Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1737 conjugacy classes of subgroups.
The subgroup lattice has 408 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 | D4 |
| 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. #17829 | C2 | semidirect |
| 4. #17828 | C2 | semidirect |
| 5. #17827 | C2 | semidirect |
| 6. #17826 | C2 | semidirect |
| 7. #17803 | C2 | semidirect |
| 8. #17793 | C2 | semidirect |
| 9. C2×#1564 | C2 | semidirect |
| 33. #1572 | C22 | semidirect |
| 34. #1571 | C22 | semidirect |
| 35. #1570 | C22 | semidirect |
| 36. #1568 | C22 | semidirect |
| 37. #1568 | C22 | semidirect |
| 38. #1567 | C22 | semidirect |
| 39. #1566 | C22 | semidirect |
| 44. #1564 | C22 | semidirect |
| 45. #1564 | C22 | semidirect |
| 46. C2×#335 | C22 | semidirect |
| 47. C2×#334 | C22 | semidirect |
| 288. C2×C8 | #131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×#41 | 2 |
| #60127 | 2 |
| #17829 | 2 |
| #17828 | 2 |
| #17827 | 2 |
| #17826 | 2 |
| #17803 | 2 |
| #17793 | 2 |
| C2×#1564 | 2 |