Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
308 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D21 |
| commutator | C22×C3×C7 | C22×C4 |
| Fitting | C3×C7×#95 | C2 |
| socle, prime radical, cosocle | C23×C3×C7 | C23 |
| hypercenter | #95 | D21 |
| hypocenter | C3×C7 | #382 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C23 ◁ #95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 13. C4×Dic21 | C4 | semidirect |
| 112. C3×C7 | #382 | semidirect |
| 149. C7 | #4650 | semidirect |
| 165. C3 | #89422 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66629 | 2 |
| C2×C4×Dic21 | 2 |
| C3×C7×#95 | 2 |
| #10620670 | 2 |
| #89422 | 3 |
| #4650 | 7 |