Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4436 conjugacy classes of subgroups; only the 3170 normal subgroups are shown.
This group has 4436 conjugacy classes of subgroups — too many to draw. Only its 3170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #36691 | C2 | semidirect |
| 3. #34552 | C2 | semidirect |
| 6. #2845 | C22 | semidirect |
| 7. #2844 | C22 | semidirect |
| 8. #2843 | C22 | semidirect |
| 12. #2402 | C22 | semidirect |
| 13. #2399 | C22 | semidirect |
| 18. C42×Q8 | C22 | semidirect |
| 23. C22×C42 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #36691 | 2 |
| #36674 | 2 |
| #34552 | 2 |
| #34012 | 2 |