Finite Group DatabasebetaFinite Group DatabasebetaOrder 336.
19 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #226 | — | 7 | 2 | no | maximal, Hall |
| 2 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | maximal |
| 3 | SL(2,𝔽3) | — | 7 | 2 | no | — |
| 4 | ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 5 | Q16 | — | 21 | 1 | no | Sylow |
| 6 | C2×C7 | — | 8 | 1 | no | — |
| 7 | Dic3 | — | 28 | 1 | no | — |
| 8 | Q8 | — | 7 | 2 | no | — |
| 9 | C8 | — | 21 | 1 | no | — |
| 10 | C7 | — | 8 | 1 | no | Sylow |
| 11 | C2×C3 | — | 28 | 1 | no | — |
| 12 | C4 | — | 21 | 1 | no | — |
| 13 | C3 | — | 28 | 1 | no | Sylow |
| 14 | C2 | GL(3,𝔽2) | 1 | 1 | yes | maximal normal, characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, cosocle, hypercenter | C2 | GL(3,𝔽2) |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #226 | 7 |
| C2×ΓL(1,𝔽8) | 8 |