Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 78 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 |
|---|---|---|
| 2. C4×C7⋊SL(2,𝔽3) | C2 | direct (N × H) |
| 3. C2×C4×C7×Q8 | C3 | semidirect |
| 4. C2×C7⋊SL(2,𝔽3) | C4 | direct (N × H) |
| 9. C4×C7×Q8 | C2×C3 | semidirect |
| 14. C7⋊SL(2,𝔽3) | C2×C4 | direct (N × H) |
| 19. C2×C7×Q8 | C3×C4 | semidirect |
| 35. C2×C4×Q8 | ΓL(1,𝔽8) | semidirect |
| 40. C7×Q8 | C2×C3×C4 | semidirect |
| 47. C2×C4×C7 | SL(2,𝔽3) | semidirect |
| 58. C4×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 69. C4×C7 | C2×SL(2,𝔽3) | semidirect |
| 77. C2×Q8 | C4×ΓL(1,𝔽8) | semidirect |
| 86. C2×C7 | C4×SL(2,𝔽3) | semidirect |
| 99. Q8 | C2×C4×ΓL(1,𝔽8) | semidirect |
| 106. C2×C4 | C7⋊SL(2,𝔽3) | direct (N × H) |
| 108. C7 | C2×C4×SL(2,𝔽3) | semidirect |
| 121. C4 | C2×C7⋊SL(2,𝔽3) | direct (N × H) |
| 123. C2 | C4×C7⋊SL(2,𝔽3) | 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 |
| C4×C7⋊SL(2,𝔽3) | 2 |
| C2×C4×C7×Q8 | 3 |
| C22×C4×ΓL(1,𝔽8) | 4 |
| C2×C4×SL(2,𝔽3) | 7 |