Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1224 conjugacy classes of subgroups; only the 201 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#380 | C2 |
| hypercenter | #380 | D3 |
| hypocenter | C3 | #2847 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C4×D3 | C2 | semidirect |
| 2. C2×#930 | C2 | semidirect |
| 3. C2×#895 | C2 | semidirect |
| 4. C2×#882 | C2 | semidirect |
| 5. C3×#380 | C2 | semidirect |
| 6. #4987 | C2 | semidirect |
| 7. #4958 | C2 | semidirect |
| 8. #4665 | C2 | semidirect |
| 15. C22×D12 | C22 | semidirect |
| 20. C2×C3×C4⋊C4 | C22 | semidirect |
| 24. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 25. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 26. C2×C4⋊Dic3 | C22 | semidirect |
| 29. C3×#95 | C22 | semidirect |
| 30. C3⋊#95 | C22 | semidirect |
| 38. #380 | D3 | semidirect |
| 44. C23×D3 | D4 | semidirect |
| 48. C22×C3×C4 | D4 | semidirect |
| 70. C2×C4×D3 | D4 | semidirect |
| 91. C2×C4⋊C4 | C2×D3 | semidirect |
| 97. #95 | C2×D3 | semidirect |
| 169. C22×C4 | D12 | semidirect |
| 183. C22×C3 | #121 | semidirect |
| 201. C2×D3 | #121 | semidirect |
| 204. C3×C4 | #120 | semidirect |
| 275. C22 | #895 | semidirect |
| 285. C4 | #882 | semidirect |
| 286. C3 | #2847 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×D3 | 2 |
| C2×#930 | 2 |
| C2×#895 | 2 |
| C2×#882 | 2 |
| C3×#380 | 2 |
| #4987 | 2 |
| #4958 | 2 |
| #4665 | 2 |
| #2847 | 3 |