Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
536 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 70 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. D4×C22⋊H(3,𝔽3) | C2 | semidirect |
| 2. C8×C22⋊H(3,𝔽3) | C2 | semidirect |
| 3. C22×C32×D8 | C3 | semidirect |
| 4. C3×A4×D8 | C3 | semidirect |
| 8. C22×C32×D4 | C2×C3 | semidirect |
| 9. C3×D4×A4 | C2×C3 | semidirect |
| 13. C22×C32×C8 | C2×C3 | semidirect |
| 14. C3×C8×A4 | C2×C3 | semidirect |
| 33. C32×D8 | A4 | semidirect |
| 37. C22⋊H(3,𝔽3) | D8 | direct (N × H) |
| 62. C32×D4 | C2×A4 | semidirect |
| 67. C32×C8 | C2×A4 | semidirect |
| 68. C22×D8 | H(3,𝔽3) | semidirect |
| 105. C22×C32 | C3×D8 | semidirect |
| 106. C3×A4 | C3×D8 | semidirect |
| 110. C22×D4 | C2×H(3,𝔽3) | semidirect |
| 114. C22×C8 | C2×H(3,𝔽3) | semidirect |
| 161. D8 | C22⋊H(3,𝔽3) | direct (N × H) |
| 190. C32 | A4×D8 | semidirect |
| 199. D4 | C2×C22⋊H(3,𝔽3) | semidirect |
| 203. C8 | C2×C22⋊H(3,𝔽3) | semidirect |
| 224. C22 | D8×H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×C22⋊H(3,𝔽3) | 2 |
| C8×C22⋊H(3,𝔽3) | 2 |
| C22×C32×D8 | 3 |
| C3×A4×D8 | 3 |
| D8×H(3,𝔽3) | 4 |