Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
34 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | SL(2,𝔽7) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×AGL(1,𝔽7) | — | 8 | 1 | no | maximal |
| 3 | #226 | — | 14 | 1 | no | — |
| 4 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 5 | AGL(1,𝔽7) | — | 8 | 2 | no | — |
| 6 | SD32 | — | 21 | 1 | no | maximal, Sylow |
| 7 | C2×D7 | — | 8 | 1 | no | — |
| 8 | #67 | — | 28 | 1 | no | maximal |
| 9 | SL(2,𝔽3) | — | 14 | 1 | no | — |
| 10 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 11 | Q16 | — | 21 | 1 | no | — |
| 12 | D8 | — | 21 | 1 | no | — |
| 13 | C16 | — | 21 | 1 | no | — |
| 14 | C2×C7 | — | 8 | 1 | no | — |
| 15 | D7 | — | 8 | 2 | no | — |
| 16 | C22×C3 | — | 28 | 1 | no | — |
| 17 | C2×D3 | — | 28 | 1 | no | — |
| 18 | Dic3 | — | 28 | 1 | no | — |
| 19 | Q8 | — | 14 | 1 | no | — |
| 20 | D4 | — | 42 | 1 | no | — |
| 21 | C8 | — | 21 | 1 | no | — |
| 22 | C7 | — | 8 | 1 | no | Sylow |
| 23 | C2×C3 | — | 28 | 1 | no | — |
| 24 | C2×C3 | — | 56 | 1 | no | — |
| 25 | D3 | — | 56 | 1 | no | — |
| 26 | C22 | — | 28 | 1 | no | — |
| 27 | C4 | — | 21 | 1 | no | — |
| 28 | C3 | — | 28 | 1 | no | Sylow |
| 29 | C2 | — | 56 | 1 | no | — |
| 30 | C2 | SO(3,𝔽7) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, hypercenter | C2 | SO(3,𝔽7) |
| commutator, cosocle, hypocenter | SL(2,𝔽7) | C2 |
| LCS | G ▷ SL(2,𝔽7) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. SL(2,𝔽7) | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,𝔽7) | 2 |
| C2×AGL(1,𝔽7) | 8 |
| SD32 | 21 |
| #67 | 28 |