Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
945 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 945 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #120 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C8×#120 |
| prime radical | #3025 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #62624 | C2 | semidirect |
| 2. #33432 | C2 | semidirect |
| 4. #12981 | C2 | semidirect |
| 5. #12915 | C2 | semidirect |
| 6. #12684 | C2 | semidirect |
| 7. #12613 | C2 | semidirect |
| 13. C23×C16 | C22 | semidirect |
| 18. #2151 | C22 | semidirect |
| 21. #2112 | C22 | semidirect |
| 22. #2108 | C22 | semidirect |
| 27. #1378 | C22 | semidirect |
| 42. C2×M32 | D4 | semidirect |
| 48. C22×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62624 | 2 |
| #33432 | 2 |
| #21483 | 2 |
| #12981 | 2 |
| #12915 | 2 |
| #12684 | 2 |
| #12613 | 2 |