Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
806 conjugacy classes of subgroups; only the 276 normal subgroups are shown.
This group has 806 conjugacy classes of subgroups — too many to draw. Only its 276 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 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #11887 | C2 | semidirect |
| 5. #11857 | C2 | semidirect |
| 6. #11743 | C2 | semidirect |
| 7. C2×#1348 | C2 | semidirect |
| 9. #3338 | C4 | semidirect |
| 11. D4×Q16 | C4 | semidirect |
| 30. C22×Q16 | C2×C4 | semidirect |
| 44. #500 | C2×C4 | semidirect |
| 45. #494 | C2×C4 | semidirect |
| 52. #467 | C2×C4 | semidirect |
| 57. C4×Q16 | C2×C4 | semidirect |
| 62. C2×#107 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62246 | 2 |
| #33486 | 2 |
| #22705 | 2 |
| #11887 | 2 |
| #11857 | 2 |
| #11743 | 2 |
| C2×#1348 | 2 |