Order 512.
1072 conjugacy classes of subgroups.
The subgroup lattice has 445 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 78. C2×#108 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3165 | 2 |
| C2×#1997 | 2 |
| #21322 | 2 |
| C2×#1990 | 2 |
| #21300 | 2 |
| C2×C4×#108 | 2 |
| #11290 | 2 |
| #11249 | 2 |
| #10857 | 2 |