Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
871 conjugacy classes of subgroups.
The subgroup lattice has 543 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #11599 |
| commutator | C2×C16 | C2×C4×C8 |
| Frattini, cosocle | C2×#1311 | C23 |
| socle | C22 | #122478 |
| prime radical | C2×#508 | C2×C8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C16 ◁ C4×#346 ◁ C8×#346 ◁ G | |
| LECS | G ▷ C2×#1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72216527 | C2 | semidirect |
| 3. #72216247 | C2 | semidirect |
| 5. #72216178 | C2 | semidirect |
| 23. #93759 | C4 | semidirect |
| 65. #7456 | C2×C4 | semidirect |
| 71. #7053 | C2×C4 | semidirect |
| 316. Q32 | #346 | semidirect |
| 320. D16 | #346 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72216527 | 2 |
| C2×#93759 | 2 |
| #72216247 | 2 |
| #72216211 | 2 |
| #72216178 | 2 |