Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
561 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 132 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 | #140751 |
| commutator | C4×C16 | C22×C8 |
| Frattini, cosocle | C2×#1311 | C23 |
| prime radical | #513082 | C4 |
| LCS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×#7053 ◁ 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. #72225222 | C2 | semidirect |
| 2. #72225212 | C2 | semidirect |
| 3. #72225169 | C2 | semidirect |
| 4. #72225141 | C2 | semidirect |
| 207. C32 | #350 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72225222 | 2 |
| #72225212 | 2 |
| #72225169 | 2 |
| #72225141 | 2 |
| #72216311 | 2 |