Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
383 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33600 | C2 | semidirect |
| 2. #13716 | C2 | semidirect |
| 3. #13713 | C2 | semidirect |
| 4. #13699 | C2 | semidirect |
| 5. #13698 | C2 | semidirect |
| 6. #13697 | C2 | semidirect |
| 7. #13666 | C2 | semidirect |
| 8. #13663 | C2 | semidirect |
| 9. #13553 | C2 | semidirect |
| 13. C2×Q64 | C22 | semidirect |
| 14. C2×SD64 | C22 | semidirect |
| 15. C2×D32 | C22 | semidirect |
| 26. #1424 | C22 | semidirect |
| 27. #1424 | C22 | semidirect |
| 28. #1423 | C22 | semidirect |
| 29. #1416 | C22 | semidirect |
| 30. #1416 | C22 | semidirect |
| 31. #1415 | C22 | semidirect |
| 32. #1415 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33600 | 2 |
| #13716 | 2 |
| #13713 | 2 |
| #13699 | 2 |
| #13698 | 2 |
| #13697 | 2 |
| #13666 | 2 |
| #13663 | 2 |
| #13553 | 2 |