Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
324 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×C4×C7×C8 | C2 | semidirect |
| 4. C7×C8×Dic3 | C2 | semidirect |
| 5. C4×C7×C3⋊C8 | C2 | semidirect |
| 14. D3×C7×C8 | C4 | direct (N × H) |
| 19. C4×C7×C8 | D3 | direct (N × H) |
| 20. C4×D3×C8 | C7 | direct (N × H) |
| 26. C4×D3×C7 | C8 | direct (N × H) |
| 28. C3×C7×C8 | C2×C4 | semidirect |
| 29. C7×C3⋊C8 | C2×C4 | semidirect |
| 40. C3×C4×C8 | C2×C7 | semidirect |
| 41. C8×Dic3 | C2×C7 | semidirect |
| 42. C4×C3⋊C8 | C2×C7 | semidirect |
| 46. C3×C4×C7 | C2×C8 | semidirect |
| 49. C7×Dic3 | C2×C8 | semidirect |
| 59. C7×C8 | C4×D3 | direct (N × H) |
| 67. D3×C8 | C4×C7 | direct (N × H) |
| 70. D3×C7 | C4×C8 | direct (N × H) |
| 75. C4×C8 | D3×C7 | direct (N × H) |
| 81. C4×C7 | D3×C8 | direct (N × H) |
| 88. C4×D3 | C7×C8 | direct (N × H) |
| 90. C3×C8 | C2×C4×C7 | semidirect |
| 91. C3⋊C8 | C2×C4×C7 | semidirect |
| 92. C3×C7 | C2×C4×C8 | semidirect |
| 107. C3×C4 | C2×C7×C8 | semidirect |
| 110. Dic3 | C2×C7×C8 | semidirect |
| 119. C8 | C4×D3×C7 | direct (N × H) |
| 120. C7 | C4×D3×C8 | direct (N × H) |
| 123. D3 | C4×C7×C8 | direct (N × H) |
| 129. C4 | D3×C7×C8 | direct (N × H) |
| 131. C3 | C2×C4×C7×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×C7×C8 | 2 |
| C42×D3×C7 | 2 |
| C3×C4×C7×C8 | 2 |
| C7×C8×Dic3 | 2 |
| C4×C7×C3⋊C8 | 2 |
| C2×C4×C7×C8 | 3 |
| C4×D3×C8 | 7 |