Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
551 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 551 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. #33838 | C2 | semidirect |
| 2. #19971 | C2 | semidirect |
| 3. D82 | C2 | semidirect |
| 4. #12517 | C2 | semidirect |
| 5. #12435 | C2 | semidirect |
| 6. #12426 | C2 | semidirect |
| 7. #12424 | C2 | semidirect |
| 8. #12320 | C2 | semidirect |
| 9. #12319 | C2 | semidirect |
| 10. C8×D16 | C2 | semidirect |
| 11. #12039 | C2 | semidirect |
| 14. #3283 | C22 | semidirect |
| 17. #2248 | C22 | semidirect |
| 18. #2248 | C22 | semidirect |
| 21. #2206 | C22 | semidirect |
| 25. #2169 | C22 | semidirect |
| 26. #1713 | C22 | semidirect |
| 29. #1669 | C22 | semidirect |
| 36. C8×D8 | C22 | semidirect |
| 37. #1371 | C22 | semidirect |
| 38. #1370 | C22 | semidirect |
| 39. #1362 | C22 | semidirect |
| 40. #1361 | C22 | semidirect |
| 42. #1331 | C22 | semidirect |
| 43. D4⋊C16 | C22 | semidirect |
| 44. C8×C16 | C22 | semidirect |
| 142. D16 | D8 | semidirect |
| 208. D8 | C2×D8 | semidirect |
| 236. C16 | C2×D8 | semidirect |
| 280. C8 | #509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33838 | 2 |
| #19971 | 2 |
| D82 | 2 |
| #12517 | 2 |
| #12435 | 2 |
| #12426 | 2 |
| #12424 | 2 |
| #12320 | 2 |
| #12319 | 2 |
| C8×D16 | 2 |
| #12039 | 2 |