Finite Group DatabasebetaFinite Group DatabasebetaOrder 168.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | AGL(1,𝔽8) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C2×A4 | — | 7 | 1 | no | maximal, Hall |
| 3 | ΓL(1,𝔽8) | — | 8 | 1 | no | maximal, Hall |
| 4 | A4 | — | 7 | 1 | no | — |
| 5 | C23 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C7 | — | 8 | 1 | no | Sylow |
| 7 | C2×C3 | — | 28 | 1 | no | — |
| 8 | C22 | — | 7 | 1 | no | — |
| 9 | C3 | — | 28 | 1 | no | Sylow |
| 10 | C2 | — | 7 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | AGL(1,𝔽8) | C3 |
| Fitting, socle | C23 | ΓL(1,𝔽8) |
| LCS | G ▷ AGL(1,𝔽8) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. AGL(1,𝔽8) | C3 | semidirect |
| 5. C23 | ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| AGL(1,𝔽8) | 3 |
| C2×A4 | 7 |
| ΓL(1,𝔽8) | 8 |