Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
624 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 138 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×C7×C4⋊C4 | C22 |
| hypercenter | C4⋊C4 | D3×D7 |
| hypocenter | C3×C7 | #550 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#66676 | C2 | semidirect |
| 2. C2×#66669 | C2 | semidirect |
| 3. #10621370 | C2 | semidirect |
| 4. C7×#895 | C2 | semidirect |
| 5. C3×#6458 | C2 | semidirect |
| 6. #10621025 | C2 | semidirect |
| 7. #10621025 | C2 | semidirect |
| 8. #10621015 | C2 | semidirect |
| 9. #10621004 | C2 | semidirect |
| 10. C4×#4245 | C2 | semidirect |
| 11. #10620970 | C2 | semidirect |
| 45. #6458 | D3 | semidirect |
| 59. Dic42 | D4 | semidirect |
| 107. #895 | D7 | semidirect |
| 128. Dic21 | #41 | semidirect |
| 245. C3×C7 | #550 | semidirect |
| 323. C7 | #5599 | semidirect |
| 351. C3 | #90290 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66676 | 2 |
| C2×#66669 | 2 |
| #10621370 | 2 |
| C7×#895 | 2 |
| C3×#6458 | 2 |
| #10621025 | 2 |
| #10621015 | 2 |
| #10621004 | 2 |
| C4×#4245 | 2 |
| #10620970 | 2 |
| #90290 | 3 |
| #5599 | 7 |