Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
722 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 722 conjugacy classes of subgroups — too many to draw. Only its 174 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×#493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ 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. #33771 | C2 | semidirect |
| 2. #33733 | C2 | semidirect |
| 3. C2×#2246 | C2 | semidirect |
| 4. #19629 | C2 | semidirect |
| 5. #12525 | C2 | semidirect |
| 6. #12515 | C2 | semidirect |
| 7. #12503 | C2 | semidirect |
| 8. C16×M16 | C2 | semidirect |
| 14. #2252 | C22 | semidirect |
| 18. #2246 | C22 | semidirect |
| 19. #2246 | C22 | semidirect |
| 20. #2240 | C22 | semidirect |
| 21. #2240 | C22 | semidirect |
| 29. #1712 | C22 | semidirect |
| 30. #1706 | C22 | semidirect |
| 32. #1312 | C22 | semidirect |
| 33. C8×C16 | C22 | semidirect |
| 160. M16 | D16 | semidirect |
| 179. C16 | AGL(1,ℤ/8) | semidirect |
| 214. C8 | C2×D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33771 | 2 |
| #33733 | 2 |
| C2×#2246 | 2 |
| #19629 | 2 |
| #12525 | 2 |
| #12515 | 2 |
| #12503 | 2 |
| C16×M16 | 2 |