Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
337 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 86 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×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | #33600 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13553 | C2 | semidirect |
| 3. C2×#1380 | C2 | semidirect |
| 4. #7419 | C2 | semidirect |
| 5. #7418 | C2 | semidirect |
| 6. C2×#508 | C4 | semidirect |
| 13. #2216 | C4 | semidirect |
| 15. #2215 | C4 | semidirect |
| 17. #2158 | C4 | semidirect |
| 28. C2×Q32 | C2×C4 | semidirect |
| 30. C2×D16 | C2×C4 | semidirect |
| 53. #366 | C2×C4 | semidirect |
| 54. #366 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33600 | 2 |
| #13553 | 2 |
| C2×#1380 | 2 |
| #7419 | 2 |
| #7418 | 2 |