Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
306 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 58 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 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C7 |
| Frattini | C2 | C22×C7×S4 |
| Fitting | C22×C7×Q8 | D3 |
| hypercenter | C23×C7 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×SL(2,𝔽3) | C2 | semidirect |
| 2. C7×#983 | C2 | direct (N × H) |
| 3. C2×C7×GL(2,𝔽3) | C2 | semidirect |
| 4. C2×C7×#226 | C2 | semidirect |
| 8. C2×C7×SL(2,𝔽3) | C22 | semidirect |
| 9. C7×GL(2,𝔽3) | C22 | semidirect |
| 10. C7×#226 | C22 | semidirect |
| 12. C22×C7×Q8 | D3 | semidirect |
| 20. C2×#983 | C7 | direct (N × H) |
| 32. C2×C7×Q8 | C2×D3 | semidirect |
| 46. C22×SL(2,𝔽3) | C2×C7 | semidirect |
| 47. #983 | C2×C7 | direct (N × H) |
| 48. C2×GL(2,𝔽3) | C2×C7 | semidirect |
| 49. C2×#226 | C2×C7 | semidirect |
| 78. C2×SL(2,𝔽3) | C22×C7 | semidirect |
| 79. GL(2,𝔽3) | C22×C7 | semidirect |
| 80. #226 | C22×C7 | semidirect |
| 86. C22×Q8 | D3×C7 | semidirect |
| 115. C2×Q8 | C2×D3×C7 | semidirect |
| 131. C2×C7 | #983 | direct (N × H) |
| 158. C7 | C2×#983 | direct (N × H) |
| 175. C2 | C7×#983 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×SL(2,𝔽3) | 2 |
| C7×#983 | 2 |
| C2×C7×GL(2,𝔽3) | 2 |
| C2×C7×#226 | 2 |
| C2×C7×#137 | 3 |
| C2×C7×#67 | 4 |
| C2×#983 | 7 |