Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1698 conjugacy classes of subgroups.
The subgroup lattice has 402 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | #336 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#1378 | C2 | direct (N × H) |
| 15. C22×M32 | C4 | semidirect |
| 16. C22×M32 | C4 | semidirect |
| 17. C22×M32 | C4 | semidirect |
| 21. C23×C16 | C4 | semidirect |
| 28. #1378 | C22 | direct (N × H) |
| 71. C2×M32 | C2×C4 | semidirect |
| 72. C2×M32 | C2×C4 | semidirect |
| 73. C2×M32 | C2×C4 | semidirect |
| 81. C22×C16 | C2×C4 | semidirect |
| 180. M32 | C22×C4 | semidirect |
| 186. C2×C16 | C22×C4 | semidirect |
| 376. C22 | #1378 | direct (N × H) |
| 399. C2 | C2×#1378 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M32 | 2 |
| C22×#405 | 2 |
| C2×#1378 | 2 |