Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2426 conjugacy classes of subgroups.
The subgroup lattice has 377 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | #95 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#1303 | C2 | direct (N × H) |
| 11. C2×#431 | C4 | semidirect |
| 15. C2×#423 | C4 | semidirect |
| 18. C22×C4⋊C8 | C4 | semidirect |
| 19. C22×C4⋊C8 | C4 | semidirect |
| 27. #1303 | C22 | direct (N × H) |
| 60. #431 | C2×C4 | semidirect |
| 65. #423 | C2×C4 | semidirect |
| 71. C2×C4⋊C8 | C2×C4 | semidirect |
| 72. C2×C4⋊C8 | C2×C4 | semidirect |
| 166. C4⋊C8 | C22×C4 | semidirect |
| 167. C4⋊C8 | C22×C4 | semidirect |
| 346. C22 | #1303 | direct (N × H) |
| 372. C2 | C2×#1303 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#431 | 2 |
| C22×#423 | 2 |
| C22×C4×C4⋊C4 | 2 |
| C2×#1303 | 2 |