Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
692 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 692 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C2×C3×C7×C22⋊C4 | C2 |
| hypercenter | C2×C22⋊C4 | D21 |
| hypocenter | C3×C7 | #534 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#4268 | C2 | semidirect |
| 2. C2×C3×C7×C22⋊C4 | C2 | semidirect |
| 3. #10621413 | C2 | semidirect |
| 4. #10621369 | C2 | semidirect |
| 5. #10621365 | C2 | semidirect |
| 8. C23×D21 | C22 | semidirect |
| 9. C22×C3×C4×C7 | C22 | semidirect |
| 12. C2×#4268 | C22 | semidirect |
| 13. C22×Dic21 | C22 | semidirect |
| 14. C2×D84 | C22 | semidirect |
| 15. C3×C7×C22⋊C4 | C22 | semidirect |
| 16. #66631 | C22 | semidirect |
| 17. C4⋊Dic21 | C22 | semidirect |
| 19. C2×C7×C22⋊C4 | D3 | semidirect |
| 25. C23×C3×C7 | D4 | semidirect |
| 40. C22×C4×C7 | C2×D3 | semidirect |
| 46. C7×C22⋊C4 | C2×D3 | semidirect |
| 50. C2×C3×C22⋊C4 | D7 | semidirect |
| 67. C23×C7 | D12 | semidirect |
| 82. C22×C3×C4 | C2×D7 | semidirect |
| 88. C3×C22⋊C4 | C2×D7 | semidirect |
| 100. C2×C22⋊C4 | D21 | semidirect |
| 113. C23×C3 | D28 | semidirect |
| 126. C3×C7 | #534 | semidirect |
| 134. C22×C4 | C2×D21 | semidirect |
| 137. C22⋊C4 | C2×D21 | semidirect |
| 158. C23 | D84 | semidirect |
| 168. C7 | #5493 | semidirect |
| 185. C3 | #90184 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#4268 | 2 |
| C2×C3×C7×C22⋊C4 | 2 |
| #10621413 | 2 |
| #10621369 | 2 |
| #10621365 | 2 |
| #90184 | 3 |
| #5493 | 7 |