Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2148 conjugacy classes of subgroups.
The subgroup lattice has 275 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C24×C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #20554 | C2 | semidirect |
| 4. #20542 | C2 | semidirect |
| 6. #20473 | C2 | semidirect |
| 7. #20472 | C2 | semidirect |
| 8. #20463 | C2 | semidirect |
| 9. #20462 | C2 | semidirect |
| 11. #2922 | C4 | semidirect |
| 59. #433 | C2×C4 | semidirect |
| 61. #431 | C2×C4 | semidirect |
| 62. #424 | C2×C4 | semidirect |
| 68. C4×M16 | C2×C4 | semidirect |
| 72. C2×C4×C8 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2922 | 2 |
| #34026 | 2 |
| #20554 | 2 |
| #20542 | 2 |
| #20532 | 2 |
| #20473 | 2 |
| #20472 | 2 |
| #20463 | 2 |
| #20462 | 2 |