Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
158 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C8 | S4 |
| commutator, hypocenter | A4 | C2×C7×C8 |
| Frattini | C4 | C2×C7×S4 |
| Fitting | C22×C7×C8 | D3 |
| socle | C23×C7 | C4×D3 |
| prime radical | C2×C7×S4 | C4 |
| cosocle | C4×A4 | C22×C7 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C7×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×C8×A4 | C2 | semidirect |
| 3. C7×A4⋊C8 | C2 | semidirect |
| 10. C22×C7×C8 | D3 | semidirect |
| 16. C8×S4 | C7 | direct (N × H) |
| 18. C7×S4 | C8 | direct (N × H) |
| 36. C8×A4 | C2×C7 | semidirect |
| 37. A4⋊C8 | C2×C7 | semidirect |
| 39. C7×A4 | C2×C8 | semidirect |
| 57. C7×C8 | S4 | direct (N × H) |
| 65. C22×C8 | D3×C7 | semidirect |
| 76. C22×C7 | D3×C8 | semidirect |
| 83. S4 | C7×C8 | direct (N × H) |
| 106. A4 | C2×C7×C8 | semidirect |
| 123. C8 | C7×S4 | direct (N × H) |
| 124. C7 | C8×S4 | direct (N × H) |
| 132. C22 | D3×C7×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C7×S4 | 2 |
| C7×C8×A4 | 2 |
| C7×A4⋊C8 | 2 |
| C7×C8×D4 | 3 |
| D3×C7×C8 | 4 |
| C8×S4 | 7 |