Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1931 conjugacy classes of subgroups.
The subgroup lattice has 352 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 | C23×C4⋊C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ 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. #17798 | C2 | semidirect |
| 4. #17796 | C2 | semidirect |
| 5. #17793 | C2 | semidirect |
| 6. C2×#1566 | C2 | semidirect |
| 7. #17788 | C2 | semidirect |
| 27. #1569 | C22 | semidirect |
| 28. #1568 | C22 | semidirect |
| 33. #1566 | C22 | semidirect |
| 34. #1565 | C22 | semidirect |
| 35. #1564 | C22 | semidirect |
| 36. C2×#334 | C22 | semidirect |
| 116. #131 | M16 | semidirect |
| 235. M16 | C2×M16 | semidirect |
| 244. C2×C8 | C2×M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60263 | 2 |
| C2×C4×#131 | 2 |
| #17798 | 2 |
| #17796 | 2 |
| #17793 | 2 |
| C2×#1566 | 2 |
| #17788 | 2 |