Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
587 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 587 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | (C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C23×C3 |
| Frattini | C2×C3 | C22×C3×S4 |
| Fitting | C22×C32×Q8 | D3 |
| socle, prime radical | C23×C32 | S4 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C23×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×#3688 | C2 | direct (N × H) |
| 4. C22×#3688 | C3 | direct (N × H) |
| 7. C3×#3688 | C22 | direct (N × H) |
| 15. C2×#3688 | C2×C3 | direct (N × H) |
| 33. #3688 | C22×C3 | direct (N × H) |
| 146. C22×C3 | #3688 | direct (N × H) |
| 173. C2×C3 | C2×#3688 | direct (N × H) |
| 177. C22 | C3×#3688 | direct (N × H) |
| 183. C3 | C22×#3688 | direct (N × H) |
| 185. C2 | C2×C3×#3688 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×Q8⋊C9 | 2 |
| C2×C3×#3688 | 2 |
| C22×C3×#216 | 3 |
| C22×#3688 | 3 |
| C22×C3×Dic9 | 4 |