Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
403 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 116 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 |
| prime radical | #13661 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×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. #33735 | C2 | semidirect |
| 2. C2×#2249 | C2 | semidirect |
| 3. #13728 | C2 | semidirect |
| 4. #13721 | C2 | semidirect |
| 5. #13690 | C2 | semidirect |
| 6. #13689 | C2 | semidirect |
| 7. #13665 | C2 | semidirect |
| 8. #13664 | C2 | semidirect |
| 9. #13662 | C2 | semidirect |
| 11. #13639 | C2 | semidirect |
| 17. C2×Q64 | C22 | semidirect |
| 19. C2×SD64 | C22 | semidirect |
| 21. #2253 | C22 | semidirect |
| 26. #2249 | C22 | semidirect |
| 31. C4×Q32 | C22 | semidirect |
| 33. C4×D16 | C22 | semidirect |
| 35. C4⋊C32 | C22 | semidirect |
| 36. #1416 | C22 | semidirect |
| 37. #1416 | C22 | semidirect |
| 42. C4×C32 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33735 | 2 |
| C2×#2249 | 2 |
| #13728 | 2 |
| #13721 | 2 |
| #13690 | 2 |
| #13689 | 2 |
| #13665 | 2 |
| #13664 | 2 |
| #13662 | 2 |
| #13661 | 2 |
| #13639 | 2 |