Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
192 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C7 | C23 |
| commutator | C22 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| socle | C22×C3×C7 | C22×C4 |
| prime radical | C23×C3×C7 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C7×C4⋊C8 | C2 | semidirect |
| 4. C3×C7×#97 | C2 | semidirect |
| 5. C3×C4×C7×C8 | C2 | semidirect |
| 6. C7×#433 | C3 | direct (N × H) |
| 15. C7×C4⋊C8 | C2×C3 | semidirect |
| 17. C7×#97 | C2×C3 | semidirect |
| 18. C4×C7×C8 | C2×C3 | semidirect |
| 19. C3×#433 | C7 | direct (N × H) |
| 36. C3×C4⋊C8 | C2×C7 | semidirect |
| 38. C3×#97 | C2×C7 | semidirect |
| 39. C3×C4×C8 | C2×C7 | semidirect |
| 47. #433 | C3×C7 | direct (N × H) |
| 67. C4⋊C8 | C2×C3×C7 | semidirect |
| 69. #97 | C2×C3×C7 | semidirect |
| 70. C4×C8 | C2×C3×C7 | semidirect |
| 86. C3×C7 | #433 | direct (N × H) |
| 112. C7 | C3×#433 | direct (N × H) |
| 123. C3 | C7×#433 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C7×#117 | 2 |
| C3×C7×C4⋊C8 | 2 |
| C3×C7×C22⋊C8 | 2 |
| C3×C7×#97 | 2 |
| C3×C4×C7×C8 | 2 |
| C7×#433 | 3 |
| C3×#433 | 7 |