Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
180 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. C22×C7⋊SL(2,𝔽3) | C2 | semidirect |
| 3. C4×C7⋊SL(2,𝔽3) | C2 | semidirect |
| 4. C7×#548 | C3 | semidirect |
| 9. C2×C7×#41 | C2×C3 | semidirect |
| 14. C4×C7×Q8 | C2×C3 | semidirect |
| 18. C7⋊SL(2,𝔽3) | D4 | semidirect |
| 46. #548 | ΓL(1,𝔽8) | semidirect |
| 73. C2×#41 | C2×ΓL(1,𝔽8) | semidirect |
| 78. C4×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 85. C22×C7 | #41⋊C3 | semidirect |
| 143. C7 | #548⋊C3 | semidirect |
| 153. C22 | #41⋊ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41⋊ΓL(1,𝔽8) | 2 |
| C22×C7⋊SL(2,𝔽3) | 2 |
| C4×C7⋊SL(2,𝔽3) | 2 |
| C7×#548 | 3 |
| C22⋊C4×ΓL(1,𝔽8) | 4 |
| #548⋊C3 | 7 |