Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
920 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 920 conjugacy classes of subgroups — too many to draw. Only its 170 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. C33×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C3×#64158 | C2 | semidirect |
| 3. C3×#64157 | C2 | semidirect |
| 4. C2×C3×#3753 | C2 | semidirect |
| 5. C3×#63660 | C2 | semidirect |
| 6. C3×#63650 | C2 | semidirect |
| 7. C3×#63649 | C2 | semidirect |
| 8. #10592898 | C3 | direct (N × H) |
| 26. C32×AGL(1,ℤ/8) | D3 | semidirect |
| 27. C32×AGL(1,ℤ/8) | C2×C3 | semidirect |
| 28. #64158 | C2×C3 | semidirect |
| 29. #64157 | C2×C3 | semidirect |
| 30. C2×#3753 | C2×C3 | semidirect |
| 34. #63660 | C2×C3 | semidirect |
| 35. #63650 | C2×C3 | semidirect |
| 36. #63649 | C2×C3 | semidirect |
| 104. C3×AGL(1,ℤ/8) | C3×D3 | semidirect |
| 105. C3×AGL(1,ℤ/8) | C3⋊D3 | semidirect |
| 225. AGL(1,ℤ/8) | C3×C3⋊D3 | semidirect |
| 231. C33 | #472 | semidirect |
| 328. C32 | #5199 | semidirect |
| 329. C32 | C3×#472 | semidirect |
| 370. C3 | C3×#5199 | semidirect |
| 371. C3 | #10592898 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×AGL(1,ℤ/8) | 2 |
| C3×#64158 | 2 |
| C3×#64157 | 2 |
| C2×C3×#3753 | 2 |
| C3×#63660 | 2 |
| C3×#63650 | 2 |
| C3×#63649 | 2 |
| #10592898 | 3 |
| C3×#5199 | 3 |