Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
504 conjugacy classes of subgroups; only the 284 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 284 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C7×#127 | C2 | semidirect |
| 2. C3×C7×#126 | C2 | semidirect |
| 3. C3×C7×#123 | C2 | semidirect |
| 4. C3×C7×#121 | C2 | semidirect |
| 5. C3×C7×#120 | C2 | semidirect |
| 6. C7×#560 | C3 | direct (N × H) |
| 10. C2×C3×C7×D4 | C22 | semidirect |
| 11. C22×C3×C4×C7 | C22 | semidirect |
| 12. C3×C7×C4⋊C4 | C22 | semidirect |
| 15. C3×C7×C22⋊C4 | C22 | semidirect |
| 16. C3×C42×C7 | C22 | semidirect |
| 17. C7×#127 | C2×C3 | semidirect |
| 18. C7×#126 | C2×C3 | semidirect |
| 19. C7×#123 | C2×C3 | semidirect |
| 20. C7×#121 | C2×C3 | semidirect |
| 21. C7×#120 | C2×C3 | semidirect |
| 22. C3×#560 | C7 | direct (N × H) |
| 36. C2×C7×D4 | C22×C3 | semidirect |
| 37. C22×C4×C7 | C22×C3 | semidirect |
| 38. C7×C4⋊C4 | C22×C3 | semidirect |
| 41. C7×C22⋊C4 | C22×C3 | semidirect |
| 42. C42×C7 | C22×C3 | semidirect |
| 43. C3×#127 | C2×C7 | semidirect |
| 44. C3×#126 | C2×C7 | semidirect |
| 45. C3×#123 | C2×C7 | semidirect |
| 46. C3×#121 | C2×C7 | semidirect |
| 47. C3×#120 | C2×C7 | semidirect |
| 56. #560 | C3×C7 | direct (N × H) |
| 70. C2×C3×D4 | C22×C7 | semidirect |
| 71. C22×C3×C4 | C22×C7 | semidirect |
| 72. C3×C4⋊C4 | C22×C7 | semidirect |
| 75. C3×C22⋊C4 | C22×C7 | semidirect |
| 76. C3×C42 | C22×C7 | semidirect |
| 81. #127 | C2×C3×C7 | semidirect |
| 82. #126 | C2×C3×C7 | semidirect |
| 83. #123 | C2×C3×C7 | semidirect |
| 84. #121 | C2×C3×C7 | semidirect |
| 85. #120 | C2×C3×C7 | semidirect |
| 104. C3×C7 | #560 | direct (N × H) |
| 108. C2×D4 | C22×C3×C7 | semidirect |
| 109. C22×C4 | C22×C3×C7 | semidirect |
| 110. C4⋊C4 | C22×C3×C7 | semidirect |
| 113. C22⋊C4 | C22×C3×C7 | semidirect |
| 114. C42 | C22×C3×C7 | semidirect |
| 137. C7 | C3×#560 | direct (N × H) |
| 150. C3 | C7×#560 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C7×#127 | 2 |
| C3×C7×#126 | 2 |
| C3×C7×#123 | 2 |
| C3×C7×#121 | 2 |
| C3×C7×#120 | 2 |
| C7×#560 | 3 |
| C3×#560 | 7 |