Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1032 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 186 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. C32×Q8×A4 | C2 | semidirect |
| 3. C32×A4⋊Q8 | C2 | semidirect |
| 4. C3×Q8×S4 | C3 | direct (N × H) |
| 11. C22×C32×Q8 | D3 | semidirect |
| 13. C3×Q8×A4 | C2×C3 | semidirect |
| 15. C3×A4⋊Q8 | C2×C3 | semidirect |
| 22. C32×S4 | Q8 | direct (N × H) |
| 26. Q8×S4 | C32 | direct (N × H) |
| 48. C32×A4 | C2×Q8 | semidirect |
| 52. C22×C3×Q8 | C3×D3 | semidirect |
| 54. Q8×A4 | C2×C32 | semidirect |
| 56. A4⋊Q8 | C2×C32 | semidirect |
| 66. C3×S4 | C3×Q8 | direct (N × H) |
| 73. C32×Q8 | S4 | direct (N × H) |
| 112. C22×C32 | D3×Q8 | semidirect |
| 115. C3×A4 | C2×C3×Q8 | semidirect |
| 125. C22×Q8 | C32×D3 | semidirect |
| 136. S4 | C32×Q8 | direct (N × H) |
| 143. C3×Q8 | C3×S4 | direct (N × H) |
| 181. C22×C3 | C3×D3×Q8 | semidirect |
| 184. A4 | C2×C32×Q8 | semidirect |
| 195. C32 | Q8×S4 | direct (N × H) |
| 202. Q8 | C32×S4 | direct (N × H) |
| 224. C22 | C32×D3×Q8 | semidirect |
| 232. C3 | C3×Q8×S4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×Q8×A4 | 2 |
| C32×C4×S4 | 2 |
| C32×A4⋊Q8 | 2 |
| C3×Q8×S4 | 3 |
| C32×D4×Q8 | 3 |
| C32×D3×Q8 | 4 |