Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
526 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#535 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #535 | D3 |
| hypocenter | C3 | #3041 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #535 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#535 | C2 | semidirect |
| 2. C2×#949 | C2 | semidirect |
| 3. C22×#213 | C2 | semidirect |
| 4. #5521 | C2 | semidirect |
| 5. #5233 | C2 | semidirect |
| 6. #5215 | C2 | semidirect |
| 7. #5072 | C2 | semidirect |
| 8. #5057 | C2 | semidirect |
| 9. #4999 | C2 | semidirect |
| 14. C3×#127 | C22 | semidirect |
| 15. C3×#124 | C22 | semidirect |
| 20. C2×#215 | C22 | semidirect |
| 23. C2×#213 | C22 | semidirect |
| 29. C4×D12 | C22 | semidirect |
| 30. #835 | C22 | semidirect |
| 31. #832 | C22 | semidirect |
| 32. #832 | C22 | semidirect |
| 33. C4⋊(C3⋊C8) | C22 | semidirect |
| 36. #535 | D3 | semidirect |
| 59. C2×D12 | D4 | semidirect |
| 87. #127 | C2×D3 | semidirect |
| 88. #124 | C2×D3 | semidirect |
| 236. C3 | #3041 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#535 | 2 |
| C2×#949 | 2 |
| C22×#213 | 2 |
| #5521 | 2 |
| #5233 | 2 |
| #5215 | 2 |
| #5072 | 2 |
| #5057 | 2 |
| #4999 | 2 |
| #3041 | 3 |