Finite Group DatabasebetaFinite Group DatabasebetaOrder 336.
34 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×AGL(1,𝔽8) | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 2 | C2×AGL(1,𝔽8) | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | AGL(1,𝔽8) | C2×C3 | 1 | 1 | yes | characteristic, direct product factor |
| 4 | C24×C3 | C7 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 5 | C2×C3×C7 | — | 8 | 1 | no | maximal |
| 6 | C23×C3 | — | 7 | 1 | no | — |
| 7 | C23×C3 | — | 7 | 1 | no | — |
| 8 | C23×C3 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 9 | C3×C7 | — | 8 | 1 | no | Hall |
| 10 | C24 | C3×C7 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C2×C7 | — | 8 | 1 | no | — |
| 12 | C22×C3 | — | 7 | 1 | no | — |
| 13 | C22×C3 | — | 7 | 3 | no | — |
| 14 | C22×C3 | — | 7 | 1 | no | — |
| 15 | C23 | — | 7 | 1 | no | — |
| 16 | C23 | — | 7 | 1 | no | — |
| 17 | C23 | C2×C3×C7 | 1 | 1 | yes | characteristic, complemented |
| 18 | C7 | — | 8 | 1 | no | Sylow |
| 19 | C2×C3 | — | 7 | 1 | no | — |
| 20 | C2×C3 | — | 7 | 1 | no | — |
| 21 | C2×C3 | AGL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 22 | C22 | — | 7 | 1 | no | — |
| 23 | C22 | — | 7 | 3 | no | — |
| 24 | C22 | — | 7 | 1 | no | — |
| 25 | C3 | C2×AGL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 26 | C2 | — | 7 | 1 | no | — |
| 27 | C2 | — | 7 | 1 | no | — |
| 28 | C2 | C3×AGL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | AGL(1,𝔽8) |
| commutator, cosocle, hypocenter | C23 | C2×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3 | C7 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×AGL(1,𝔽8) | C2 | direct (N × H) |
| 2. C2×AGL(1,𝔽8) | C3 | direct (N × H) |
| 3. AGL(1,𝔽8) | C2×C3 | direct (N × H) |
| 4. C24×C3 | C7 | semidirect |
| 8. C23×C3 | C2×C7 | semidirect |
| 10. C24 | C3×C7 | semidirect |
| 17. C23 | C2×C3×C7 | semidirect |
| 21. C2×C3 | AGL(1,𝔽8) | direct (N × H) |
| 25. C3 | C2×AGL(1,𝔽8) | direct (N × H) |
| 28. C2 | 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 |
|---|---|
| C3×AGL(1,𝔽8) | 2 |
| C2×AGL(1,𝔽8) | 3 |
| C24×C3 | 7 |
| C2×C3×C7 | 8 |