Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
4544 conjugacy classes of subgroups; only the 725 normal subgroups are shown.
This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 725 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C22×C3×C22⋊C4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C22×C22⋊C4 | D3 |
| hypocenter | C3 | C22×#120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×D3 | C2 | semidirect |
| 2. C23×#67 | C2 | semidirect |
| 3. C23×D12 | C2 | semidirect |
| 4. C22×C3×C22⋊C4 | C2 | semidirect |
| 5. C22×(C2×D3)⋊C4 | C2 | semidirect |
| 6. C2×#882 | C2 | direct (N × H) |
| 12. C24×D3 | C22 | semidirect |
| 13. C24×D3 | C22 | semidirect |
| 14. C23×C3×C4 | C22 | semidirect |
| 16. C22×#67 | C22 | semidirect |
| 17. C23×Dic3 | C22 | semidirect |
| 19. C22×D12 | C22 | semidirect |
| 20. C2×C3×C22⋊C4 | C22 | semidirect |
| 21. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 22. #882 | C22 | direct (N × H) |
| 28. C22×C22⋊C4 | D3 | semidirect |
| 30. C24×C3 | D4 | semidirect |
| 40. C23×D3 | D4 | semidirect |
| 42. C23×D3 | C23 | semidirect |
| 43. C23×D3 | C23 | semidirect |
| 45. C22×C3×C4 | C23 | semidirect |
| 47. C2×#67 | C23 | semidirect |
| 49. C22×Dic3 | C23 | semidirect |
| 51. C2×D12 | C23 | semidirect |
| 52. C3×C22⋊C4 | C23 | semidirect |
| 53. (C2×D3)⋊C4 | C23 | semidirect |
| 65. C23×C4 | C2×D3 | semidirect |
| 68. C2×C22⋊C4 | C2×D3 | semidirect |
| 73. C23×C3 | C2×D4 | semidirect |
| 85. C22×D3 | C2×D4 | semidirect |
| 87. C22×D3 | C24 | semidirect |
| 89. C2×C3×C4 | C24 | semidirect |
| 92. C2×Dic3 | C24 | semidirect |
| 106. C24 | D12 | semidirect |
| 116. C22×C4 | C22×D3 | semidirect |
| 118. C22⋊C4 | C22×D3 | semidirect |
| 121. C22×C3 | #120 | semidirect |
| 124. C22×C3 | C22×D4 | semidirect |
| 133. C2×D3 | C22×D4 | semidirect |
| 151. C23 | C2×D12 | semidirect |
| 160. C2×C4 | C23×D3 | semidirect |
| 162. C2×C3 | C2×#120 | semidirect |
| 177. C22 | #882 | direct (N × H) |
| 179. C22 | C22×D12 | semidirect |
| 184. C3 | C22×#120 | semidirect |
| 188. C2 | C2×#882 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×D3 | 2 |
| C23×#67 | 2 |
| C23×D12 | 2 |
| C22×C3×C22⋊C4 | 2 |
| C22×(C2×D3)⋊C4 | 2 |
| C2×#882 | 2 |
| C22×#120 | 3 |