Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
768 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 768 conjugacy classes of subgroups — too many to draw. Only its 150 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 |
|---|---|---|
| 2. C22×C32×C42 | C3 | semidirect |
| 3. C3×C42×A4 | C3 | semidirect |
| 5. C4×C22⋊H(3,𝔽3) | C4 | direct (N × H) |
| 21. C22×C32×C4 | C3×C4 | semidirect |
| 22. C3×C4×A4 | C3×C4 | semidirect |
| 26. C32×C42 | A4 | semidirect |
| 28. C22⋊H(3,𝔽3) | C42 | direct (N × H) |
| 48. C22×C42 | H(3,𝔽3) | semidirect |
| 74. C22×C32 | C3×C42 | semidirect |
| 76. C3×A4 | C3×C42 | semidirect |
| 79. C32×C4 | C4×A4 | semidirect |
| 112. C22×C4 | C4×H(3,𝔽3) | semidirect |
| 115. C42 | C22⋊H(3,𝔽3) | direct (N × H) |
| 136. C32 | C42×A4 | semidirect |
| 157. C22 | C42×H(3,𝔽3) | semidirect |
| 160. C4 | C4×C22⋊H(3,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C22⋊H(3,𝔽3) | 2 |
| C22×C32×C42 | 3 |
| C3×C42×A4 | 3 |
| C42×H(3,𝔽3) | 4 |