Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
210 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 45 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 | C2×A4×D7 |
| commutator, cosocle | C2×C7×Q8 | C22×C3 |
| Fitting | C4×C7×Q8 | C2×C3 |
| socle | C22×C7 | C22×A4 |
| hypercenter | C2×C4 | A4×D7 |
| hypocenter | C7×Q8 | C3×D4 |
| LCS | G ▷ C2×C7×Q8 ▷ C7×Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D7×SL(2,𝔽3) | C2 | semidirect |
| 2. C4×C7×SL(2,𝔽3) | C2 | semidirect |
| 3. Q8×D28 | C3 | semidirect |
| 7. C2×Q8×D7 | C2×C3 | semidirect |
| 8. C4×C7×Q8 | C2×C3 | semidirect |
| 16. C7×SL(2,𝔽3) | D4 | semidirect |
| 29. C4×SL(2,𝔽3) | D7 | semidirect |
| 38. C7×Q8 | C3×D4 | semidirect |
| 43. D28 | SL(2,𝔽3) | direct (N × H) |
| 58. C4×Q8 | C3×D7 | semidirect |
| 63. C2×D7 | C2×SL(2,𝔽3) | semidirect |
| 67. C4×C7 | C2×SL(2,𝔽3) | semidirect |
| 72. SL(2,𝔽3) | D28 | direct (N × H) |
| 98. Q8 | C3×D28 | semidirect |
| 106. C7 | D4×SL(2,𝔽3) | semidirect |
| 119. C4 | D7×SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D7×SL(2,𝔽3) | 2 |
| C4×C7×SL(2,𝔽3) | 2 |
| Q8×D28 | 3 |
| C2×C3×D28 | 4 |
| D4×SL(2,𝔽3) | 7 |