = [[3,3]] 
48 
Intl  Geo  Orb.  Schön.  Con.  Cox.  Ord.  Fund. domain 

532  5.3  532  I  I  [5,3]^{+}  60  
532/m  53  *532  I_{h}  ±I  [5,3]  120 
See also
 Crystallographic point group
 Triangle group
 List of planar symmetry groups
 Point groups in two dimensions
Notes
References
 Peter R. Cromwell, Polyhedra (1997), Appendix I
 Sands, Donald E. (1993). "Crystal Systems and Geometry". Introduction to Crystallography. Mineola, New York: Dover Publications, Inc. p. 165. ISBN 0486678393.
 On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 9781568811345
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, ISBN 9781568812205
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [2]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
 N.W. Johnson: Geometries and Transformations, (2018) ISBN 9781107103405 Chapter 11: Finite symmetry groups, Table 11.4 Finite Groups of Isometries in 3space