Finite Group DatabasebetaFinite Group DatabasebetaOrder 1056.
188 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×D11 |
| commutator, socle, cosocle | C22×C3×C11 | C23 |
| Fitting | C2×C3×C4×C11 | C22 |
| hypercenter | C2×C4 | D3×D11 |
| hypocenter | C3×C11 | #121 |
| LCS | G ▷ C22×C3×C11 ▷ C3×C11 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C11×D12 | C2 | semidirect |
| 2. C2×#63140 | C2 | semidirect |
| 3. C2×#63140 | C2 | semidirect |
| 4. C2×D3×Dic11 | C2 | semidirect |
| 5. #10587201 | C2 | semidirect |
| 6. C3×(C2×D11)⋊C4 | C2 | semidirect |
| 7. (C2×D3)⋊Dic11 | C2 | semidirect |
| 25. (C2×D11)⋊C4 | D3 | semidirect |
| 33. C2×D3×C11 | D4 | semidirect |
| 39. Dic33 | D4 | semidirect |
| 61. C2×D12 | D11 | semidirect |
| 80. C3×C11 | #121 | semidirect |
| 115. C2×D3 | C22⋊D11 | semidirect |
| 120. C11 | #884 | semidirect |
| 152. C3 | #66992 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C11×D12 | 2 |
| C2×#63140 | 2 |
| C2×D3×Dic11 | 2 |
| #10587201 | 2 |
| C3×(C2×D11)⋊C4 | 2 |
| (C2×D3)⋊Dic11 | 2 |
| #66992 | 3 |
| #884 | 11 |