Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1424 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 545 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D12 |
| commutator, cosocle | C3×C4 | C25 |
| Frattini | C4 | C24×D3 |
| Fitting | C2×C3×#131 | C2 |
| socle | C2×C3 | C23×D4 |
| hypercenter | C2×#131 | D3 |
| hypocenter | C3 | #3584 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ C2×#131 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5964 | C2 | semidirect |
| 2. C2×C3×#131 | C2 | semidirect |
| 3. #5753 | C2 | semidirect |
| 4. #5752 | C2 | semidirect |
| 5. #5751 | C2 | semidirect |
| 6. #5748 | C2 | semidirect |
| 7. C2×#904 | C2 | semidirect |
| 10. #1010 | C22 | semidirect |
| 11. #1009 | C22 | semidirect |
| 15. C2×#235 | C22 | semidirect |
| 16. C3×#131 | C22 | semidirect |
| 17. C3×#131 | C22 | semidirect |
| 18. C2×C3×M16 | C22 | semidirect |
| 19. C22×C3×C8 | C22 | semidirect |
| 20. #909 | C22 | semidirect |
| 21. #908 | C22 | semidirect |
| 22. C2×Dic12 | C22 | semidirect |
| 23. #904 | C22 | semidirect |
| 24. #904 | C22 | semidirect |
| 25. #904 | C22 | semidirect |
| 26. #904 | C22 | semidirect |
| 27. C2×D24 | C22 | semidirect |
| 28. C2×#204 | C22 | semidirect |
| 35. C2×#131 | D3 | semidirect |
| 80. #131 | C2×D3 | semidirect |
| 81. #131 | C2×D3 | semidirect |
| 82. C2×M16 | C2×D3 | semidirect |
| 83. C22×C8 | C2×D3 | semidirect |
| 174. C3 | #3584 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5964 | 2 |
| C2×C3×#131 | 2 |
| #5753 | 2 |
| #5752 | 2 |
| #5751 | 2 |
| #5748 | 2 |
| C2×#904 | 2 |
| #3584 | 3 |