Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
424 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 187 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, prime radical, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C4×C7×Q8 | C2 |
| hypercenter | C4×Q8 | D21 |
| hypocenter | C3×C7 | Q82 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 22. C3×C7×Q8 | Q8 | semidirect |
| 29. Dic42 | Q8 | direct (N × H) |
| 51. C7×Q8 | Dic6 | semidirect |
| 82. C3×Q8 | Dic14 | semidirect |
| 90. C3×C7 | Q82 | semidirect |
| 111. Q8 | Dic42 | direct (N × H) |
| 117. C7 | Q8×Dic6 | semidirect |
| 127. C3 | Q8×Dic14 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×C7×Q8 | 2 |
| Q8×Dic21 | 2 |
| #10621371 | 2 |
| #10621352 | 2 |
| C4×Dic42 | 2 |
| Q8×Dic14 | 3 |
| Q8×Dic6 | 7 |