Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7880 conjugacy classes of subgroups; only the 7730 normal subgroups are shown.
This group has 7880 conjugacy classes of subgroups — too many to draw. Only its 7730 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C24 |
| commutator | C2 | C25×C8 |
| Frattini, cosocle | C8 | C26 |
| socle | C22 | C24×C8 |
| prime radical | C2×#585 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #62624 | C2 | direct (N × H) |
| 3. C2×#3407 | C2 | semidirect |
| 4. C22×#504 | C2 | semidirect |
| 10. #3407 | C22 | semidirect |
| 11. C2×#504 | C22 | semidirect |
| 12. C2×#504 | C22 | semidirect |
| 13. C2×#504 | C22 | semidirect |
| 14. C22×M32 | C22 | semidirect |
| 15. C23×C16 | C22 | semidirect |
| 28. #504 | C23 | semidirect |
| 29. #504 | C23 | semidirect |
| 31. C2×M32 | C23 | semidirect |
| 34. C22×C16 | C23 | semidirect |
| 99. C2 | #62624 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3573 | 2 |
| #62624 | 2 |
| C2×#3407 | 2 |
| C22×#504 | 2 |