Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
430 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 85 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. Q8×C22⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. D4×C22⋊ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×C4×C22⋊ΓL(1,𝔽8) | C2 | semidirect |
| 4. C22×C7×#41 | C3 | semidirect |
| 13. C22×C7×Q8 | C2×C3 | semidirect |
| 14. C22×C7×D4 | C2×C3 | semidirect |
| 15. C23×C4×C7 | C2×C3 | semidirect |
| 29. C7×#41 | A4 | semidirect |
| 44. C22⋊ΓL(1,𝔽8) | #41 | direct (N × H) |
| 48. C22×#41 | ΓL(1,𝔽8) | semidirect |
| 56. C7×Q8 | C2×A4 | semidirect |
| 60. C7×D4 | C2×A4 | semidirect |
| 69. C2×C4×C7 | C2×A4 | semidirect |
| 80. C22×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 81. C22×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 82. C23×C4 | C2×ΓL(1,𝔽8) | semidirect |
| 90. C22×C7 | C3×#41 | semidirect |
| 109. #41 | C22⋊ΓL(1,𝔽8) | direct (N × H) |
| 137. Q8 | C2×C22⋊ΓL(1,𝔽8) | semidirect |
| 141. D4 | C2×C22⋊ΓL(1,𝔽8) | semidirect |
| 150. C2×C4 | C2×C22⋊ΓL(1,𝔽8) | semidirect |
| 151. C7 | A4×#41 | semidirect |
| 161. C22 | #41×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C22⋊ΓL(1,𝔽8) | 2 |
| D4×C22⋊ΓL(1,𝔽8) | 2 |
| C2×C4×C22⋊ΓL(1,𝔽8) | 2 |
| C22×C7×#41 | 3 |
| #41×ΓL(1,𝔽8) | 4 |
| A4×#41 | 7 |