Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
584 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 152 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 |
| hypercenter | C42 | D3×D7 |
| hypocenter | C3×C7 | #547 |
| 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×#66670 | C2 | semidirect |
| 2. C4×D84 | C2 | semidirect |
| 3. C7×#869 | C2 | semidirect |
| 4. C3×C4×D28 | C2 | semidirect |
| 5. #10621012 | C2 | semidirect |
| 6. #10620975 | C2 | semidirect |
| 7. D7×C4⋊Dic3 | C2 | semidirect |
| 8. #10620945 | C2 | semidirect |
| 34. C4×D28 | D3 | semidirect |
| 47. C3×D28 | D4 | semidirect |
| 75. #869 | D7 | semidirect |
| 85. C3×C4×C7 | #41 | semidirect |
| 101. D28 | D12 | semidirect |
| 145. C4×C7 | #237 | semidirect |
| 163. C3×C7 | #547 | semidirect |
| 196. C3×C4 | #1161 | semidirect |
| 214. C7 | #5555 | semidirect |
| 231. C4 | #66670 | semidirect |
| 233. C3 | #90417 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66670 | 2 |
| C4×D84 | 2 |
| C7×#869 | 2 |
| C3×C4×D28 | 2 |
| #10621012 | 2 |
| #10620975 | 2 |
| D7×C4⋊Dic3 | 2 |
| #10620945 | 2 |
| #90417 | 3 |
| #5555 | 7 |