Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
601 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 601 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #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. C2×#2228 | C2 | semidirect |
| 2. #21723 | C2 | semidirect |
| 4. #13507 | C2 | semidirect |
| 6. #13131 | C2 | semidirect |
| 7. #13130 | C2 | semidirect |
| 8. #12880 | C2 | semidirect |
| 9. #12811 | C2 | semidirect |
| 48. #499 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2228 | 2 |
| #21723 | 2 |
| #21697 | 2 |
| #13507 | 2 |
| #13407 | 2 |
| #13131 | 2 |
| #13130 | 2 |
| #12880 | 2 |
| #12811 | 2 |