Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1570 conjugacy classes of subgroups.
The subgroup lattice has 358 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #17825 | C2 | semidirect |
| 4. #17824 | C2 | semidirect |
| 5. #17806 | C2 | semidirect |
| 6. #17803 | C2 | semidirect |
| 7. #17795 | C2 | semidirect |
| 8. #17793 | C2 | semidirect |
| 29. #1574 | C22 | semidirect |
| 30. #1573 | C22 | semidirect |
| 31. #1572 | C22 | semidirect |
| 32. #1572 | C22 | semidirect |
| 33. #1571 | C22 | semidirect |
| 34. #1570 | C22 | semidirect |
| 35. #1569 | C22 | semidirect |
| 36. #1568 | C22 | semidirect |
| 37. #1568 | C22 | semidirect |
| 38. #1567 | C22 | semidirect |
| 39. #1566 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60277 | 2 |
| C2×#2922 | 2 |
| #17825 | 2 |
| #17824 | 2 |
| #17806 | 2 |
| #17803 | 2 |
| #17795 | 2 |
| #17793 | 2 |