Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
396 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3199 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #32652 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #29270 | C2 | semidirect |
| 2. #13525 | C2 | semidirect |
| 3. #13513 | C2 | semidirect |
| 4. #13466 | C2 | semidirect |
| 5. #13266 | C2 | semidirect |
| 7. #13169 | C2 | semidirect |
| 9. #12947 | C2 | semidirect |
| 40. C2×Q32 | Q8 | semidirect |
| 41. C2×SD32 | Q8 | semidirect |
| 42. C2×D16 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #29270 | 2 |
| #13525 | 2 |
| #13513 | 2 |
| #13466 | 2 |
| #13266 | 2 |
| #13241 | 2 |
| #13169 | 2 |
| #13016 | 2 |
| #12947 | 2 |