Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
408 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 86 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. 2−1+4⋊ΓL(1,𝔽8) | C2 | direct (N × H) |
| 2. C2×#41⋊ΓL(1,𝔽8) | C2 | semidirect |
| 3. C22×C7⋊SL(2,𝔽3) | C2 | semidirect |
| 4. C2×C7×2−1+4 | C3 | semidirect |
| 5. #41⋊ΓL(1,𝔽8) | C22 | semidirect |
| 7. C2×C7⋊SL(2,𝔽3) | C22 | semidirect |
| 10. C7×2−1+4 | C2×C3 | semidirect |
| 13. C2×C7×#41 | C2×C3 | semidirect |
| 15. C22×C7×Q8 | C2×C3 | semidirect |
| 26. C7×#41 | C22×C3 | semidirect |
| 33. C2×C7×Q8 | C22×C3 | semidirect |
| 47. C2×2−1+4 | ΓL(1,𝔽8) | semidirect |
| 75. 2−1+4 | C2×ΓL(1,𝔽8) | semidirect |
| 78. C2×#41 | C2×ΓL(1,𝔽8) | semidirect |
| 80. C22×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 100. #41 | C22×ΓL(1,𝔽8) | semidirect |
| 107. C2×Q8 | C22×ΓL(1,𝔽8) | semidirect |
| 116. C2×C7 | 2−1+4⋊C3 | semidirect |
| 141. C7 | C2×2−1+4⋊C3 | semidirect |
| 157. C2 | 2−1+4⋊ΓL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+4⋊ΓL(1,𝔽8) | 2 |
| C2×#41⋊ΓL(1,𝔽8) | 2 |
| C22×C7⋊SL(2,𝔽3) | 2 |
| C2×C7×2−1+4 | 3 |
| C2×D4×ΓL(1,𝔽8) | 4 |
| C2×2−1+4⋊C3 | 7 |