Order 512.
3826 conjugacy classes of subgroups.
The subgroup lattice has 422 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 71. C2×C4×Q8 | Q8 | semidirect |
| 72. C2×C4×Q8 | Q8 | semidirect |
| 310. Q8 | #395 | semidirect |
| 375. C2×C4 | #557 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#557 | 2 |
| C2×#2351 | 2 |
| C2×Q8×C4⋊C4 | 2 |
| #15230 | 2 |
| #15228 | 2 |
| #15218 | 2 |
| #15202 | 2 |
| #15128 | 2 |
| C4×#395 | 2 |
| Q8×#95 | 2 |