Finite Group DatabasebetaFinite Group DatabasebetaOrder 336.
36 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×Dic6 | C2 | 1 | 2 | yes | maximal |
| 2 | C3×C7×C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C7×Q16 | — | 3 | 1 | no | maximal, Hall |
| 4 | C3×C4×C7 | C22 | 1 | 1 | yes | characteristic |
| 5 | C7×Dic3 | — | 2 | 2 | no | — |
| 6 | C7×Q8 | — | 3 | 2 | no | — |
| 7 | C7×C8 | D3 | 1 | 1 | yes | characteristic |
| 8 | Dic12 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 9 | C2×C3×C7 | D4 | 1 | 1 | yes | characteristic |
| 10 | C4×C7 | — | 6 | 2 | no | — |
| 11 | C4×C7 | C2×D3 | 1 | 1 | yes | characteristic |
| 12 | Dic6 | C2×C7 | 1 | 2 | yes | — |
| 13 | C3×C8 | C2×C7 | 1 | 1 | yes | characteristic |
| 14 | C3×C7 | Q16 | 1 | 1 | yes | characteristic, complemented, Hall |
| 15 | Q16 | — | 3 | 1 | no | Sylow |
| 16 | C2×C7 | D12 | 1 | 1 | yes | characteristic |
| 17 | C3×C4 | C22×C7 | 1 | 1 | yes | characteristic |
| 18 | Dic3 | — | 2 | 2 | no | — |
| 19 | Q8 | — | 3 | 2 | no | — |
| 20 | C8 | D3×C7 | 1 | 1 | yes | characteristic |
| 21 | C7 | Dic12 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 22 | C2×C3 | C7×D4 | 1 | 1 | yes | characteristic |
| 23 | C4 | — | 6 | 2 | no | — |
| 24 | C4 | C2×D3×C7 | 1 | 1 | yes | characteristic |
| 25 | C3 | C7×Q16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 26 | C2 | C7×D12 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. Dic12 | C7 | direct (N × H) |
| 14. C3×C7 | Q16 | semidirect |
| 21. C7 | Dic12 | direct (N × H) |
| 25. C3 | C7×Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×Dic6 | 2 |
| C3×C7×C8 | 2 |
| C7×Q16 | 3 |
| Dic12 | 7 |