Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
1780 conjugacy classes of subgroups; only the 568 normal subgroups are shown.
This group has 1780 conjugacy classes of subgroups — too many to draw. Only its 568 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×D3×D7 |
| commutator, cosocle | C2×C3×C7 | C25 |
| Frattini | C2 | C23×D3×D7 |
| Fitting | C24×C3×C7 | C22 |
| socle | C23×C3×C7 | C23 |
| hypercenter | C24 | D3×D7 |
| hypocenter | C3×C7 | C22×#41 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C23 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#4268 | C2 | semidirect |
| 2. C22×C7×#67 | C2 | semidirect |
| 3. C23×C3×Dic7 | C2 | semidirect |
| 4. C22×#4248 | C2 | semidirect |
| 5. C22×#4247 | C2 | semidirect |
| 6. C22×D21⋊C4 | C2 | semidirect |
| 7. C2×#66686 | C2 | direct (N × H) |
| 8. C22×D3×Dic7 | C2 | semidirect |
| 13. C2×#4268 | C22 | semidirect |
| 15. C2×C7×#67 | C22 | semidirect |
| 19. C22×C3×Dic7 | C22 | semidirect |
| 20. C2×#4248 | C22 | semidirect |
| 21. C2×#4247 | C22 | semidirect |
| 22. C2×D21⋊C4 | C22 | semidirect |
| 23. #66686 | C22 | direct (N × H) |
| 24. C2×D3×Dic7 | C22 | semidirect |
| 29. C23×Dic7 | D3 | semidirect |
| 41. #4268 | C23 | semidirect |
| 43. C7×#67 | C23 | semidirect |
| 47. C2×C3×Dic7 | C23 | semidirect |
| 48. #4248 | C23 | semidirect |
| 49. #4247 | C23 | semidirect |
| 50. D21⋊C4 | C23 | semidirect |
| 51. D3×Dic7 | C23 | semidirect |
| 63. C22×Dic7 | C2×D3 | semidirect |
| 69. C22×#67 | D7 | semidirect |
| 76. C22×C3×C7 | #41 | semidirect |
| 103. C2×Dic7 | C22×D3 | semidirect |
| 113. C2×#67 | C2×D7 | semidirect |
| 122. C2×C3×C7 | C2×#41 | semidirect |
| 137. C22×C7 | #235 | semidirect |
| 157. #67 | C22×D7 | semidirect |
| 164. C3×C7 | C22×#41 | semidirect |
| 182. C2×C7 | C2×#235 | semidirect |
| 186. C22×C3 | #1159 | semidirect |
| 215. C7 | C22×#235 | semidirect |
| 217. C2×C3 | C2×#1159 | semidirect |
| 226. C22 | #66686 | direct (N × H) |
| 233. C3 | C22×#1159 | semidirect |
| 237. C2 | C2×#66686 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#4268 | 2 |
| C22×C7×#67 | 2 |
| C23×C3×Dic7 | 2 |
| C22×#4248 | 2 |
| C22×#4247 | 2 |
| C22×D21⋊C4 | 2 |
| C2×#66686 | 2 |
| C22×D3×Dic7 | 2 |
| C22×#1159 | 3 |
| C22×#235 | 7 |