Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
350 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×C22⋊A4 |
| commutator, Fitting, cosocle, hypocenter | C22×C7×Q8 | C2×C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C22×C7×Q8 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C22⋊SL(2,𝔽3) | C2 | semidirect |
| 2. C22×Q8×D7 | C3 | semidirect |
| 5. C22×C7×Q8 | C2×C3 | semidirect |
| 21. Q8×D7 | A4 | semidirect |
| 25. C22⋊SL(2,𝔽3) | D7 | direct (N × H) |
| 33. C22×D7 | SL(2,𝔽3) | semidirect |
| 35. C7×Q8 | C2×A4 | semidirect |
| 49. C22×Q8 | C3×D7 | semidirect |
| 53. C22×C7 | C2×SL(2,𝔽3) | semidirect |
| 81. D7 | C22⋊SL(2,𝔽3) | direct (N × H) |
| 92. Q8 | A4×D7 | semidirect |
| 97. C7 | C2×C22⋊SL(2,𝔽3) | semidirect |
| 108. C22 | D7×SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C22⋊SL(2,𝔽3) | 2 |
| C22×Q8×D7 | 3 |
| C2×A4×D7 | 4 |
| D7×SL(2,𝔽3) | 4 |
| C2×C22⋊SL(2,𝔽3) | 7 |