Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
6520 conjugacy classes of subgroups; only the 3301 normal subgroups are shown.
This group has 6520 conjugacy classes of subgroups — too many to draw. Only its 3301 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D3 |
| commutator, cosocle | C2×C3 | C26 |
| Frattini | C2 | C25×D3 |
| Fitting | C24×C3×C4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C24×C4 | D3 |
| hypocenter | C3 | C23×#41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C3×C4 | C2 | semidirect |
| 2. C23×#67 | C2 | semidirect |
| 3. C22×#235 | C2 | direct (N × H) |
| 4. C23×D12 | C2 | semidirect |
| 5. C23×C4×D3 | C2 | semidirect |
| 6. C23×Dic6 | C2 | semidirect |
| 12. C23×C3×C4 | C22 | semidirect |
| 13. C22×#67 | C22 | semidirect |
| 15. C2×#235 | C22 | direct (N × H) |
| 16. C22×D12 | C22 | semidirect |
| 17. C22×C4×D3 | C22 | semidirect |
| 18. C22×Dic6 | C22 | semidirect |
| 24. C24×C4 | D3 | semidirect |
| 32. C22×C3×C4 | C23 | semidirect |
| 33. C2×#67 | C23 | semidirect |
| 35. #235 | C23 | direct (N × H) |
| 36. C2×D12 | C23 | semidirect |
| 37. C2×C4×D3 | C23 | semidirect |
| 38. C2×Dic6 | C23 | semidirect |
| 49. C23×C4 | C2×D3 | semidirect |
| 51. C23×C3 | #41 | semidirect |
| 58. C2×C3×C4 | C24 | semidirect |
| 59. #67 | C24 | semidirect |
| 61. D12 | C24 | semidirect |
| 62. C4×D3 | C24 | semidirect |
| 63. Dic6 | C24 | semidirect |
| 77. C22×C4 | C22×D3 | semidirect |
| 79. C22×C3 | C2×#41 | semidirect |
| 90. C23 | #235 | direct (N × H) |
| 100. C2×C4 | C23×D3 | semidirect |
| 102. C2×C3 | C22×#41 | semidirect |
| 108. C22 | C2×#235 | direct (N × H) |
| 114. C3 | C23×#41 | semidirect |
| 117. C2 | C22×#235 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×C4 | 2 |
| C23×#67 | 2 |
| C22×#235 | 2 |
| C23×D12 | 2 |
| C23×C4×D3 | 2 |
| C23×Dic6 | 2 |
| C23×#41 | 3 |