Order 512.
1738 conjugacy classes of subgroups.
The subgroup lattice has 426 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #33196 | C2 | semidirect |
| 4. #29996 | C2 | semidirect |
| 5. #29287 | C2 | semidirect |
| 6. #28283 | C2 | semidirect |
| 57. #533 | Q8 | semidirect |
| 373. C2×C4 | #475 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #36645 | 2 |
| C4×#533 | 2 |
| #33196 | 2 |
| #29996 | 2 |
| #29287 | 2 |
| #28283 | 2 |
| #24838 | 2 |
| #21959 | 2 |
| #21885 | 2 |
| C2×#1842 | 2 |