Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5426 conjugacy classes of subgroups; only the 1898 normal subgroups are shown.
This group has 5426 conjugacy classes of subgroups — too many to draw. Only its 1898 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13796 | C2 | semidirect |
| 4. #13753 | C2 | semidirect |
| 8. C2×C4×C22⋊C4 | C4 | semidirect |
| 11. C2×#376 | C22 | semidirect |
| 13. #1434 | C22 | semidirect |
| 15. C4×#95 | C22 | semidirect |
| 26. C22×C42 | C2×C4 | semidirect |
| 29. C4×C22⋊C4 | C2×C4 | semidirect |
| 30. C4×C22⋊C4 | C2×C4 | semidirect |
| 32. C2×#95 | C2×C4 | semidirect |
| 109. C23 | #376 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×C22⋊C4 | 2 |
| #13796 | 2 |
| #13794 | 2 |
| #13753 | 2 |