Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
324 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 324 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×#41×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C2×SD16×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×D8×ΓL(1,𝔽8) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#121 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 6. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 7. ΓL(1,𝔽8)×C22⋊C8 | C2 | semidirect |
| 8. C7×#449 | C3 | semidirect |
| 24. C2×C7×#41 | C2×C3 | semidirect |
| 25. C2×C7×SD16 | C2×C3 | semidirect |
| 26. C2×C7×D8 | C2×C3 | semidirect |
| 27. C7×#121 | C2×C3 | semidirect |
| 28. C7×#103 | C2×C3 | semidirect |
| 29. C7×#102 | C2×C3 | semidirect |
| 30. C7×C22⋊C8 | C2×C3 | semidirect |
| 36. Q8×ΓL(1,𝔽8) | D4 | semidirect |
| 42. D4×ΓL(1,𝔽8) | D4 | semidirect |
| 89. #449 | ΓL(1,𝔽8) | direct (N × H) |
| 94. C7×Q8 | C3×D4 | semidirect |
| 100. C7×D4 | C3×D4 | semidirect |
| 131. C2×#41 | C2×ΓL(1,𝔽8) | semidirect |
| 132. C2×SD16 | C2×ΓL(1,𝔽8) | semidirect |
| 133. C2×D8 | C2×ΓL(1,𝔽8) | semidirect |
| 134. #121 | C2×ΓL(1,𝔽8) | semidirect |
| 135. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 136. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 137. C22⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 173. ΓL(1,𝔽8) | #449 | direct (N × H) |
| 214. Q8 | D4×ΓL(1,𝔽8) | semidirect |
| 220. D4 | D4×ΓL(1,𝔽8) | semidirect |
| 230. C7 | C3×#449 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×ΓL(1,𝔽8) | 2 |
| C2×SD16×ΓL(1,𝔽8) | 2 |
| C2×D8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#121 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| C7×#449 | 3 |
| C3×#449 | 7 |