Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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×#126 | C2 |
| prime radical | C23×D21 | C22 |
| hypercenter | #126 | D21 |
| hypocenter | C3×C7 | C2×#126 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C7×#126 | C2 | semidirect |
| 2. #10621381 | C2 | semidirect |
| 3. C4⋊C4×D21 | C2 | semidirect |
| 5. #10621362 | C2 | semidirect |
| 6. #10621359 | C2 | semidirect |
| 7. C42×D21 | C2 | semidirect |
| 12. C3×C7×C4⋊C4 | C22 | semidirect |
| 14. C3×C42×C7 | C22 | semidirect |
| 18. C4⋊Dic21 | C22 | semidirect |
| 19. #66629 | C22 | semidirect |
| 20. #66629 | C22 | semidirect |
| 21. C4×Dic21 | C22 | semidirect |
| 22. C7×#126 | D3 | semidirect |
| 43. C7×C4⋊C4 | C2×D3 | semidirect |
| 45. C42×C7 | C2×D3 | semidirect |
| 53. C3×#126 | D7 | semidirect |
| 83. C3×C4⋊C4 | C2×D7 | semidirect |
| 85. C3×C42 | C2×D7 | semidirect |
| 96. D21 | #126 | direct (N × H) |
| 100. #126 | D21 | direct (N × H) |
| 123. C3×C7 | C2×#126 | semidirect |
| 130. C4⋊C4 | C2×D21 | semidirect |
| 136. C42 | C2×D21 | semidirect |
| 160. C7 | D3×#126 | semidirect |
| 174. C3 | D7×#126 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C7×#126 | 2 |
| #10621381 | 2 |
| C4⋊C4×D21 | 2 |
| C22⋊C4×D21 | 2 |
| #10621362 | 2 |
| #10621359 | 2 |
| C42×D21 | 2 |
| D7×#126 | 3 |
| D3×#126 | 7 |