Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
373 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1804 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8565 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12882 | C2 | semidirect |
| 5. #8268 | C2 | semidirect |
| 6. #7181 | C2 | semidirect |
| 7. #7180 | C2 | semidirect |
| 41. #442 | C8 | semidirect |
| 42. #441 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20512 | 2 |
| #12882 | 2 |
| #11604 | 2 |
| #11602 | 2 |
| #8268 | 2 |
| #7181 | 2 |
| #7180 | 2 |