Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
469 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 138 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 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#2263 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #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 |
|---|---|---|
| 2. C22×SD64 | C2 | semidirect |
| 3. #33721 | C2 | semidirect |
| 4. #33456 | C2 | semidirect |
| 5. #13689 | C2 | semidirect |
| 6. #13687 | C2 | semidirect |
| 7. #13628 | C2 | semidirect |
| 8. #13618 | C2 | semidirect |
| 9. #13615 | C2 | semidirect |
| 24. #2249 | C22 | semidirect |
| 25. #2247 | C22 | semidirect |
| 31. C4⋊C32 | C22 | semidirect |
| 32. #1416 | C22 | semidirect |
| 33. #1416 | C22 | semidirect |
| 34. #1415 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2263 | 2 |
| C22×SD64 | 2 |
| #33721 | 2 |
| #33456 | 2 |
| #13689 | 2 |
| #13687 | 2 |
| #13628 | 2 |
| #13618 | 2 |
| #13615 | 2 |