Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
540 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 144 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 | C22×D3×D7 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C42×C7 | C22 |
| prime radical | C2×#66669 | C2 |
| hypercenter | C42 | D3×D7 |
| hypocenter | C3×C7 | #536 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D3×Dic14 | C2 | semidirect |
| 3. #10621352 | C2 | semidirect |
| 4. C7×#872 | C2 | semidirect |
| 5. C3×#6438 | C2 | semidirect |
| 6. #10621019 | C2 | semidirect |
| 7. #10620984 | C2 | semidirect |
| 8. #10620951 | C2 | semidirect |
| 9. #10620926 | C2 | semidirect |
| 36. #6438 | D3 | semidirect |
| 84. #872 | D7 | semidirect |
| 179. C3×C7 | #536 | semidirect |
| 233. C7 | #5675 | semidirect |
| 250. C3 | #90217 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×Dic14 | 2 |
| C2×#66669 | 2 |
| #10621352 | 2 |
| C7×#872 | 2 |
| C3×#6438 | 2 |
| #10621019 | 2 |
| #10620984 | 2 |
| #10620951 | 2 |
| #10620926 | 2 |
| #90217 | 3 |
| #5675 | 7 |