Finite Group DatabasebetaFinite Group DatabasebetaOrder 336.
38 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C7×A4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22×Dic7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C7×A4 | C4 | 1 | 1 | yes | characteristic, complemented |
| 4 | C3×Dic7 | — | 4 | 1 | no | maximal |
| 5 | C23×C7 | C2×C3 | 1 | 1 | yes | characteristic |
| 6 | C2×Dic7 | — | 3 | 1 | no | — |
| 7 | C2×Dic7 | — | 3 | 1 | no | — |
| 8 | C4×A4 | — | 7 | 1 | no | maximal, Hall |
| 9 | C2×C3×C7 | — | 4 | 1 | no | — |
| 10 | C22×C7 | — | 3 | 1 | no | — |
| 11 | C22×C7 | — | 3 | 1 | no | — |
| 12 | C22×C7 | C3×C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | Dic7 | — | 3 | 1 | no | — |
| 14 | Dic7 | A4 | 1 | 1 | yes | characteristic, direct product factor |
| 15 | C2×A4 | D7 | 1 | 1 | yes | characteristic |
| 16 | C3×C7 | — | 4 | 1 | no | Hall |
| 17 | C22×C4 | — | 7 | 1 | no | Sylow |
| 18 | C2×C7 | — | 3 | 1 | no | — |
| 19 | C2×C7 | — | 3 | 1 | no | — |
| 20 | C2×C7 | C2×A4 | 1 | 1 | yes | characteristic |
| 21 | A4 | Dic7 | 1 | 1 | yes | characteristic, direct product factor |
| 22 | C3×C4 | — | 28 | 1 | no | — |
| 23 | C23 | C3×D7 | 1 | 1 | yes | characteristic |
| 24 | C2×C4 | — | 21 | 1 | no | — |
| 25 | C2×C4 | — | 21 | 1 | no | — |
| 26 | C7 | C4×A4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 27 | C2×C3 | — | 4 | 1 | no | — |
| 28 | C22 | — | 3 | 1 | no | — |
| 29 | C22 | — | 3 | 1 | no | — |
| 30 | C22 | C3×Dic7 | 1 | 1 | yes | characteristic, complemented |
| 31 | C4 | — | 7 | 1 | no | — |
| 32 | C4 | — | 21 | 1 | no | — |
| 33 | C3 | — | 4 | 1 | no | Sylow |
| 34 | C2 | — | 3 | 1 | no | — |
| 35 | C2 | — | 3 | 1 | no | — |
| 36 | C2 | A4×D7 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×D7 |
| commutator, hypocenter | C22×C7 | C3×C4 |
| Fitting, socle, cosocle | C23×C7 | C2×C3 |
| prime radical | C2×C7×A4 | C2 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×Dic7 | C3 | semidirect |
| 3. C7×A4 | C4 | semidirect |
| 12. C22×C7 | C3×C4 | semidirect |
| 14. Dic7 | A4 | direct (N × H) |
| 21. A4 | Dic7 | direct (N × H) |
| 26. C7 | C4×A4 | semidirect |
| 30. C22 | C3×Dic7 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×A4 | 2 |
| C22×Dic7 | 3 |
| C3×Dic7 | 4 |
| C4×A4 | 7 |