Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
250 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C7 | S4 |
| commutator, hypocenter | A4 | C42×C7 |
| Frattini | C22 | C2×C7×S4 |
| Fitting | C23×C4×C7 | D3 |
| socle | C24×C7 | C2×D3 |
| prime radical | C22×C7×A4 | C22 |
| cosocle | C22×A4 | C22×C7 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C4×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C4×C7×A4 | C4 | semidirect |
| 6. C7×SL(2,ℤ/4) | C4 | direct (N × H) |
| 13. C4×SL(2,ℤ/4) | C7 | direct (N × H) |
| 24. C22×C4×C7 | Dic3 | semidirect |
| 33. C7×A4 | C42 | semidirect |
| 55. C4×A4 | C4×C7 | semidirect |
| 56. SL(2,ℤ/4) | C4×C7 | direct (N × H) |
| 72. C22×C7 | C4×Dic3 | semidirect |
| 77. C4×C7 | SL(2,ℤ/4) | direct (N × H) |
| 89. C22×C4 | C7×Dic3 | semidirect |
| 101. A4 | C42×C7 | semidirect |
| 121. C7 | C4×SL(2,ℤ/4) | direct (N × H) |
| 131. C22 | C4×C7×Dic3 | semidirect |
| 136. C4 | C7×SL(2,ℤ/4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C7×A4 | 2 |
| C2×C7×SL(2,ℤ/4) | 2 |
| C4×C7×C22⋊C4 | 3 |
| C4×C7×Dic3 | 4 |
| C4×SL(2,ℤ/4) | 7 |