Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4426 conjugacy classes of subgroups.
The subgroup lattice has 488 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| 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 |
|---|---|---|
| 5. #13804 | C2 | semidirect |
| 6. C4⋊C42 | C2 | semidirect |
| 9. #13759 | C2 | semidirect |
| 11. #13755 | C2 | semidirect |
| 12. C42×C4⋊C4 | C2 | semidirect |
| 19. #2285 | C4 | semidirect |
| 25. C4×#117 | C4 | semidirect |
| 26. C2×C4×C4⋊C4 | C4 | semidirect |
| 100. #384 | C2×C4 | semidirect |
| 101. #382 | C2×C4 | semidirect |
| 111. C4×C4⋊C4 | C2×C4 | semidirect |
| 112. C4×C4⋊C4 | C2×C4 | semidirect |
| 113. C4×C4⋊C4 | C2×C4 | semidirect |
| 114. C4×C4⋊C4 | C2×C4 | semidirect |
| 119. #376 | C2×C4 | semidirect |
| 132. C43 | C2×C4 | semidirect |
| 167. #117 | C4⋊C4 | direct (N × H) |
| 213. C2×C42 | C42 | semidirect |
| 337. C4⋊C4 | C2×C4⋊C4 | semidirect |
| 338. C4⋊C4 | #117 | direct (N × H) |
| 354. C42 | C2×C4⋊C4 | semidirect |
| 416. C2×C4 | C4×C4⋊C4 | semidirect |
| 476. C4 | C4×#117 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2285 | 2 |
| C2×C4×#117 | 2 |
| C22×C4×C4⋊C4 | 2 |
| #13804 | 2 |
| C4⋊C42 | 2 |
| #13802 | 2 |
| C22⋊C4×C4⋊C4 | 2 |
| #13759 | 2 |
| #13756 | 2 |
| #13755 | 2 |
| C42×C4⋊C4 | 2 |