Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
672 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 138 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 | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ 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×#2219 | C2 | semidirect |
| 4. #13496 | C2 | semidirect |
| 5. #13394 | C2 | semidirect |
| 6. #13315 | C2 | semidirect |
| 7. #13066 | C2 | semidirect |
| 8. #12838 | C2 | semidirect |
| 10. C22×Q32 | C22 | semidirect |
| 18. C2×#368 | C22 | semidirect |
| 19. C2×#367 | C22 | semidirect |
| 20. C2×#362 | C22 | semidirect |
| 21. C2×#358 | C22 | semidirect |
| 30. #1387 | C22 | semidirect |
| 31. #1381 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2219 | 2 |
| #33333 | 2 |
| #32943 | 2 |
| #13496 | 2 |
| #13394 | 2 |
| #13315 | 2 |
| #13066 | 2 |
| #12838 | 2 |