Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
180 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×A4 |
| commutator, cosocle | C2×Q8 | C22×C3×C7 |
| Frattini | C22 | C22×C7×A4 |
| Fitting | C7×#548 | C3 |
| hypercenter | C7×C22⋊C4 | A4 |
| hypocenter | Q8 | C3×C7×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22×C7 ◁ C7×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#41⋊C3 | C2 | semidirect |
| 2. C22×C7×SL(2,𝔽3) | C2 | semidirect |
| 3. C4×C7×SL(2,𝔽3) | C2 | semidirect |
| 4. C7×#548 | C3 | semidirect |
| 9. C2×C7×#41 | C2×C3 | semidirect |
| 14. C4×C7×Q8 | C2×C3 | semidirect |
| 16. #548⋊C3 | C7 | direct (N × H) |
| 20. C7×SL(2,𝔽3) | D4 | semidirect |
| 37. C2×#41⋊C3 | C2×C7 | semidirect |
| 38. C22×SL(2,𝔽3) | C2×C7 | semidirect |
| 39. C4×SL(2,𝔽3) | C2×C7 | semidirect |
| 46. #548 | C3×C7 | semidirect |
| 73. C2×#41 | C2×C3×C7 | semidirect |
| 78. C4×Q8 | C2×C3×C7 | semidirect |
| 85. C22×C7 | #41⋊C3 | semidirect |
| 96. SL(2,𝔽3) | C7×D4 | semidirect |
| 143. C7 | #548⋊C3 | direct (N × H) |
| 153. C22 | C7×#41⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#41⋊C3 | 2 |
| C22×C7×SL(2,𝔽3) | 2 |
| C4×C7×SL(2,𝔽3) | 2 |
| C7×#548 | 3 |
| C3×C7×C22⋊C4 | 4 |
| #548⋊C3 | 7 |