Order 512.
3642 conjugacy classes of subgroups.
The subgroup lattice has 407 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #14663 | C2 | semidirect |
| 5. #14663 | C2 | semidirect |
| 6. #14346 | C2 | semidirect |
| 7. #14289 | C2 | semidirect |
| 10. #13961 | C2 | semidirect |
| 14. C2×#533 | C4 | semidirect |
| 61. C2×C4⋊Q8 | C2×C4 | semidirect |
| 62. C2×C4⋊Q8 | C2×C4 | semidirect |
| 65. C2×#125 | C2×C4 | semidirect |
| 76. C2×C4×Q8 | C2×C4 | semidirect |
| 77. C2×C4×Q8 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#533 | 2 |
| C2×#2283 | 2 |
| C22×#383 | 2 |
| #14663 | 2 |
| #14346 | 2 |
| #14289 | 2 |
| #14281 | 2 |
| #14159 | 2 |
| #13961 | 2 |
| #13958 | 2 |