Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
540 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 540 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 | C3×C7×#117 | C2 |
| prime radical | C2×#66728 | C2 |
| hypercenter | #117 | D21 |
| hypocenter | C3×C7 | #536 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×Dic42 | C2 | semidirect |
| 3. C3×C7×#117 | C2 | semidirect |
| 4. #10621380 | C2 | semidirect |
| 5. #10621369 | C2 | semidirect |
| 6. #10621358 | C2 | semidirect |
| 7. #10621352 | C2 | semidirect |
| 26. C7×#117 | D3 | semidirect |
| 64. C3×#117 | D7 | semidirect |
| 120. #117 | D21 | semidirect |
| 146. C3×C7 | #536 | semidirect |
| 191. C7 | #5526 | semidirect |
| 206. C3 | #90217 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66728 | 2 |
| C22×Dic42 | 2 |
| C3×C7×#117 | 2 |
| #10621380 | 2 |
| #10621369 | 2 |
| #10621358 | 2 |
| #10621352 | 2 |
| #90217 | 3 |
| #5526 | 7 |