Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
750 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 750 conjugacy classes of subgroups — too many to draw. Only its 136 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 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C2×C32×Q8 | C22 |
| hypercenter | C2×Q8 | D32 |
| hypocenter | C32 | #535 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3832 | C2 | semidirect |
| 2. C2×#3802 | C2 | semidirect |
| 3. C3×#947 | C2 | semidirect |
| 4. #63923 | C2 | semidirect |
| 5. #63912 | C2 | semidirect |
| 6. #63911 | C2 | semidirect |
| 7. #63874 | C2 | semidirect |
| 10. C2×C32×Q8 | C22 | semidirect |
| 13. C2×C4⋊(C3⋊D3) | C22 | semidirect |
| 15. C2×C4×C3⋊D3 | C22 | semidirect |
| 17. C2×#626 | C22 | semidirect |
| 20. C3×(C2×D3)⋊C4 | C22 | semidirect |
| 21. C3×C4×Dic3 | C22 | semidirect |
| 22. #3724 | C22 | semidirect |
| 28. #947 | D3 | semidirect |
| 46. C4×C3⋊D3 | D4 | semidirect |
| 54. C2×C3×Q8 | C2×D3 | semidirect |
| 74. (C2×D3)⋊C4 | C2×D3 | semidirect |
| 76. C4×Dic3 | C2×D3 | semidirect |
| 134. C2×Q8 | D32 | semidirect |
| 170. C32 | #535 | semidirect |
| 211. C3 | #5674 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3832 | 2 |
| C2×#3802 | 2 |
| C3×#947 | 2 |
| #63923 | 2 |
| #63912 | 2 |
| #63911 | 2 |
| #63874 | 2 |
| #5674 | 3 |