Finite Group DatabasebetaFinite Group DatabasebetaOrder 336.
68 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C24×C7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C7×A4 | — | 4 | 5 | no | maximal |
| 3 | C23×C7 | — | 3 | 5 | no | — |
| 4 | C22⋊A4 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 5 | C22×C7 | — | 3 | 10 | no | — |
| 6 | C22×C7 | A4 | 1 | 5 | yes | complemented |
| 7 | C3×C7 | — | 16 | 1 | no | Hall |
| 8 | C24 | C3×C7 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C2×C7 | — | 3 | 5 | no | — |
| 10 | A4 | — | 4 | 5 | no | — |
| 11 | C23 | — | 3 | 5 | no | — |
| 12 | C7 | C22⋊A4 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 13 | C22 | — | 3 | 10 | no | — |
| 14 | C22 | C7×A4 | 1 | 5 | yes | complemented |
| 15 | C3 | — | 16 | 1 | no | Sylow |
| 16 | C2 | — | 3 | 5 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | C22⋊A4 |
| commutator, cosocle, hypocenter | C24 | C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C24×C7 | C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C7 | C3 | semidirect |
| 4. C22⋊A4 | C7 | direct (N × H) |
| 6. C22×C7 | A4 | semidirect |
| 8. C24 | C3×C7 | semidirect |
| 12. C7 | C22⋊A4 | direct (N × H) |
| 14. C22 | C7×A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C7 | 3 |
| C7×A4 | 4 |
| C22⋊A4 | 7 |