Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1680 conjugacy classes of subgroups.
The subgroup lattice has 433 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. #22595 | C2 | semidirect |
| 3. #22591 | C2 | semidirect |
| 4. #22589 | C2 | semidirect |
| 6. #21920 | C2 | semidirect |
| 7. #21918 | C2 | semidirect |
| 8. #20781 | C2 | semidirect |
| 9. #20780 | C2 | semidirect |
| 28. #1917 | C22 | semidirect |
| 29. #1917 | C22 | semidirect |
| 30. #1916 | C22 | semidirect |
| 31. #1916 | C22 | semidirect |
| 34. #1840 | C22 | semidirect |
| 35. #1839 | C22 | semidirect |
| 36. #1832 | C22 | semidirect |
| 37. #1831 | C22 | semidirect |
| 45. #1768 | C22 | semidirect |
| 46. #1767 | C22 | semidirect |
| 47. C4×C4⋊C8 | C22 | semidirect |
| 52. #1746 | C22 | semidirect |
| 53. #1745 | C22 | semidirect |
| 54. #1744 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #34012 | 2 |
| #22595 | 2 |
| #22591 | 2 |
| #22589 | 2 |
| #21931 | 2 |
| #21920 | 2 |
| #21918 | 2 |
| #20781 | 2 |
| #20780 | 2 |
| C2×#1817 | 2 |
| #20528 | 2 |