Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1519 conjugacy classes of subgroups.
The subgroup lattice has 478 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2920 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #18072 | C2 | semidirect |
| 4. #18068 | C2 | semidirect |
| 5. #18062 | C2 | semidirect |
| 6. #18058 | C2 | semidirect |
| 7. #17991 | C2 | semidirect |
| 8. #17972 | C2 | semidirect |
| 39. #1593 | C22 | semidirect |
| 40. #1591 | C22 | semidirect |
| 41. #1587 | C22 | semidirect |
| 42. #1584 | C22 | semidirect |
| 43. #1579 | C22 | semidirect |
| 44. C8×Q16 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3170 | 2 |
| #60251 | 2 |
| #18072 | 2 |
| #18068 | 2 |
| #18062 | 2 |
| #18058 | 2 |
| #17991 | 2 |
| #17972 | 2 |
| C2×#1568 | 2 |
| #17496 | 2 |
| #17323 | 2 |