Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
763 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 763 conjugacy classes of subgroups — too many to draw. Only its 272 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#357 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33974 | C2 | semidirect |
| 3. #33411 | C2 | semidirect |
| 4. #13622 | C2 | semidirect |
| 5. C2×#1418 | C2 | semidirect |
| 7. #3417 | C4 | semidirect |
| 9. #3415 | C4 | semidirect |
| 15. C2×M64 | C22 | semidirect |
| 17. #2158 | C22 | semidirect |
| 18. C2×#367 | C22 | semidirect |
| 19. #1418 | C22 | semidirect |
| 20. #1418 | C22 | semidirect |
| 33. #508 | C2×C4 | semidirect |
| 35. C2×Q32 | C2×C4 | semidirect |
| 37. C2×D16 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62635 | 2 |
| #33974 | 2 |
| #33411 | 2 |
| #13622 | 2 |
| C2×#1418 | 2 |