Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
668 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 171 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 | C23×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C7×#117 | C2 |
| hypercenter | #117 | D21 |
| hypocenter | C3×C7 | #535 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#66728 | C2 | semidirect |
| 2. C22×D84 | C2 | semidirect |
| 3. C3×C7×#117 | C2 | semidirect |
| 4. #10621378 | C2 | semidirect |
| 5. #10621365 | C2 | semidirect |
| 6. #10621358 | C2 | semidirect |
| 7. #10621357 | C2 | semidirect |
| 16. C2×D84 | C22 | semidirect |
| 17. C2×D84 | C22 | semidirect |
| 18. C2×C4×D21 | C22 | semidirect |
| 19. C2×Dic42 | C22 | semidirect |
| 20. C3×C7×C4⋊C4 | C22 | semidirect |
| 22. C3×C42×C7 | C22 | semidirect |
| 23. #66631 | C22 | semidirect |
| 26. C7×#117 | D3 | semidirect |
| 37. C2×C3×C4×C7 | D4 | semidirect |
| 60. C7×C4⋊C4 | C2×D3 | semidirect |
| 62. C42×C7 | C2×D3 | semidirect |
| 66. C3×#117 | D7 | semidirect |
| 89. C2×C4×C7 | D12 | semidirect |
| 112. C3×C4⋊C4 | C2×D7 | semidirect |
| 114. C3×C42 | C2×D7 | semidirect |
| 127. #117 | D21 | semidirect |
| 145. C2×C3×C4 | D28 | semidirect |
| 157. C3×C7 | #535 | semidirect |
| 169. C4⋊C4 | C2×D21 | semidirect |
| 172. C42 | C2×D21 | semidirect |
| 203. C2×C4 | D84 | semidirect |
| 207. C7 | #5525 | semidirect |
| 225. C3 | #90216 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66728 | 2 |
| C22×D84 | 2 |
| C3×C7×#117 | 2 |
| #10621378 | 2 |
| #10621365 | 2 |
| #10621358 | 2 |
| #10621357 | 2 |
| #90216 | 3 |
| #5525 | 7 |