Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
523 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 523 conjugacy classes of subgroups — too many to draw. Only its 128 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×#466 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33216 | C2 | semidirect |
| 3. #13530 | C2 | semidirect |
| 4. #13526 | C2 | semidirect |
| 5. #13505 | C2 | semidirect |
| 6. #13408 | C2 | semidirect |
| 7. #13383 | C2 | semidirect |
| 9. #13366 | C2 | semidirect |
| 10. #13365 | C2 | semidirect |
| 11. #12845 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33216 | 2 |
| #33182 | 2 |
| #13530 | 2 |
| #13526 | 2 |
| #13505 | 2 |
| #13408 | 2 |
| #13383 | 2 |
| #13380 | 2 |
| #13366 | 2 |
| #13365 | 2 |
| #12845 | 2 |