Finite Group DatabasebetaFinite Group DatabasebetaOrder 224.
72 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×AGL(1,𝔽8) | C2 | 1 | 3 | yes | maximal, direct product factor |
| 2 | AGL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic, direct product factor |
| 3 | C25 | C7 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | C22×C7 | — | 8 | 1 | no | maximal |
| 5 | C24 | — | 7 | 3 | no | — |
| 6 | C24 | — | 7 | 1 | no | — |
| 7 | C24 | C2×C7 | 1 | 3 | yes | complemented |
| 8 | C2×C7 | — | 8 | 3 | no | — |
| 9 | C23 | — | 7 | 3 | no | — |
| 10 | C23 | — | 7 | 3 | no | — |
| 11 | C23 | — | 7 | 9 | no | — |
| 12 | C23 | — | 7 | 6 | no | — |
| 13 | C23 | — | 7 | 1 | no | — |
| 14 | C23 | C22×C7 | 1 | 1 | yes | characteristic, complemented |
| 15 | C7 | — | 8 | 1 | no | Sylow |
| 16 | C22 | — | 7 | 1 | no | — |
| 17 | C22 | — | 7 | 9 | no | — |
| 18 | C22 | — | 7 | 6 | no | — |
| 19 | C22 | — | 7 | 3 | no | — |
| 20 | C22 | — | 7 | 3 | no | — |
| 21 | C22 | AGL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 22 | C2 | — | 7 | 3 | no | — |
| 23 | C2 | — | 7 | 1 | no | — |
| 24 | C2 | C2×AGL(1,𝔽8) | 1 | 3 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | AGL(1,𝔽8) |
| commutator, cosocle, hypocenter | C23 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C25 | C7 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×AGL(1,𝔽8) | C2 | direct (N × H) |
| 2. AGL(1,𝔽8) | C22 | direct (N × H) |
| 3. C25 | C7 | semidirect |
| 7. C24 | C2×C7 | semidirect |
| 14. C23 | C22×C7 | semidirect |
| 21. C22 | AGL(1,𝔽8) | direct (N × H) |
| 24. C2 | C2×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 |
|---|---|
| C2×AGL(1,𝔽8) | 2 |
| C25 | 7 |
| C22×C7 | 8 |