Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
384 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C3×C42×C7 | C22 |
| prime radical | C22×D21 | C23 |
| hypercenter | C42 | D3×D7 |
| hypocenter | C3×C7 | #541 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#869 | C2 | semidirect |
| 3. C3×#6433 | C2 | semidirect |
| 5. #10620906 | C2 | semidirect |
| 7. #10620897 | C2 | semidirect |
| 28. #6433 | D3 | semidirect |
| 58. #869 | D7 | semidirect |
| 119. C3×C7 | #541 | semidirect |
| 154. C7 | #5698 | semidirect |
| 167. C3 | #90422 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10621355 | 2 |
| C7×#869 | 2 |
| C3×#6433 | 2 |
| #10620945 | 2 |
| #10620906 | 2 |
| #10620899 | 2 |
| #10620897 | 2 |
| #90422 | 3 |
| #5698 | 7 |