Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
605 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 605 conjugacy classes of subgroups — too many to draw. Only its 145 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 |
|---|---|---|
| 15. #88550 | C4 | direct (N × H) |
| 59. C4×H(3,𝔽3) | Q16 | semidirect |
| 123. C32×C4 | #216 | semidirect |
| 135. H(3,𝔽3) | C4×Q16 | semidirect |
| 195. C32 | C4×#216 | semidirect |
| 221. C4 | #88550 | 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 |
| C2×#88550 | 2 |
| C4×#6229 | 2 |
| #11720583 | 2 |
| #11720532 | 2 |
| #11720529 | 2 |
| C4×#6180 | 2 |
| C3×C4×#216 | 3 |