Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
774 conjugacy classes of subgroups.
The subgroup lattice has 484 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33418 | C2 | semidirect |
| 3. C2×#1564 | C2 | semidirect |
| 4. #11656 | C2 | semidirect |
| 5. #11653 | C2 | semidirect |
| 6. #11652 | C2 | semidirect |
| 7. #11598 | C2 | semidirect |
| 41. #1334 | C22 | semidirect |
| 42. #1333 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60673 | 2 |
| #33418 | 2 |
| C2×#1564 | 2 |
| #11656 | 2 |
| #11653 | 2 |
| #11652 | 2 |
| #11598 | 2 |