Finite Group DatabasebetaFinite Group DatabasebetaOrder 336.
40 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | GL(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 2 | C2×S4 | — | 7 | 2 | no | maximal, Hall |
| 3 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | maximal |
| 4 | C2×A4 | — | 7 | 2 | no | — |
| 5 | S4 | — | 7 | 2 | no | — |
| 6 | S4 | — | 7 | 2 | no | — |
| 7 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 8 | C2×D4 | — | 21 | 1 | no | Sylow |
| 9 | C2×C7 | — | 8 | 1 | no | — |
| 10 | C2×D3 | — | 28 | 1 | no | — |
| 11 | A4 | — | 7 | 2 | no | — |
| 12 | C23 | — | 7 | 2 | no | — |
| 13 | D4 | — | 21 | 1 | no | — |
| 14 | D4 | — | 21 | 2 | no | — |
| 15 | D4 | — | 21 | 1 | no | — |
| 16 | C2×C4 | — | 21 | 1 | no | — |
| 17 | C7 | — | 8 | 1 | no | Sylow |
| 18 | C2×C3 | — | 28 | 1 | no | — |
| 19 | D3 | — | 28 | 1 | no | — |
| 20 | D3 | — | 28 | 1 | no | — |
| 21 | C22 | — | 7 | 2 | no | — |
| 22 | C22 | — | 21 | 2 | no | — |
| 23 | C22 | — | 21 | 1 | no | — |
| 24 | C4 | — | 21 | 1 | no | — |
| 25 | C4 | — | 21 | 1 | no | — |
| 26 | C3 | — | 28 | 1 | no | Sylow |
| 27 | C2 | — | 21 | 1 | no | — |
| 28 | C2 | — | 21 | 1 | no | — |
| 29 | C2 | GL(3,𝔽2) | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2 | GL(3,𝔽2) |
| commutator, hypocenter | GL(3,𝔽2) | C2 |
| Frattini | C1 | G |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. GL(3,𝔽2) | C2 | direct (N × H) |
| 29. C2 | GL(3,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| GL(3,𝔽2) | 2 |
| C2×S4 | 7 |
| C2×ΓL(1,𝔽8) | 8 |