Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
996 conjugacy classes of subgroups; only the 552 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 552 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | D3×D7 |
| commutator, hypocenter | C3×C7 | C22×C42 |
| Frattini | C22 | C22×D3×D7 |
| Fitting, socle, prime radical | C24×C3×C7 | C22 |
| cosocle | C22×C3×C7 | C24 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×Dic3×Dic7 | C2 | direct (N × H) |
| 7. C22×Dic21 | C4 | semidirect |
| 9. C22×C7×Dic3 | C4 | semidirect |
| 11. C22×C3×Dic7 | C4 | semidirect |
| 13. Dic3×Dic7 | C22 | direct (N × H) |
| 23. C2×Dic21 | C2×C4 | semidirect |
| 25. C2×C7×Dic3 | C2×C4 | semidirect |
| 27. C2×C3×Dic7 | C2×C4 | semidirect |
| 34. C22×Dic7 | Dic3 | direct (N × H) |
| 41. C22×C3×C7 | C42 | semidirect |
| 46. Dic21 | C22×C4 | semidirect |
| 47. C3×Dic7 | C22×C4 | semidirect |
| 48. C7×Dic3 | C22×C4 | semidirect |
| 58. C2×Dic7 | C2×Dic3 | direct (N × H) |
| 65. C22×Dic3 | Dic7 | direct (N × H) |
| 68. C2×C3×C7 | C2×C42 | semidirect |
| 76. C22×C7 | C4×Dic3 | semidirect |
| 83. Dic7 | C22×Dic3 | direct (N × H) |
| 92. C2×Dic3 | C2×Dic7 | direct (N × H) |
| 94. C3×C7 | C22×C42 | semidirect |
| 103. C2×C7 | C2×C4×Dic3 | semidirect |
| 107. C22×C3 | C4×Dic7 | semidirect |
| 114. Dic3 | C22×Dic7 | direct (N × H) |
| 125. C7 | C22×C4×Dic3 | semidirect |
| 126. C2×C3 | C2×C4×Dic7 | semidirect |
| 130. C22 | Dic3×Dic7 | direct (N × H) |
| 138. C3 | C22×C4×Dic7 | semidirect |
| 139. C2 | C2×Dic3×Dic7 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Dic21 | 2 |
| C23×C7×Dic3 | 2 |
| C23×C3×Dic7 | 2 |
| C2×Dic3×Dic7 | 2 |
| C22×C4×Dic7 | 3 |
| C22×C4×Dic3 | 7 |