Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
282 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D32 |
| commutator, Frattini, cosocle | C32 | C24 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #504 ◁ #2258 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33984 | C2 | semidirect |
| 2. #33982 | C2 | semidirect |
| 3. #13743 | C2 | semidirect |
| 4. #13742 | C2 | semidirect |
| 5. #13741 | C2 | semidirect |
| 6. C2×SD128 | C2 | semidirect |
| 7. #13737 | C2 | semidirect |
| 13. #2262 | C22 | semidirect |
| 14. C2×Q64 | C22 | semidirect |
| 17. Q128 | C22 | semidirect |
| 18. SD128 | C22 | semidirect |
| 19. SD128 | C22 | semidirect |
| 20. SD128 | C22 | semidirect |
| 21. D64 | C22 | semidirect |
| 22. M128 | C22 | semidirect |
| 23. C2×C64 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33984 | 2 |
| #33982 | 2 |
| #13743 | 2 |
| #13742 | 2 |
| #13741 | 2 |
| C2×SD128 | 2 |
| #13737 | 2 |