Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
788 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 788 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C25 | D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2216 | C2 | semidirect |
| 4. #13493 | C2 | semidirect |
| 5. #13383 | C2 | semidirect |
| 6. #13306 | C2 | semidirect |
| 7. #13060 | C2 | semidirect |
| 8. #12834 | C2 | semidirect |
| 10. C22×Q32 | C22 | semidirect |
| 17. C2×#366 | C22 | semidirect |
| 18. C2×#366 | C22 | semidirect |
| 19. C2×#358 | C22 | semidirect |
| 20. C2×#358 | C22 | semidirect |
| 28. #1380 | C22 | semidirect |
| 29. #1380 | C22 | semidirect |
| 137. C24 | Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2216 | 2 |
| C22×C22⋊C16 | 2 |
| #32931 | 2 |
| #13493 | 2 |
| #13383 | 2 |
| #13306 | 2 |
| #13060 | 2 |
| #12834 | 2 |