Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
516 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 159 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. C2×C4⋊C4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×C22⋊C4×ΓL(1,𝔽8) | C2 | semidirect |
| 4. C2×C42×ΓL(1,𝔽8) | C2 | semidirect |
| 5. C7×#390 | C3 | semidirect |
| 9. C2×D4×ΓL(1,𝔽8) | C4 | semidirect |
| 17. C2×C7×C4⋊C4 | C2×C3 | semidirect |
| 18. C2×C7×C22⋊C4 | C2×C3 | semidirect |
| 19. C2×C42×C7 | C2×C3 | semidirect |
| 35. C2×C4×ΓL(1,𝔽8) | D4 | semidirect |
| 39. C2×C7×D4 | C3×C4 | semidirect |
| 62. C4×ΓL(1,𝔽8) | C22⋊C4 | semidirect |
| 63. #390 | ΓL(1,𝔽8) | direct (N × H) |
| 78. C2×C4×C7 | C3×D4 | semidirect |
| 96. C2×C4⋊C4 | C2×ΓL(1,𝔽8) | semidirect |
| 97. C2×C22⋊C4 | C2×ΓL(1,𝔽8) | semidirect |
| 98. C2×C42 | C2×ΓL(1,𝔽8) | semidirect |
| 111. C4×C7 | C3×C22⋊C4 | semidirect |
| 127. ΓL(1,𝔽8) | #390 | direct (N × H) |
| 131. C2×D4 | C4×ΓL(1,𝔽8) | semidirect |
| 171. C2×C4 | D4×ΓL(1,𝔽8) | semidirect |
| 172. C7 | C3×#390 | semidirect |
| 191. C4 | C22⋊C4×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D4×ΓL(1,𝔽8) | 2 |
| C2×C4⋊C4×ΓL(1,𝔽8) | 2 |
| C2×C22⋊C4×ΓL(1,𝔽8) | 2 |
| C2×C42×ΓL(1,𝔽8) | 2 |
| C7×#390 | 3 |
| C3×#390 | 7 |