Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3090 conjugacy classes of subgroups.
The subgroup lattice has 424 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C22×D4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #21169 | C2 | direct (N × H) |
| 6. C2×#1949 | C2 | semidirect |
| 7. C2×#1948 | C2 | semidirect |
| 8. C2×#1947 | C2 | semidirect |
| 19. C22×C4⋊C8 | C4 | semidirect |
| 40. #1949 | C22 | semidirect |
| 41. #1948 | C22 | semidirect |
| 42. #1947 | C22 | semidirect |
| 96. C2×C4⋊C8 | C2×C4 | semidirect |
| 359. C2×C4 | #427 | semidirect |
| 411. C4 | C2×#427 | semidirect |
| 417. C2 | #21169 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#427 | 2 |
| C23×C4⋊C8 | 2 |
| C2×#2285 | 2 |
| C2×C4×#117 | 2 |
| #21169 | 2 |
| C2×#1949 | 2 |
| C2×#1948 | 2 |
| C2×#1947 | 2 |
| C2×#1735 | 2 |