Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
237 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 61 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. C32×C8×Q8 | C3 | semidirect |
| 3. C3×C8×SL(2,𝔽3) | C3 | semidirect |
| 11. Q8⋊H(3,𝔽3) | C8 | direct (N × H) |
| 34. C32×Q8 | C3×C8 | semidirect |
| 39. C3×SL(2,𝔽3) | C3×C8 | semidirect |
| 42. C32×C8 | SL(2,𝔽3) | semidirect |
| 43. C8×Q8 | H(3,𝔽3) | semidirect |
| 116. C32 | C8×SL(2,𝔽3) | semidirect |
| 118. Q8 | C8×H(3,𝔽3) | semidirect |
| 123. C8 | Q8⋊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 |
|---|---|
| C4×Q8⋊H(3,𝔽3) | 2 |
| C32×C8×Q8 | 3 |
| C3×C8×SL(2,𝔽3) | 3 |
| C2×C8×H(3,𝔽3) | 4 |