Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C43 | C7 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | AGL(1,𝔽8) | — | 8 | 1 | no | maximal |
| 3 | C2×C42 | — | 7 | 1 | no | — |
| 4 | C22×C4 | — | 7 | 1 | no | — |
| 5 | C42 | — | 7 | 4 | no | — |
| 6 | C23 | AGL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 7 | C2×C4 | — | 7 | 6 | no | — |
| 8 | C7 | — | 64 | 1 | no | Sylow |
| 9 | C22 | — | 7 | 1 | no | — |
| 10 | C4 | — | 7 | 4 | no | — |
| 11 | C2 | — | 7 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C43 | C7 |
| Frattini, socle | C23 | AGL(1,𝔽8) |
| LCS | G ▷ C43 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C43 | C7 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C43 | 7 |
| AGL(1,𝔽8) | 8 |