Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1034 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 282 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | C23 |
| commutator, socle | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #2388 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#504 | C2 | semidirect |
| 2. C2×D4×C16 | C2 | semidirect |
| 4. #12977 | C2 | semidirect |
| 5. #12913 | C2 | semidirect |
| 6. #12680 | C2 | semidirect |
| 7. C16×C22⋊C4 | C2 | semidirect |
| 38. C22×C16 | D4 | semidirect |
| 157. C23 | #504 | semidirect |
| 174. C2×C4 | #504 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#504 | 2 |
| C2×D4×C16 | 2 |
| #21442 | 2 |
| #12977 | 2 |
| #12913 | 2 |
| #12680 | 2 |
| C16×C22⋊C4 | 2 |