Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
564 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 126 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 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33840 | C2 | semidirect |
| 2. #19975 | C2 | semidirect |
| 3. D82 | C2 | semidirect |
| 4. #12515 | C2 | semidirect |
| 5. #12433 | C2 | semidirect |
| 6. #12425 | C2 | semidirect |
| 7. #12424 | C2 | semidirect |
| 8. #12317 | C2 | semidirect |
| 9. #12316 | C2 | semidirect |
| 10. C8×SD32 | C2 | semidirect |
| 11. #12039 | C2 | semidirect |
| 14. #3287 | C22 | semidirect |
| 17. #2246 | C22 | semidirect |
| 18. #2246 | C22 | semidirect |
| 20. #2207 | C22 | semidirect |
| 25. #2169 | C22 | semidirect |
| 27. #1712 | C22 | semidirect |
| 31. #1667 | C22 | semidirect |
| 34. C8×Q16 | C22 | semidirect |
| 37. #1372 | C22 | semidirect |
| 38. #1370 | C22 | semidirect |
| 39. #1361 | C22 | semidirect |
| 40. #1361 | C22 | semidirect |
| 41. #1332 | C22 | semidirect |
| 43. D4⋊C16 | C22 | semidirect |
| 44. C8×C16 | C22 | semidirect |
| 141. SD32 | D8 | semidirect |
| 195. Q16 | C2×D8 | semidirect |
| 239. C16 | C2×D8 | semidirect |
| 285. C8 | #509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33840 | 2 |
| #19975 | 2 |
| D82 | 2 |
| #12515 | 2 |
| #12433 | 2 |
| #12425 | 2 |
| #12424 | 2 |
| #12317 | 2 |
| #12316 | 2 |
| C8×SD32 | 2 |
| #12039 | 2 |