Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2606 conjugacy classes of subgroups.
The subgroup lattice has 401 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C24×C4 |
| prime radical | C22×#41 | C2×C4 |
| 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. #22734 | C2 | semidirect |
| 4. Q8×C4⋊C8 | C2 | semidirect |
| 5. #22696 | C2 | semidirect |
| 6. D4×C4⋊C8 | C2 | semidirect |
| 7. #21349 | C2 | semidirect |
| 8. #20808 | C2 | semidirect |
| 9. C2×C4×C4⋊C8 | C2 | semidirect |
| 48. C4×#41 | C8 | semidirect |
| 138. C4×Q8 | C2×C8 | semidirect |
| 139. C4×Q8 | C2×C8 | semidirect |
| 145. C4×D4 | C2×C8 | semidirect |
| 146. C4×D4 | C2×C8 | semidirect |
| 151. #117 | C2×C8 | semidirect |
| 166. C2×C42 | C2×C8 | semidirect |
| 201. C4⋊C8 | #41 | direct (N × H) |
| 211. #41 | C4⋊C8 | direct (N × H) |
| 308. Q8 | C2×C4⋊C8 | semidirect |
| 309. D4 | C2×C4⋊C8 | semidirect |
| 337. C2×C4 | C2×C4⋊C8 | semidirect |
| 389. C4 | C8×#41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×#41 | 2 |
| C42×#41 | 2 |
| #22734 | 2 |
| Q8×C4⋊C8 | 2 |
| #22696 | 2 |
| D4×C4⋊C8 | 2 |
| #21349 | 2 |
| #20808 | 2 |
| C2×C4×C4⋊C8 | 2 |