Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
1284 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 1284 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×#127 | C2 |
| hypercenter | #127 | C3⋊D3 |
| hypocenter | C32 | D42 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×C3⋊D3 | C2 | semidirect |
| 2. C32×#127 | C2 | semidirect |
| 3. #64147 | C2 | semidirect |
| 4. #64146 | C2 | semidirect |
| 5. C4×C4⋊(C3⋊D3) | C2 | semidirect |
| 7. C23×C3⋊D3 | C22 | semidirect |
| 8. C2×C32×D4 | C22 | semidirect |
| 9. C2×C32×D4 | C22 | semidirect |
| 10. C2×#638 | C22 | semidirect |
| 13. C2×C4×C3⋊D3 | C22 | semidirect |
| 14. C32×C42 | C22 | semidirect |
| 15. #3757 | C22 | semidirect |
| 16. #3752 | C22 | semidirect |
| 17. C4⋊(C3⋊Dic3) | C22 | semidirect |
| 19. C3×#127 | D3 | semidirect |
| 33. C4⋊(C3⋊D3) | D4 | semidirect |
| 37. C2×C3×D4 | C2×D3 | semidirect |
| 38. C2×C3×D4 | C2×D3 | semidirect |
| 43. C3×C42 | C2×D3 | semidirect |
| 53. C2×C3⋊D3 | C2×D4 | semidirect |
| 55. C32×C4 | C2×D4 | semidirect |
| 58. #127 | C3⋊D3 | semidirect |
| 82. C2×D4 | C2×C3⋊D3 | semidirect |
| 83. C2×D4 | C2×C3⋊D3 | semidirect |
| 88. C42 | C2×C3⋊D3 | semidirect |
| 97. C3×C4 | D3×D4 | semidirect |
| 99. C32 | D42 | semidirect |
| 125. C4 | D4×C3⋊D3 | semidirect |
| 126. C3 | #5717 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×C3⋊D3 | 2 |
| C32×#127 | 2 |
| #64147 | 2 |
| #64146 | 2 |
| C4×C4⋊(C3⋊D3) | 2 |
| #5717 | 3 |