Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
547 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 547 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#625 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C22×D32 |
| Fitting | C32×#41 | C22 |
| socle | C2×C32 | C22×D4 |
| hypercenter | #41 | D32 |
| hypocenter | C32 | #577 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#1010 | C2 | semidirect |
| 2. C3×#1009 | C2 | semidirect |
| 3. #64157 | C2 | semidirect |
| 4. #63827 | C2 | semidirect |
| 5. C2×#3715 | C2 | semidirect |
| 6. #63823 | C2 | semidirect |
| 7. #63822 | C2 | semidirect |
| 16. C3×#235 | C22 | semidirect |
| 18. C2×C3×Dic6 | C22 | semidirect |
| 19. #3748 | C22 | semidirect |
| 20. C2×C3⋊(C3⋊C8) | C22 | semidirect |
| 21. #3718 | C22 | semidirect |
| 22. #3715 | C22 | semidirect |
| 23. #3715 | C22 | semidirect |
| 24. #3715 | C22 | semidirect |
| 25. #3713 | C22 | semidirect |
| 28. #1010 | D3 | semidirect |
| 29. #1009 | D3 | semidirect |
| 75. #235 | C2×D3 | semidirect |
| 77. C2×Dic6 | C2×D3 | semidirect |
| 187. C32 | #577 | semidirect |
| 224. C3 | #5836 | semidirect |
| 225. C3 | #5837 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#1010 | 2 |
| C3×#1009 | 2 |
| #64157 | 2 |
| #63827 | 2 |
| C2×#3715 | 2 |
| #63823 | 2 |
| #63822 | 2 |
| #5837 | 3 |
| #5836 | 3 |