Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
571 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 571 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72225471 |
| commutator | C4×C32 | C22×C4 |
| Frattini, cosocle | C4×M64 | C23 |
| prime radical | #72216261 | C2 |
| LCS | G ▷ C4×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ C8×M64 ◁ G | |
| LECS | G ▷ C4×M64 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72225470 | C2 | semidirect |
| 2. #72225468 | C2 | semidirect |
| 3. #72225395 | C2 | semidirect |
| 4. #72225313 | C2 | semidirect |
| 11. #151732 | C4 | semidirect |
| 12. #151724 | C4 | semidirect |
| 15. #151715 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72225470 | 2 |
| #72225468 | 2 |
| #72225395 | 2 |
| #72225313 | 2 |
| #72216261 | 2 |