Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
316 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C7 | C22⋊C4 |
| commutator | C22 | C22×C3×C4×C7 |
| Frattini, cosocle | C23 | C23×C3×C7 |
| socle | C2×C3×C7 | C2×C22⋊C4 |
| prime radical | C2×C3×C7×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C7 ◁ C22×C3×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C7×#117 | C2 | semidirect |
| 3. C3×C7×#99 | C2 | semidirect |
| 4. C7×#410 | C3 | direct (N × H) |
| 7. C2×C3×C7×Q8 | C4 | semidirect |
| 8. C2×C3×C7×D4 | C4 | semidirect |
| 10. C22×C3×C4×C7 | C4 | semidirect |
| 16. C7×#117 | C2×C3 | semidirect |
| 17. C7×#99 | C2×C3 | semidirect |
| 18. C3×#410 | C7 | direct (N × H) |
| 34. C2×C7×Q8 | C3×C4 | semidirect |
| 35. C2×C7×D4 | C3×C4 | semidirect |
| 37. C22×C4×C7 | C3×C4 | semidirect |
| 43. C3×#117 | C2×C7 | semidirect |
| 44. C3×#99 | C2×C7 | semidirect |
| 55. #410 | C3×C7 | direct (N × H) |
| 71. C2×C3×Q8 | C4×C7 | semidirect |
| 72. C2×C3×D4 | C4×C7 | semidirect |
| 74. C22×C3×C4 | C4×C7 | semidirect |
| 84. #117 | C2×C3×C7 | semidirect |
| 85. #99 | C2×C3×C7 | semidirect |
| 109. C3×C7 | #410 | direct (N × H) |
| 112. C2×Q8 | C3×C4×C7 | semidirect |
| 113. C2×D4 | C3×C4×C7 | semidirect |
| 115. C22×C4 | C3×C4×C7 | semidirect |
| 147. C7 | C3×#410 | direct (N × H) |
| 162. C3 | C7×#410 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C7×#41 | 2 |
| C3×C7×#117 | 2 |
| C3×C7×#99 | 2 |
| C7×#410 | 3 |
| C3×#410 | 7 |