Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
535 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 535 conjugacy classes of subgroups — too many to draw. Only its 120 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 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13331 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33248 | C2 | semidirect |
| 2. #21296 | C2 | semidirect |
| 3. #13458 | C2 | semidirect |
| 4. #13413 | C2 | semidirect |
| 5. #13372 | C2 | semidirect |
| 7. #13249 | C2 | semidirect |
| 8. #13155 | C2 | semidirect |
| 9. #13138 | C2 | semidirect |
| 10. #13025 | C2 | semidirect |
| 11. #12596 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33248 | 2 |
| #21296 | 2 |
| #13458 | 2 |
| #13413 | 2 |
| #13372 | 2 |
| #13331 | 2 |
| #13249 | 2 |
| #13155 | 2 |
| #13138 | 2 |
| #13025 | 2 |
| #12596 | 2 |