Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
310 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 111 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×D509 |
| commutator, socle, cosocle | C22×C509 | C24 |
| Fitting | C4⋊Q8×C509 | C2 |
| hypercenter | C4⋊Q8 | D509 |
| hypocenter | C509 | #532 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22550862 | C2 | semidirect |
| 2. #30654212 | C2 | semidirect |
| 3. #30654204 | C2 | semidirect |
| 4. #30654190 | C2 | semidirect |
| 5. #30654177 | C2 | semidirect |
| 6. C42×D509 | C2 | semidirect |
| 7. C4⋊Q8×C509 | C2 | semidirect |
| 30. C4×D509 | D4 | semidirect |
| 42. C4×C509 | #41 | semidirect |
| 49. C509 | #532 | semidirect |
| 52. C4⋊Q8 | D509 | semidirect |
| 94. C4 | #22550862 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22550862 | 2 |
| #30654212 | 2 |
| #30654204 | 2 |
| #30654190 | 2 |
| #30654177 | 2 |
| C42×D509 | 2 |
| C4⋊Q8×C509 | 2 |
| #532 | 509 |