Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
228 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 228 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 |
|---|---|---|
| 1. C2×Q16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#107 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 6. C7×#467 | C3 | semidirect |
| 16. C2×C7×Q16 | C2×C3 | semidirect |
| 19. C7×#107 | C2×C3 | semidirect |
| 20. C7×#103 | C2×C3 | semidirect |
| 49. C22×ΓL(1,𝔽8) | Q16 | semidirect |
| 56. #467 | ΓL(1,𝔽8) | direct (N × H) |
| 81. C2×Q16 | C2×ΓL(1,𝔽8) | semidirect |
| 84. #107 | C2×ΓL(1,𝔽8) | semidirect |
| 85. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 88. C22×C7 | C3×Q16 | semidirect |
| 106. ΓL(1,𝔽8) | #467 | direct (N × H) |
| 140. C7 | C3×#467 | semidirect |
| 147. C22 | Q16×ΓL(1,𝔽8) | 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 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C22⋊Q8 | 2 |
| ΓL(1,𝔽8)×#107 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| C7×#467 | 3 |
| C3×#467 | 7 |