Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1638 conjugacy classes of subgroups.
The subgroup lattice has 441 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #21928 | C2 | semidirect |
| 3. #21916 | C2 | semidirect |
| 4. #21912 | C2 | semidirect |
| 6. #21437 | C2 | semidirect |
| 7. #20814 | C2 | semidirect |
| 10. #20469 | C2 | semidirect |
| 32. #1917 | C22 | semidirect |
| 33. #1917 | C22 | semidirect |
| 34. #1916 | C22 | semidirect |
| 35. #1916 | C22 | semidirect |
| 36. #1843 | C22 | semidirect |
| 37. #1832 | C22 | semidirect |
| 47. #1768 | C22 | semidirect |
| 48. #1768 | C22 | semidirect |
| 53. #1746 | C22 | semidirect |
| 54. #1726 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2283 | 2 |
| #21928 | 2 |
| #21916 | 2 |
| #21912 | 2 |
| #21904 | 2 |
| #21437 | 2 |
| #20814 | 2 |
| #20715 | 2 |
| #20714 | 2 |
| #20469 | 2 |