Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
454 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3058 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #27662 |
| prime radical | #532 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33414 | C2 | semidirect |
| 4. #13519 | C2 | semidirect |
| 5. #13499 | C2 | semidirect |
| 6. #13170 | C2 | semidirect |
| 7. #12877 | C2 | semidirect |
| 8. #12841 | C2 | semidirect |
| 9. #12839 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33435 | 2 |
| #33414 | 2 |
| #27693 | 2 |
| #13519 | 2 |
| #13499 | 2 |
| #13170 | 2 |
| #12877 | 2 |
| #12841 | 2 |
| #12839 | 2 |