Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
456 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 155 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, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C4×C7×Q8 | C2 |
| prime radical | C2×D84 | C22 |
| hypercenter | C4×Q8 | D21 |
| hypocenter | C3×C7 | #554 |
| 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 |
|---|---|---|
| 1. C3×C4×C7×Q8 | C2 | semidirect |
| 2. #10621429 | C2 | semidirect |
| 3. #10621379 | C2 | semidirect |
| 4. C4⋊C4×D21 | C2 | semidirect |
| 7. #10621353 | C2 | semidirect |
| 8. #10621352 | C2 | semidirect |
| 26. C4×C7×Q8 | D3 | semidirect |
| 44. D84 | Q8 | semidirect |
| 64. C3×C4×Q8 | D7 | semidirect |
| 121. C4×Q8 | D21 | semidirect |
| 149. C3×C7 | #554 | semidirect |
| 194. C7 | #5579 | semidirect |
| 210. C3 | #90270 | 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 |
| #10621429 | 2 |
| #10621379 | 2 |
| C4⋊C4×D21 | 2 |
| C4×D84 | 2 |
| #10621353 | 2 |
| #10621352 | 2 |
| #90270 | 3 |
| #5579 | 7 |