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, hypercenter | C2×C4×C7 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C7 |
| Frattini | C2 | C22×C7×S4 |
| Fitting | C2×C7×#41 | D3 |
| socle | C22×C7 | C2×S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#41⋊C3 | C2 | semidirect |
| 2. C7×#985 | C2 | direct (N × H) |
| 3. C2×C7×GL(2,𝔽3) | C2 | semidirect |
| 4. C2×C7×#226 | C2 | semidirect |
| 7. C7×#41⋊C3 | C22 | semidirect |
| 9. C7×GL(2,𝔽3) | C22 | semidirect |
| 10. C7×#226 | C22 | semidirect |
| 12. C2×C7×#41 | D3 | semidirect |
| 20. C2×#985 | C7 | direct (N × H) |
| 29. C7×#41 | C2×D3 | semidirect |
| 46. C2×#41⋊C3 | C2×C7 | semidirect |
| 47. #985 | C2×C7 | direct (N × H) |
| 48. C2×GL(2,𝔽3) | C2×C7 | semidirect |
| 49. C2×#226 | C2×C7 | semidirect |
| 76. #41⋊C3 | C22×C7 | semidirect |
| 78. GL(2,𝔽3) | C22×C7 | semidirect |
| 79. #226 | C22×C7 | semidirect |
| 84. C2×#41 | D3×C7 | semidirect |
| 110. #41 | C2×D3×C7 | semidirect |
| 129. C2×C7 | #985 | direct (N × H) |
| 155. C7 | C2×#985 | direct (N × H) |
| 171. C2 | C7×#985 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#41⋊C3 | 2 |
| C7×#985 | 2 |
| C2×C7×GL(2,𝔽3) | 2 |
| C2×C7×#226 | 2 |
| C2×C7×#135 | 3 |
| C2×C4×D3×C7 | 4 |
| C2×#985 | 7 |