Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
390 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C23×C3×C7 |
| Frattini | C2 | C23×C7×A4 |
| Fitting | C7×#585 | C3 |
| socle | C2×C7 | C23×A4 |
| hypercenter | C7×#41 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C4×C7 ◁ C7×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×2−1+4⋊C3 | C2 | semidirect |
| 2. C7×#994 | C2 | semidirect |
| 3. C2×C7×#41⋊C3 | C2 | semidirect |
| 4. C7×#585 | C3 | semidirect |
| 6. C7×#41⋊C3 | C22 | semidirect |
| 7. C7×#41⋊C3 | C22 | semidirect |
| 8. C2×C7×SL(2,𝔽3) | C22 | semidirect |
| 10. C7×2−1+4 | C2×C3 | semidirect |
| 12. C7×H(5,𝔽2) | C2×C3 | semidirect |
| 15. C2×C7×#41 | C2×C3 | semidirect |
| 16. #5945 | C7 | direct (N × H) |
| 40. 2−1+4⋊C3 | C2×C7 | semidirect |
| 41. #994 | C2×C7 | semidirect |
| 42. C2×#41⋊C3 | C2×C7 | semidirect |
| 46. #585 | C3×C7 | semidirect |
| 67. #41⋊C3 | C22×C7 | semidirect |
| 68. #41⋊C3 | C22×C7 | semidirect |
| 69. C2×SL(2,𝔽3) | C22×C7 | semidirect |
| 73. 2−1+4 | C2×C3×C7 | semidirect |
| 75. H(5,𝔽2) | C2×C3×C7 | semidirect |
| 78. C2×#41 | C2×C3×C7 | semidirect |
| 138. C7 | #5945 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×2−1+4⋊C3 | 2 |
| C7×#994 | 2 |
| C2×C7×#41⋊C3 | 2 |
| C7×#585 | 3 |
| C3×C7×#41 | 4 |
| #5945 | 7 |