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, direct product factor, Hall |
| 2 | C23×C3 | C7 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C3×C7 | — | 8 | 1 | no | maximal, Hall |
| 4 | C22×C3 | — | 7 | 1 | no | — |
| 5 | C23 | C3×C7 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C7 | — | 8 | 1 | no | Sylow |
| 7 | C2×C3 | — | 7 | 1 | no | — |
| 8 | C22 | — | 7 | 1 | no | — |
| 9 | C3 | AGL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 10 | C2 | — | 7 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | AGL(1,𝔽8) |
| commutator, cosocle, hypocenter | C23 | C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3 | C7 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. AGL(1,𝔽8) | C3 | direct (N × H) |
| 2. C23×C3 | C7 | semidirect |
| 5. C23 | C3×C7 | semidirect |
| 9. C3 | AGL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| AGL(1,𝔽8) | 3 |
| C23×C3 | 7 |
| C3×C7 | 8 |