Order 512.
2078 conjugacy classes of subgroups.
The subgroup lattice has 427 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1831 | C2 | semidirect |
| 4. #14687 | C2 | semidirect |
| 5. #10849 | C2 | semidirect |
| 6. #10806 | C2 | semidirect |
| 17. C42×D4 | C4 | semidirect |
| 288. C42 | #102 | semidirect |
| 323. D4 | #382 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C42×D4 | 2 |
| C2×#1831 | 2 |
| C2×#1802 | 2 |
| #14687 | 2 |
| #10849 | 2 |
| #10806 | 2 |
| #10788 | 2 |