Finite Group DatabasebetaFinite Group DatabasebetaOrder 336.
268 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×ΓL(1,𝔽8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×AGL(1,𝔽7) | C2 | 1 | 14 | yes | maximal, direct product factor |
| 3 | C23×D7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 4 | C22×ΓL(1,𝔽8) | C22 | 1 | 7 | yes | complemented |
| 5 | C2×AGL(1,𝔽7) | C22 | 1 | 28 | yes | direct product factor |
| 6 | C23×C7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C22×D7 | C2×C3 | 1 | 14 | yes | complemented |
| 8 | C24×C3 | — | 7 | 1 | no | maximal, Hall |
| 9 | C2×ΓL(1,𝔽8) | C23 | 1 | 7 | yes | complemented |
| 10 | AGL(1,𝔽7) | C23 | 1 | 8 | yes | direct product factor |
| 11 | C22×C7 | C22×C3 | 1 | 7 | yes | complemented |
| 12 | C2×D7 | C22×C3 | 1 | 28 | yes | complemented |
| 13 | C23×C3 | — | 7 | 14 | no | — |
| 14 | C23×C3 | — | 7 | 1 | no | — |
| 15 | ΓL(1,𝔽8) | C24 | 1 | 1 | yes | characteristic, complemented, Hall |
| 16 | C24 | — | 7 | 1 | no | Sylow |
| 17 | C2×C7 | C23×C3 | 1 | 7 | yes | complemented |
| 18 | D7 | C23×C3 | 1 | 8 | yes | complemented |
| 19 | C22×C3 | — | 7 | 28 | no | — |
| 20 | C22×C3 | — | 7 | 7 | no | — |
| 21 | C23 | — | 7 | 14 | no | — |
| 22 | C23 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, direct product factor |
| 23 | C7 | C24×C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 24 | C2×C3 | — | 7 | 8 | no | — |
| 25 | C2×C3 | — | 7 | 7 | no | — |
| 26 | C22 | — | 7 | 28 | no | — |
| 27 | C22 | C2×AGL(1,𝔽7) | 1 | 7 | yes | direct product factor |
| 28 | C3 | — | 7 | 1 | no | Sylow |
| 29 | C2 | — | 7 | 8 | no | — |
| 30 | C2 | C22×AGL(1,𝔽7) | 1 | 7 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C7 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C22×AGL(1,𝔽7) | C2 | direct (N × H) |
| 3. C23×D7 | C3 | semidirect |
| 4. C22×ΓL(1,𝔽8) | C22 | semidirect |
| 5. C2×AGL(1,𝔽7) | C22 | direct (N × H) |
| 6. C23×C7 | C2×C3 | semidirect |
| 7. C22×D7 | C2×C3 | semidirect |
| 9. C2×ΓL(1,𝔽8) | C23 | semidirect |
| 10. AGL(1,𝔽7) | C23 | direct (N × H) |
| 11. C22×C7 | C22×C3 | semidirect |
| 12. C2×D7 | C22×C3 | semidirect |
| 15. ΓL(1,𝔽8) | C24 | semidirect |
| 17. C2×C7 | C23×C3 | semidirect |
| 18. D7 | C23×C3 | semidirect |
| 22. C23 | AGL(1,𝔽7) | direct (N × H) |
| 23. C7 | C24×C3 | semidirect |
| 27. C22 | C2×AGL(1,𝔽7) | direct (N × H) |
| 30. C2 | C22×AGL(1,𝔽7) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×ΓL(1,𝔽8) | 2 |
| C22×AGL(1,𝔽7) | 2 |
| C23×D7 | 3 |
| C24×C3 | 7 |