Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
495 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 104 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 | #72225435 |
| commutator | C4×C32 | C22×C4 |
| Frattini, cosocle | C8×C32 | C23 |
| prime radical | #72216261 | C2 |
| LCS | G ▷ C4×C32 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ C8×M64 ◁ G | |
| LECS | G ▷ C8×C32 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72225433 | C2 | semidirect |
| 2. #72225432 | C2 | semidirect |
| 3. #72225395 | C2 | semidirect |
| 4. #72225352 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72225433 | 2 |
| #72225432 | 2 |
| #72225395 | 2 |
| #72225352 | 2 |
| #72216261 | 2 |