Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
498 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3192 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #29237 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ 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 |
|---|---|---|
| 2. #21164 | C2 | semidirect |
| 3. #13519 | C2 | semidirect |
| 4. #13466 | C2 | semidirect |
| 5. #13115 | C2 | semidirect |
| 8. #12944 | C2 | semidirect |
| 9. #12932 | C2 | semidirect |
| 10. #12924 | C2 | semidirect |
| 11. #12786 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21550 | 2 |
| #21164 | 2 |
| #13519 | 2 |
| #13466 | 2 |
| #13115 | 2 |
| #13008 | 2 |
| #12998 | 2 |
| #12944 | 2 |
| #12932 | 2 |
| #12924 | 2 |
| #12786 | 2 |