Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
372 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 99 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, socle, cosocle | C23×C3×C7 | C23 |
| Fitting | C3×C7×#95 | C2 |
| prime radical | C23×D21 | C22 |
| hypercenter | #95 | D21 |
| hypocenter | C3×C7 | #396 |
| LCS | G ▷ C23×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C23 ◁ #95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#66629 | C2 | semidirect |
| 3. C3×C7×#95 | C2 | semidirect |
| 12. C7×#95 | D3 | semidirect |
| 31. C3×#95 | D7 | semidirect |
| 62. #95 | D21 | semidirect |
| 81. C3×C7 | #396 | semidirect |
| 109. C7 | #4676 | semidirect |
| 122. C3 | #89448 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66631 | 2 |
| C2×#66629 | 2 |
| C3×C7×#95 | 2 |
| #89448 | 3 |
| #4676 | 7 |