Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
10282 conjugacy classes of subgroups; only the 4002 normal subgroups are shown.
This group has 10282 conjugacy classes of subgroups — too many to draw. Only its 4002 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C22×#483 | C2 | direct (N × H) |
| 4. C23×#106 | C2 | semidirect |
| 5. C23×#103 | C2 | semidirect |
| 20. C2×#483 | C22 | direct (N × H) |
| 21. C22×#106 | C22 | semidirect |
| 22. C22×#103 | C22 | semidirect |
| 51. #483 | C23 | direct (N × H) |
| 52. C2×#106 | C23 | semidirect |
| 53. C2×#103 | C23 | semidirect |
| 94. #106 | C24 | semidirect |
| 95. #103 | C24 | semidirect |
| 140. C23 | #483 | direct (N × H) |
| 168. C22 | C2×#483 | direct (N × H) |
| 181. C2 | C22×#483 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C22⋊Q8 | 2 |
| C24×C4⋊C4 | 2 |
| C22×#483 | 2 |
| C23×#106 | 2 |
| C23×#103 | 2 |
| C23×C22⋊C8 | 2 |