Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
678 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 678 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21683 |
| prime radical | #23794 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33441 | C2 | semidirect |
| 2. #33432 | C2 | semidirect |
| 4. #13200 | C2 | semidirect |
| 5. #13198 | C2 | semidirect |
| 6. #12902 | C2 | semidirect |
| 7. #12542 | C2 | semidirect |
| 58. #346 | D4 | semidirect |
| 59. C4×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33441 | 2 |
| #33432 | 2 |
| #23794 | 2 |
| #13200 | 2 |
| #13198 | 2 |
| #12902 | 2 |
| #12542 | 2 |