Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1812 conjugacy classes of subgroups; only the 277 normal subgroups are shown.
This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 277 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 | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C22×C3×C22⋊C4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C22×C22⋊C4 | D3 |
| hypocenter | C3 | C2×#392 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#67 | C2 | semidirect |
| 2. C23×D12 | C2 | semidirect |
| 3. C22×C3×C22⋊C4 | C2 | semidirect |
| 4. C22×(C2×D3)⋊C4 | C2 | semidirect |
| 5. C2×C3⋊#95 | C2 | semidirect |
| 6. #4960 | C2 | direct (N × H) |
| 15. C22×#67 | C22 | semidirect |
| 19. C22×D12 | C22 | semidirect |
| 21. C2×C3×C22⋊C4 | C22 | semidirect |
| 23. C2×(C2×D3)⋊C4 | C22 | semidirect |
| 24. C3⋊#95 | C22 | semidirect |
| 28. C22×C22⋊C4 | D3 | semidirect |
| 35. C24×C3 | D4 | semidirect |
| 44. C23×D3 | D4 | semidirect |
| 49. C22×C3×C4 | D4 | semidirect |
| 58. C22×Dic3 | D4 | semidirect |
| 80. C2×C22⋊C4 | C2×D3 | semidirect |
| 95. C23×C3 | C2×D4 | semidirect |
| 111. C22×D3 | C2×D4 | semidirect |
| 117. C2×C3×C4 | C2×D4 | semidirect |
| 122. C2×Dic3 | C2×D4 | semidirect |
| 136. C24 | D12 | semidirect |
| 153. C22×C4 | #67 | semidirect |
| 209. C23 | C2×D12 | semidirect |
| 226. C2×C4 | C2×#67 | semidirect |
| 228. C2×C3 | #392 | semidirect |
| 262. C3 | C2×#392 | semidirect |
| 265. C2 | #4960 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#67 | 2 |
| C23×D12 | 2 |
| C22×C3×C22⋊C4 | 2 |
| C22×(C2×D3)⋊C4 | 2 |
| C2×C3⋊#95 | 2 |
| #4960 | 2 |
| C2×#392 | 3 |