Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
781 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 781 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×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. C2×#3160 | C2 | semidirect |
| 2. C2×#2231 | C2 | semidirect |
| 4. #13457 | C2 | semidirect |
| 5. #13371 | C2 | semidirect |
| 6. #13273 | C2 | semidirect |
| 8. #12930 | C2 | semidirect |
| 9. #12788 | C2 | semidirect |
| 54. #499 | D4 | semidirect |
| 56. #467 | Q8 | semidirect |
| 57. #466 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3160 | 2 |
| C2×#2231 | 2 |
| #21544 | 2 |
| #13457 | 2 |
| #13371 | 2 |
| #13273 | 2 |
| #13002 | 2 |
| #12930 | 2 |
| #12788 | 2 |