Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
400 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 179 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 | #557 |
| 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 |
| 50. C7×Q8 | Dic6 | semidirect |
| 80. C3×Q8 | Dic14 | semidirect |
| 87. C3×C7 | #557 | semidirect |
| 107. Q8 | Dic42 | semidirect |
| 113. C7 | #5570 | semidirect |
| 123. C3 | #90261 | 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 |
| #10621373 | 2 |
| #10621352 | 2 |
| C4×Dic42 | 2 |
| #90261 | 3 |
| #5570 | 7 |