Full octahedral group/List of all subgroups

The list of subgroups as shown in Full octahedral group, but including lists of all 48 permutations of the respective example solid.

The group itself

Oh S4 × C2 [4,3]

Subgroups of order 24

Td S4 green orange [3,3]
solids

O S4 blue red [4,3]+
solids

Th A4 × C2 [3+,4]
solids

Subgroups of order 16

D4h Dih4 × C2 [2,4]
solids

Subgroups of order 12

T A4 [3,3]+
solids submatrix

D3d Dih6 [2+,6]
solids submatrix

Subgroups of order 8

D2h C23 white [2,2]
solids

D2h C23 green [2,2]
solids

C4h C4 × C2 [4+,2]
solids

C4v Dih4 green red [4]
solids submatrix Permutations of the square for comparison Cube permutations that leave the z-axis fixed

D2d Dih4 blue orange [2+,4]
solids

D2d Dih4 green orange [2+,4]
solids

D4 Dih4 blue red [2,4]+
solids

Subgroups of order 6

C3v S3 green [3]
solids

D3 S3 blue [2,3]+
solids

S6 C6 [2+,6+]
solids

Subgroups of order 4

S4 C4 orange [2+,4+]
solids

C4 C4 red [4]+
solids

C2h = D1d V inv white [2+,2]
solids

C2h = D1d V inv green [2+,2]
solids

C2v V green blue yellow [2]
solids

C2v V yellow white [2]
solids

C2v V green white [2]
solids

D2 V blue white [2,2]+
solids

D2 V white [2,2]+
solids

Subgroups of order 3

C3 C3 [3]+
solids

Subgroups of order 2

S2 C2 inv [2+,2+]
solids

Cs = C1v C2 yellow [ ]
solids

Cs = C1v C2 green [ ]
solids

C2 C2 white [2]+
solids

C2 C2 blue [2]+
solids

The trivial group

C1 C1 [ ]+