Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7498 conjugacy classes of subgroups.
The subgroup lattice has 474 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| 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 |
|---|---|---|
| 5. #13802 | C2 | direct (N × H) |
| 6. C2×#1436 | C2 | semidirect |
| 7. C2×#1434 | C2 | semidirect |
| 9. C2×C4×#95 | C2 | semidirect |
| 18. C2×#380 | C4 | semidirect |
| 21. C2×C4×C22⋊C4 | C4 | semidirect |
| 24. C22×#95 | C4 | semidirect |
| 34. #1436 | C22 | semidirect |
| 35. #1434 | C22 | semidirect |
| 37. C4×#95 | C22 | semidirect |
| 58. C22×C4⋊C4 | C2×C4 | semidirect |
| 76. C22×C42 | C2×C4 | semidirect |
| 79. #380 | C2×C4 | semidirect |
| 81. C4×C22⋊C4 | C2×C4 | semidirect |
| 88. C2×#95 | C2×C4 | semidirect |
| 89. C2×#95 | C2×C4 | semidirect |
| 90. C2×#95 | C2×C4 | semidirect |
| 144. C23×C4 | C42 | semidirect |
| 172. C2×C4⋊C4 | C22×C4 | semidirect |
| 179. C2×C22⋊C4 | C4⋊C4 | semidirect |
| 191. C2×C42 | C22×C4 | semidirect |
| 198. #95 | C22×C4 | semidirect |
| 199. #95 | C22×C4 | semidirect |
| 300. C22×C4 | C2×C42 | semidirect |
| 320. C22⋊C4 | C2×C4⋊C4 | semidirect |
| 463. C2 | #13802 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#380 | 2 |
| C22×C4×C22⋊C4 | 2 |
| C23×#95 | 2 |
| #13802 | 2 |
| C2×#1436 | 2 |
| C2×#1434 | 2 |
| C2×C22⋊#95 | 2 |
| C2×C4×#95 | 2 |