Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
3000 conjugacy classes of subgroups; only the 725 normal subgroups are shown.
This group has 3000 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 | C23×C3×D4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | C22×#127 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×D4 | C2 | semidirect |
| 2. C23×#67 | C2 | semidirect |
| 3. C23×D12 | C2 | semidirect |
| 4. C2×#940 | C2 | direct (N × H) |
| 5. C22×C4×Dic3 | C2 | semidirect |
| 10. C22×C3×D4 | C22 | semidirect |
| 13. C22×#67 | C22 | semidirect |
| 16. C22×D12 | C22 | semidirect |
| 17. #940 | C22 | direct (N × H) |
| 18. C2×C4×Dic3 | C22 | semidirect |
| 21. C23×D4 | D3 | semidirect |
| 32. C2×C3×D4 | C23 | semidirect |
| 33. C22×C3×C4 | D4 | semidirect |
| 36. C2×#67 | C23 | semidirect |
| 37. C22×Dic3 | D4 | semidirect |
| 40. C2×D12 | C23 | semidirect |
| 41. C4×Dic3 | C23 | semidirect |
| 49. C22×D4 | C2×D3 | semidirect |
| 66. C2×C3×C4 | C2×D4 | semidirect |
| 69. C2×Dic3 | C2×D4 | semidirect |
| 84. C2×D4 | C22×D3 | semidirect |
| 87. C22×C4 | #67 | semidirect |
| 93. C22×C3 | #127 | semidirect |
| 100. C3×C4 | C22×D4 | semidirect |
| 101. Dic3 | C22×D4 | semidirect |
| 118. C2×C4 | C2×#67 | semidirect |
| 121. C2×C3 | C2×#127 | semidirect |
| 131. C22 | #940 | direct (N × H) |
| 136. C4 | C22×#67 | semidirect |
| 137. C3 | C22×#127 | semidirect |
| 140. C2 | C2×#940 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×D4 | 2 |
| C23×#67 | 2 |
| C23×D12 | 2 |
| C2×#940 | 2 |
| C22×C4×Dic3 | 2 |
| C22×#127 | 3 |