Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
767 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 767 conjugacy classes of subgroups — too many to draw. Only its 247 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×C9×A4 | C2 | direct (N × H) |
| 4. C23×Q8×C9 | C3 | semidirect |
| 5. C2×Q8×C22⋊C9 | C3 | semidirect |
| 14. C22×Q8×C9 | C2×C3 | semidirect |
| 16. Q8×C22⋊C9 | C2×C3 | semidirect |
| 18. C2×C9×A4 | Q8 | direct (N × H) |
| 23. C2×Q8×A4 | C9 | direct (N × H) |
| 36. C2×Q8×C9 | A4 | direct (N × H) |
| 43. C9×A4 | C2×Q8 | direct (N × H) |
| 50. Q8×A4 | C2×C9 | direct (N × H) |
| 59. C23×C9 | C3×Q8 | semidirect |
| 61. C2×C22⋊C9 | C3×Q8 | semidirect |
| 66. Q8×C9 | C2×A4 | direct (N × H) |
| 74. C23×Q8 | C3×C9 | semidirect |
| 103. C22×C9 | C2×C3×Q8 | semidirect |
| 105. C22⋊C9 | C2×C3×Q8 | semidirect |
| 112. C22×Q8 | C2×C3×C9 | semidirect |
| 120. C2×A4 | Q8×C9 | direct (N × H) |
| 140. C2×C9 | Q8×A4 | direct (N × H) |
| 149. C2×Q8 | C9×A4 | direct (N × H) |
| 164. A4 | C2×Q8×C9 | direct (N × H) |
| 170. C9 | C2×Q8×A4 | direct (N × H) |
| 174. C23 | C3×Q8×C9 | semidirect |
| 178. Q8 | C2×C9×A4 | direct (N × H) |
| 199. C22 | C2×C3×Q8×C9 | semidirect |
| 208. C2 | Q8×C9×A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C9×A4 | 2 |
| C2×C4×C9×A4 | 2 |
| C2×C3×Q8×A4 | 3 |
| C23×Q8×C9 | 3 |
| C2×Q8×C22⋊C9 | 3 |
| C2×C3×Q8×C9 | 4 |