Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
544 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 146 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 | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #28263 | C2 | semidirect |
| 2. #13526 | C2 | semidirect |
| 3. #13505 | C2 | semidirect |
| 4. #13474 | C2 | semidirect |
| 5. #13263 | C2 | semidirect |
| 7. #13165 | C2 | semidirect |
| 9. #12935 | C2 | semidirect |
| 41. C2×D16 | Q8 | semidirect |
| 143. C2×Q8 | D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28263 | 2 |
| #13526 | 2 |
| #13505 | 2 |
| #13474 | 2 |
| #13263 | 2 |
| #13235 | 2 |
| #13165 | 2 |
| #13012 | 2 |
| #12935 | 2 |