Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
396 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 124 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 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C3×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ C22×C3×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×C7×D4 | C2 | semidirect |
| 2. C2×C3×C7×D8 | C2 | semidirect |
| 3. C3×C7×#121 | C2 | semidirect |
| 4. C3×C7×#102 | C2 | semidirect |
| 5. C3×C7×C22⋊C8 | C2 | semidirect |
| 6. C7×#447 | C3 | direct (N × H) |
| 12. C2×C3×C7×D4 | C22 | semidirect |
| 15. C2×C3×C7×C8 | C22 | semidirect |
| 16. C3×C7×C4⋊C4 | C22 | semidirect |
| 18. C22×C7×D4 | C2×C3 | semidirect |
| 19. C2×C7×D8 | C2×C3 | semidirect |
| 20. C7×#121 | C2×C3 | semidirect |
| 21. C7×#102 | C2×C3 | semidirect |
| 22. C7×C22⋊C8 | C2×C3 | semidirect |
| 23. C3×#447 | C7 | direct (N × H) |
| 34. C3×C7×D4 | D4 | semidirect |
| 45. C2×C7×D4 | C22×C3 | semidirect |
| 48. C2×C7×C8 | C22×C3 | semidirect |
| 49. C7×C4⋊C4 | C22×C3 | semidirect |
| 51. C22×C3×D4 | C2×C7 | semidirect |
| 52. C2×C3×D8 | C2×C7 | semidirect |
| 53. C3×#121 | C2×C7 | semidirect |
| 54. C3×#102 | C2×C7 | semidirect |
| 55. C3×C22⋊C8 | C2×C7 | semidirect |
| 65. C22×C3×C7 | D8 | semidirect |
| 70. #447 | C3×C7 | direct (N × H) |
| 81. C7×D4 | C3×D4 | semidirect |
| 92. C2×C3×D4 | C22×C7 | semidirect |
| 95. C2×C3×C8 | C22×C7 | semidirect |
| 96. C3×C4⋊C4 | C22×C7 | semidirect |
| 104. C22×D4 | C2×C3×C7 | semidirect |
| 105. C2×D8 | C2×C3×C7 | semidirect |
| 106. #121 | C2×C3×C7 | semidirect |
| 107. #102 | C2×C3×C7 | semidirect |
| 108. C22⋊C8 | C2×C3×C7 | semidirect |
| 118. C22×C7 | C3×D8 | semidirect |
| 133. C3×D4 | C7×D4 | semidirect |
| 139. C3×C7 | #447 | direct (N × H) |
| 145. C2×D4 | C22×C3×C7 | semidirect |
| 148. C2×C8 | C22×C3×C7 | semidirect |
| 149. C4⋊C4 | C22×C3×C7 | semidirect |
| 166. C22×C3 | C7×D8 | semidirect |
| 181. D4 | C3×C7×D4 | semidirect |
| 187. C7 | C3×#447 | direct (N × H) |
| 203. C22 | C3×C7×D8 | semidirect |
| 208. C3 | C7×#447 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C7×D4 | 2 |
| C2×C3×C7×D8 | 2 |
| C3×C7×#121 | 2 |
| C3×C7×#102 | 2 |
| C3×C7×C22⋊C8 | 2 |
| C7×#447 | 3 |
| C3×#447 | 7 |