Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
504 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 120 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 | #560 |
| 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. #10621357 | C2 | semidirect |
| 2. C7×#876 | C2 | semidirect |
| 3. C3×#6439 | C2 | semidirect |
| 4. #10621038 | C2 | semidirect |
| 5. #10621021 | C2 | semidirect |
| 6. #10620995 | C2 | semidirect |
| 7. #10620982 | C2 | semidirect |
| 11. C2×D84 | C22 | semidirect |
| 14. C2×D21⋊C4 | C22 | semidirect |
| 15. C3×C42×C7 | C22 | semidirect |
| 17. C7×Dic3⋊C4 | C22 | semidirect |
| 19. C3×Dic7⋊C4 | C22 | semidirect |
| 21. #66575 | C22 | semidirect |
| 27. #6439 | D3 | semidirect |
| 51. C42×C7 | C2×D3 | semidirect |
| 55. Dic7⋊C4 | C2×D3 | semidirect |
| 61. #876 | D7 | semidirect |
| 96. C3×C42 | C2×D7 | semidirect |
| 100. Dic3⋊C4 | C2×D7 | semidirect |
| 137. C3×C7 | #560 | semidirect |
| 150. C42 | D3×D7 | semidirect |
| 180. C7 | #5711 | semidirect |
| 197. C3 | #90402 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10621357 | 2 |
| C7×#876 | 2 |
| C3×#6439 | 2 |
| #10621038 | 2 |
| #10621021 | 2 |
| #10620995 | 2 |
| #10620982 | 2 |
| #90402 | 3 |
| #5711 | 7 |