Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
891 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 891 conjugacy classes of subgroups — too many to draw. Only its 213 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. C4×C8×H(3,𝔽3) | C2 | semidirect |
| 4. C8×#1084 | C2 | semidirect |
| 5. C4×#6180 | C2 | semidirect |
| 14. C8×#272 | C4 | direct (N × H) |
| 17. C32×C4×C8 | D3 | semidirect |
| 25. C4×#272 | C8 | direct (N × H) |
| 27. C8×H(3,𝔽3) | C2×C4 | semidirect |
| 28. #6180 | C2×C4 | semidirect |
| 42. C4×H(3,𝔽3) | C2×C8 | semidirect |
| 45. #1084 | C2×C8 | semidirect |
| 63. C32×C8 | C4×D3 | semidirect |
| 68. #272 | C4×C8 | direct (N × H) |
| 88. C32×C4 | D3×C8 | semidirect |
| 96. C4×C8 | #272 | direct (N × H) |
| 97. H(3,𝔽3) | C2×C4×C8 | semidirect |
| 141. C32 | C4×D3×C8 | semidirect |
| 150. C8 | C4×#272 | direct (N × H) |
| 162. C4 | C8×#272 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×#272 | 2 |
| C42×#272 | 2 |
| C4×C8×H(3,𝔽3) | 2 |
| C8×#1084 | 2 |
| C4×#6180 | 2 |
| C3×C4×D3×C8 | 3 |