Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 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×#518 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #518 | D3 |
| hypocenter | C3 | #3041 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #518 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#518 | C2 | semidirect |
| 2. C2×#932 | C2 | semidirect |
| 3. C22×#213 | C2 | semidirect |
| 4. #5525 | C2 | semidirect |
| 5. #5016 | C2 | semidirect |
| 6. #5015 | C2 | semidirect |
| 7. #5002 | C2 | semidirect |
| 8. #5001 | C2 | semidirect |
| 9. #4999 | C2 | semidirect |
| 15. C3×C4×D4 | C22 | semidirect |
| 19. C2×#214 | C22 | semidirect |
| 23. C2×#213 | C22 | semidirect |
| 28. #875 | C22 | semidirect |
| 29. #874 | C22 | semidirect |
| 30. #810 | C22 | semidirect |
| 31. #809 | C22 | semidirect |
| 32. #809 | C22 | semidirect |
| 33. C4⋊(C3⋊C8) | C22 | semidirect |
| 37. #518 | D3 | semidirect |
| 47. C2×C3×D4 | D4 | semidirect |
| 92. C4×D4 | C2×D3 | semidirect |
| 145. C2×D4 | D12 | 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×#518 | 2 |
| C2×#932 | 2 |
| C22×#213 | 2 |
| #5525 | 2 |
| #5016 | 2 |
| #5015 | 2 |
| #5002 | 2 |
| #5001 | 2 |
| #4999 | 2 |
| #3041 | 3 |