Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
364 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 112 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 | C2×D3×D7 |
| commutator, socle, prime radical, cosocle | C23×C3×C7 | C23 |
| Fitting | C22×C3×C4×C7 | C22 |
| hypercenter | C22×C4 | D3×D7 |
| hypocenter | C3×C7 | #395 |
| LCS | G ▷ C23×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 33. C2×C7×Dic3 | Q8 | semidirect |
| 36. C2×C3×Dic7 | Q8 | semidirect |
| 75. C2×Dic7 | Dic6 | semidirect |
| 119. C2×Dic3 | Dic14 | semidirect |
| 121. C3×C7 | #395 | semidirect |
| 161. C7 | #4656 | semidirect |
| 177. C3 | #89751 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Dic42 | 2 |
| C2×C7×Dic3⋊C4 | 2 |
| C2×Dic7⋊Dic3 | 2 |
| C3×C7⋊#95 | 2 |
| #10620529 | 2 |
| #89751 | 3 |
| #4656 | 7 |