Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
224 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 81 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×Q16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. C4×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 5. ΓL(1,𝔽8)×C4⋊C8 | C2 | semidirect |
| 6. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 8. C7×#464 | C3 | semidirect |
| 22. C2×C7×Q16 | C2×C3 | semidirect |
| 25. C4×C7×Q8 | C2×C3 | semidirect |
| 26. C7×C4⋊C8 | C2×C3 | semidirect |
| 27. C7×#103 | C2×C3 | semidirect |
| 76. #464 | ΓL(1,𝔽8) | direct (N × H) |
| 109. C2×Q16 | C2×ΓL(1,𝔽8) | semidirect |
| 112. C4×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 113. C4⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 114. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 142. ΓL(1,𝔽8) | #464 | direct (N × H) |
| 186. C7 | C3×#464 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q16×ΓL(1,𝔽8) | 2 |
| C2×SD16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#124 | 2 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| C7×#464 | 3 |
| C3×#464 | 7 |