Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
248 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 248 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×SD16×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C22×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×C22⋊Q8 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#106 | C2 | semidirect |
| 6. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 7. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 8. C7×#465 | C3 | semidirect |
| 22. C2×C7×SD16 | C2×C3 | semidirect |
| 23. C22×C7×C8 | C2×C3 | semidirect |
| 24. C7×C22⋊Q8 | C2×C3 | semidirect |
| 26. C7×#106 | C2×C3 | semidirect |
| 27. C7×#103 | C2×C3 | semidirect |
| 28. C7×#102 | C2×C3 | semidirect |
| 45. C8×ΓL(1,𝔽8) | D4 | semidirect |
| 71. C22×ΓL(1,𝔽8) | SD16 | semidirect |
| 79. #465 | ΓL(1,𝔽8) | direct (N × H) |
| 95. C7×C8 | C3×D4 | semidirect |
| 114. C2×SD16 | C2×ΓL(1,𝔽8) | semidirect |
| 115. C22×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 116. C22⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 118. #106 | C2×ΓL(1,𝔽8) | semidirect |
| 119. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 120. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 126. C22×C7 | C3×SD16 | semidirect |
| 150. ΓL(1,𝔽8) | #465 | direct (N × H) |
| 197. C8 | D4×ΓL(1,𝔽8) | semidirect |
| 198. C7 | C3×#465 | semidirect |
| 209. C22 | 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 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C22⋊Q8 | 2 |
| ΓL(1,𝔽8)×#121 | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| C7×#465 | 3 |
| C3×#465 | 7 |