Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
650 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D42 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23173 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33443 | C2 | semidirect |
| 3. #13197 | C2 | semidirect |
| 4. #12982 | C2 | semidirect |
| 5. C4×#364 | C2 | semidirect |
| 50. #364 | D4 | semidirect |
| 51. #364 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33443 | 2 |
| #21705 | 2 |
| #13197 | 2 |
| #12982 | 2 |
| C4×#364 | 2 |