Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
710 conjugacy classes of subgroups; only the 259 normal subgroups are shown.
This group has 710 conjugacy classes of subgroups — too many to draw. Only its 259 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 |
|---|---|---|
| 3. C3×C4×C8×C9 | C2 | semidirect |
| 4. #11720538 | C2 | semidirect |
| 5. C4×C3⋊(C9⋊C8) | C2 | semidirect |
| 15. #88523 | C4 | direct (N × H) |
| 23. C4×C8×C9 | D3 | semidirect |
| 33. C3×C8×C9 | C2×C4 | semidirect |
| 34. C3⋊(C9⋊C8) | C2×C4 | semidirect |
| 56. C3×C4×C9 | M16 | semidirect |
| 64. C3×C4×C8 | D9 | semidirect |
| 86. C8×C9 | C4×D3 | semidirect |
| 118. C4×C9 | ΓL(1,𝔽25) | semidirect |
| 126. C4×C8 | C3⋊D9 | semidirect |
| 127. C3×C9 | C4×M16 | semidirect |
| 142. C3×C8 | C4×D9 | semidirect |
| 166. C3×C4 | #3663 | semidirect |
| 175. C9 | C4×ΓL(1,𝔽25) | semidirect |
| 184. C8 | C4×C3⋊D9 | semidirect |
| 196. C4 | #88523 | direct (N × H) |
| 198. C3 | C4×#3663 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88523 | 2 |
| C42×C3⋊D9 | 2 |
| C3×C4×C8×C9 | 2 |
| #11720538 | 2 |
| C4×C3⋊(C9⋊C8) | 2 |
| C4×#3743 | 3 |
| C4×#3663 | 3 |