Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
684 conjugacy classes of subgroups; only the 364 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 364 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #135778 |
| commutator | C8×C16 | C22×C4 |
| Frattini, cosocle | C2×C8×C16 | C23 |
| prime radical | C23 | #12517 |
| LCS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×C162 ◁ G | |
| LECS | G ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 12. #93664 | C4 | semidirect |
| 13. C16×C32 | C4 | semidirect |
| 96. C32 | #367 | semidirect |
| 114. C16 | #1426 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72216298 | 2 |
| #72216239 | 2 |
| C16×M64 | 2 |
| #72216197 | 2 |