Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
128 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C4×C7 |
| Frattini | C4 | C2×C7×S4 |
| Fitting | C7×#131 | D3 |
| socle | C2×C7 | C4×S4 |
| prime radical | C7×#985 | C2 |
| cosocle | #41⋊C3 | C22×C7 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C7×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#131⋊C3 | C2 | semidirect |
| 3. C7×GU(2,𝔽3) | C2 | semidirect |
| 6. C7×GL(2,𝔽3) | C4 | semidirect |
| 7. C7×#226 | C4 | semidirect |
| 11. C7×#131 | D3 | semidirect |
| 16. #5404 | C7 | direct (N × H) |
| 35. #131⋊C3 | C2×C7 | semidirect |
| 36. GU(2,𝔽3) | C2×C7 | semidirect |
| 55. GL(2,𝔽3) | C4×C7 | semidirect |
| 56. #226 | C4×C7 | semidirect |
| 62. #131 | D3×C7 | semidirect |
| 110. C7 | #5404 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#985 | 2 |
| C7×#131⋊C3 | 2 |
| C7×GU(2,𝔽3) | 2 |
| C7×#443 | 3 |
| D3×C7×C8 | 4 |
| #5404 | 7 |