Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
240 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 87 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. ΓL(1,𝔽8)×C4⋊Q8 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×C4⋊C8 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 6. C7×#461 | C3 | semidirect |
| 18. C7×C4⋊Q8 | C2×C3 | semidirect |
| 20. C7×C4⋊C8 | C2×C3 | semidirect |
| 21. C7×#103 | C2×C3 | semidirect |
| 57. C4×ΓL(1,𝔽8) | SD16 | semidirect |
| 60. #461 | ΓL(1,𝔽8) | direct (N × H) |
| 88. C4⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 90. C4⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 91. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 99. C4×C7 | C3×SD16 | semidirect |
| 114. ΓL(1,𝔽8) | #461 | direct (N × H) |
| 151. C7 | C3×#461 | semidirect |
| 163. C4 | SD16×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SD16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C4⋊Q8 | 2 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| C7×#461 | 3 |
| C3×#461 | 7 |