Order 4096.
703 conjugacy classes of subgroups.
The subgroup lattice has 440 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #93269 |
| commutator | C4×C16 | C82 |
| Frattini, cosocle | #72216182 | C22 |
| socle | C22 | #72216542 |
| prime radical | C23 | #92938 |
| LCS | G ▷ C4×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C16 ◁ C8×#346 ◁ #72216182 ◁ G | |
| LECS | G ▷ #72216182 ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 36. #93876 | C8 | semidirect |
| 39. #93665 | C8 | semidirect |
| 41. C16×C32 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72227194 | 2 |
| #72227185 | 2 |
| #72216231 | 2 |