Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
178 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 54 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 | C2×C7×C8 |
| Frattini | C4 | C2×C7×S4 |
| Fitting | C23×C4×C7 | D3 |
| socle | C24×C7 | Dic3 |
| prime radical | C22×C7×A4 | C4 |
| cosocle | C4×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 |
|---|---|---|
| 2. C7×A4⋊C8 | C2 | direct (N × H) |
| 13. C2×A4⋊C8 | C7 | direct (N × H) |
| 14. C2×C7×A4 | C8 | semidirect |
| 32. A4⋊C8 | C2×C7 | direct (N × H) |
| 34. C7×A4 | C2×C8 | semidirect |
| 41. C23×C7 | C3⋊C8 | semidirect |
| 72. C22×C7 | C2×C3⋊C8 | semidirect |
| 78. C2×A4 | C7×C8 | semidirect |
| 99. C2×C7 | A4⋊C8 | direct (N × H) |
| 102. A4 | C2×C7×C8 | semidirect |
| 108. C23 | C7×C3⋊C8 | semidirect |
| 121. C7 | C2×A4⋊C8 | direct (N × H) |
| 131. C22 | C2×C7×C3⋊C8 | semidirect |
| 141. C2 | C7×A4⋊C8 | 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 |
| C7×A4⋊C8 | 2 |
| C2×C7×C22⋊C8 | 3 |
| C2×C7×C3⋊C8 | 4 |
| C2×A4⋊C8 | 7 |