Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
560 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 160 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 | C22×D3×D7 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C42×C7 | C22 |
| prime radical | C2×D84 | C22 |
| hypercenter | C42 | D3×D7 |
| hypocenter | C3×C7 | D4×Q8 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D21⋊Q8 | C2 | semidirect |
| 3. C4×C7×Dic6 | C2 | semidirect |
| 4. C3×#6432 | C2 | semidirect |
| 5. #10621031 | C2 | semidirect |
| 6. #10620991 | C2 | semidirect |
| 8. #10620952 | C2 | semidirect |
| 34. #6432 | D3 | semidirect |
| 41. D84 | Q8 | semidirect |
| 45. C7×Dic6 | D4 | semidirect |
| 74. C4×Dic6 | D7 | semidirect |
| 78. C2×D21 | C2×Q8 | semidirect |
| 81. C3×C4×C7 | C2×Q8 | semidirect |
| 138. C4×C7 | D3×Q8 | semidirect |
| 154. Dic6 | D28 | semidirect |
| 155. C3×C7 | D4×Q8 | semidirect |
| 184. C3×C4 | Q8×D7 | semidirect |
| 202. C7 | #5727 | semidirect |
| 216. C4 | D21⋊Q8 | semidirect |
| 218. C3 | Q8×D28 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D21⋊Q8 | 2 |
| C4×D84 | 2 |
| C4×C7×Dic6 | 2 |
| C3×#6432 | 2 |
| #10621031 | 2 |
| #10620991 | 2 |
| #10620990 | 2 |
| #10620952 | 2 |
| Q8×D28 | 3 |
| #5727 | 7 |