Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
336 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2226 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33730 | C2 | semidirect |
| 3. #13710 | C2 | semidirect |
| 4. #13709 | C2 | semidirect |
| 5. #13638 | C2 | semidirect |
| 6. C2×#1426 | C2 | semidirect |
| 7. #13630 | C2 | semidirect |
| 16. #2254 | C22 | semidirect |
| 17. #2253 | C22 | semidirect |
| 26. #1426 | C22 | semidirect |
| 27. #1426 | C22 | semidirect |
| 28. #1426 | C22 | semidirect |
| 29. #1422 | C22 | semidirect |
| 38. #508 | Q8 | semidirect |
| 40. C2×Q32 | 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 |
|---|---|
| #33730 | 2 |
| #33465 | 2 |
| #13710 | 2 |
| #13709 | 2 |
| #13638 | 2 |
| C2×#1426 | 2 |
| #13630 | 2 |
| #13622 | 2 |
| C2×#1418 | 2 |