Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
619 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 619 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×#626 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C22×D32 |
| Fitting | C32×#41 | C22 |
| socle | C2×C32 | C22×D4 |
| hypercenter | #41 | D32 |
| hypocenter | C32 | #576 |
| 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. #64366 | C2 | semidirect |
| 2. C3×#1009 | C2 | semidirect |
| 3. C3×#948 | C2 | semidirect |
| 4. #63830 | C2 | semidirect |
| 5. #63829 | C2 | semidirect |
| 6. #63825 | C2 | semidirect |
| 7. C2×#3714 | C2 | semidirect |
| 13. #3828 | C22 | semidirect |
| 14. C2×C4⋊(C3⋊D3) | C22 | semidirect |
| 17. C3×#235 | C22 | semidirect |
| 18. C2×C3×D12 | C22 | semidirect |
| 19. C3×#208 | C22 | semidirect |
| 20. C2×C3×C3⋊C8 | C22 | semidirect |
| 21. #3719 | C22 | semidirect |
| 22. #3717 | C22 | semidirect |
| 23. #3716 | C22 | semidirect |
| 24. #3714 | C22 | semidirect |
| 25. #3714 | C22 | semidirect |
| 30. #1009 | D3 | semidirect |
| 34. #948 | D3 | semidirect |
| 77. #235 | C2×D3 | semidirect |
| 81. C2×D12 | C2×D3 | semidirect |
| 89. #208 | C2×D3 | semidirect |
| 90. C2×C3⋊C8 | C2×D3 | semidirect |
| 199. C32 | #576 | semidirect |
| 238. C3 | #5752 | semidirect |
| 239. C3 | #5835 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64366 | 2 |
| C3×#1009 | 2 |
| C3×#948 | 2 |
| #63830 | 2 |
| #63829 | 2 |
| #63825 | 2 |
| C2×#3714 | 2 |
| #5835 | 3 |
| #5752 | 3 |