Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
516 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | D3×D7 |
| commutator, hypocenter | C3×C7 | C43 |
| Frattini | C23 | C2×D3×D7 |
| Fitting | C22×C3×C4×C7 | C22 |
| socle, prime radical, cosocle | C23×C3×C7 | C23 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 10. C4×Dic21 | C4 | semidirect |
| 11. C4×C7×Dic3 | C4 | semidirect |
| 12. C3×C4×Dic7 | C4 | semidirect |
| 13. Dic3×Dic7 | C4 | direct (N × H) |
| 36. C4×Dic7 | Dic3 | direct (N × H) |
| 45. C3×C4×C7 | C42 | semidirect |
| 46. Dic21 | C42 | semidirect |
| 47. C3×Dic7 | C42 | semidirect |
| 48. C7×Dic3 | C42 | semidirect |
| 67. C4×Dic3 | Dic7 | direct (N × H) |
| 81. C4×C7 | C4×Dic3 | semidirect |
| 83. Dic7 | C4×Dic3 | direct (N × H) |
| 94. C3×C7 | C43 | semidirect |
| 112. C3×C4 | C4×Dic7 | semidirect |
| 114. Dic3 | C4×Dic7 | direct (N × H) |
| 125. C7 | C42×Dic3 | semidirect |
| 137. C4 | Dic3×Dic7 | direct (N × H) |
| 138. C3 | C42×Dic7 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×Dic21 | 2 |
| C2×C4×C7×Dic3 | 2 |
| C2×C3×C4×Dic7 | 2 |
| C2×Dic3×Dic7 | 2 |
| C42×Dic7 | 3 |
| C42×Dic3 | 7 |