Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
531 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 531 conjugacy classes of subgroups — too many to draw. Only its 134 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 | C2×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2966 ◁ 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. C2×#2246 | C2 | semidirect |
| 2. #33480 | C2 | semidirect |
| 3. #19353 | C2 | semidirect |
| 4. #12442 | C2 | semidirect |
| 5. #12439 | C2 | semidirect |
| 6. #12433 | C2 | semidirect |
| 7. #12430 | C2 | semidirect |
| 8. #12024 | C2 | semidirect |
| 9. #11964 | C2 | semidirect |
| 19. #2183 | C22 | semidirect |
| 20. Q8×C16 | C22 | semidirect |
| 21. #2169 | C22 | semidirect |
| 24. D4×C16 | C22 | semidirect |
| 31. #1668 | C22 | semidirect |
| 32. #1667 | C22 | semidirect |
| 35. #1372 | C22 | semidirect |
| 36. #1371 | C22 | semidirect |
| 37. #1361 | C22 | semidirect |
| 38. #1361 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2246 | 2 |
| #33480 | 2 |
| #19353 | 2 |
| #12442 | 2 |
| #12439 | 2 |
| #12433 | 2 |
| #12430 | 2 |
| #12024 | 2 |
| #11964 | 2 |