Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
605 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 605 conjugacy classes of subgroups — too many to draw. Only its 169 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 |
|---|---|---|
| 8. Q16×H(3,𝔽3) | C4 | semidirect |
| 32. C32×Q16 | Dic3 | semidirect |
| 45. #1084 | Q16 | direct (N × H) |
| 97. H(3,𝔽3) | C4×Q16 | semidirect |
| 120. Q16 | #1084 | direct (N × H) |
| 141. C32 | Dic3×Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q16×H(3,𝔽3) | 2 |
| Q8×#1084 | 2 |
| #11720583 | 2 |
| #11720551 | 2 |
| C8×#1084 | 2 |
| C3×Dic3×Q16 | 3 |