Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
440 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | A4×D7 |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Frattini | C2 | C22×A4×D7 |
| Fitting | C22×C7×Q8 | C2×C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×#66662 | C2 | direct (N × H) |
| 3. C22×#1161 | C3 | semidirect |
| 5. C2×C7×SL(2,𝔽3) | C22 | semidirect |
| 6. #66662 | C22 | direct (N × H) |
| 9. C2×#1161 | C2×C3 | semidirect |
| 15. C7×SL(2,𝔽3) | C23 | semidirect |
| 23. #1161 | C22×C3 | semidirect |
| 29. C22×SL(2,𝔽3) | D7 | semidirect |
| 49. C2×SL(2,𝔽3) | C2×D7 | semidirect |
| 57. C22×C7 | #41⋊C3 | semidirect |
| 66. SL(2,𝔽3) | C22×D7 | semidirect |
| 80. C2×C7 | C2×#41⋊C3 | semidirect |
| 99. C7 | C22×#41⋊C3 | semidirect |
| 104. C22 | #66662 | direct (N × H) |
| 111. C2 | C2×#66662 | 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 |
| C2×#66662 | 2 |
| C22×#1161 | 3 |
| C22×C3×Dic7 | 4 |
| C22×#41⋊C3 | 7 |