Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4554 conjugacy classes of subgroups.
The subgroup lattice has 378 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #13806 | C2 | direct (N × H) |
| 15. C2×#383 | C4 | semidirect |
| 17. C2×C4×C4⋊C4 | C4 | semidirect |
| 19. C2×#376 | C4 | semidirect |
| 63. #383 | C2×C4 | semidirect |
| 64. C4×C4⋊C4 | C2×C4 | semidirect |
| 65. #376 | C2×C4 | semidirect |
| 150. C2×C42 | C42 | semidirect |
| 262. C42 | C2×C42 | semidirect |
| 368. C2 | #13806 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#383 | 2 |
| C22×C4×C4⋊C4 | 2 |
| C22×#376 | 2 |
| #13806 | 2 |
| C2×#1436 | 2 |
| C2×#1434 | 2 |
| C2×C4×#95 | 2 |