Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
566 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 566 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 | C22 | #2328 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22277 | C2 | semidirect |
| 4. #9779 | C2 | semidirect |
| 6. #9300 | C2 | semidirect |
| 7. #9291 | C2 | semidirect |
| 8. #8791 | C2 | semidirect |
| 9. #8142 | C2 | semidirect |
| 22. #1722 | C4 | semidirect |
| 23. #1713 | C4 | semidirect |
| 24. #1709 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22282 | 2 |
| #22277 | 2 |
| #19662 | 2 |
| #9779 | 2 |
| #9760 | 2 |
| #9300 | 2 |
| #9291 | 2 |
| #8791 | 2 |
| #8142 | 2 |