Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
459 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 135 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×#6188 | C2 | direct (N × H) |
| 7. C2×#6188 | C4 | direct (N × H) |
| 13. C2×C4×H(3,𝔽3) | C8 | semidirect |
| 15. C2×#1084 | C8 | semidirect |
| 20. #6188 | C2×C4 | direct (N × H) |
| 31. C4×H(3,𝔽3) | C2×C8 | semidirect |
| 33. #1084 | C2×C8 | semidirect |
| 48. C2×H(3,𝔽3) | C4×C8 | semidirect |
| 75. H(3,𝔽3) | C2×C4×C8 | semidirect |
| 127. C2×C4 | #6188 | direct (N × H) |
| 145. C4 | C2×#6188 | direct (N × H) |
| 148. C2 | C4×#6188 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#6188 | 2 |
| C2×C4×#1084 | 2 |
| C4×#6188 | 2 |
| C2×C3×C4×C8 | 9 |