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 | #525 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #2929 ◁ 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. #13520 | C2 | semidirect |
| 5. #13498 | C2 | semidirect |
| 6. #13173 | C2 | semidirect |
| 7. #12878 | C2 | semidirect |
| 8. #12840 | C2 | semidirect |
| 9. #12838 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33433 | 2 |
| #33414 | 2 |
| #27691 | 2 |
| #13520 | 2 |
| #13498 | 2 |
| #13173 | 2 |
| #12878 | 2 |
| #12840 | 2 |
| #12838 | 2 |