Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 105 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. C4×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#107 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 7. ΓL(1,𝔽8)×#97 | C2 | semidirect |
| 8. C7×#440 | C3 | semidirect |
| 12. SD16×ΓL(1,𝔽8) | C4 | semidirect |
| 22. C4×C7×Q8 | C2×C3 | semidirect |
| 24. C7×#107 | C2×C3 | semidirect |
| 25. C7×#103 | C2×C3 | semidirect |
| 27. C7×#97 | C2×C3 | semidirect |
| 31. Q8×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 42. C8×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 46. C7×SD16 | C3×C4 | semidirect |
| 73. #440 | ΓL(1,𝔽8) | direct (N × H) |
| 76. C7×Q8 | C2×C3×C4 | semidirect |
| 87. C7×C8 | C2×C3×C4 | semidirect |
| 105. C4×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 107. #107 | C2×ΓL(1,𝔽8) | semidirect |
| 108. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 110. #97 | C2×ΓL(1,𝔽8) | semidirect |
| 136. ΓL(1,𝔽8) | #440 | direct (N × H) |
| 140. SD16 | C4×ΓL(1,𝔽8) | semidirect |
| 166. Q8 | C2×C4×ΓL(1,𝔽8) | semidirect |
| 177. C8 | C2×C4×ΓL(1,𝔽8) | semidirect |
| 178. C7 | C3×#440 | 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 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#107 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| ΓL(1,𝔽8)×#97 | 2 |
| C7×#440 | 3 |
| C3×#440 | 7 |