Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
620 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 148 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 | C22×D3×D7 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C2×C3×C7×D4 | C22 |
| hypercenter | C2×D4 | D3×D7 |
| hypocenter | C3×C7 | #532 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#1159 | C2 | semidirect |
| 2. C2×#66677 | C2 | semidirect |
| 3. #10621423 | C2 | semidirect |
| 4. C7×#936 | C2 | semidirect |
| 5. #10621114 | C2 | semidirect |
| 6. #10621112 | C2 | semidirect |
| 7. C4×Dic3×D7 | C2 | semidirect |
| 8. #10620947 | C2 | semidirect |
| 9. #10620936 | C2 | semidirect |
| 31. C2×#1159 | D3 | semidirect |
| 55. C3×C4×D7 | D4 | semidirect |
| 85. #936 | D7 | semidirect |
| 96. Dic21 | #41 | semidirect |
| 120. C4×D7 | #67 | semidirect |
| 188. C3×C7 | #532 | semidirect |
| 246. C7 | #5832 | semidirect |
| 265. C3 | #90359 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#1159 | 2 |
| C2×#66677 | 2 |
| #10621423 | 2 |
| C7×#936 | 2 |
| #10621114 | 2 |
| #10621112 | 2 |
| C4×Dic3×D7 | 2 |
| #10620947 | 2 |
| #10620936 | 2 |
| #90359 | 3 |
| #5832 | 7 |