Finite Group DatabasebetaFinite Group DatabasebetaOrder 6561.
24033 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C93 | C3 | 1 | 40 | yes | maximal |
| 2 | C32×C92 | C32 | 1 | 130 | yes | — |
| 3 | C93 | C9 | 1 | 1080 | yes | direct product factor |
| 4 | C33×C9 | C33 | 1 | 40 | yes | — |
| 5 | C3×C92 | C3×C9 | 1 | 4680 | yes | — |
| 6 | C34 | C34 | 1 | 1 | yes | characteristic |
| 7 | C32×C9 | C32×C9 | 1 | 1560 | yes | — |
| 8 | C92 | C92 | 1 | 10530 | yes | direct product factor |
| 9 | C33 | C33×C9 | 1 | 40 | yes | — |
| 10 | C3×C9 | C3×C92 | 1 | 4680 | yes | — |
| 11 | C32 | C32×C92 | 1 | 130 | yes | — |
| 12 | C9 | C93 | 1 | 1080 | yes | direct product factor |
| 13 | C3 | C3×C93 | 1 | 40 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, socle, prime radical, cosocle | C34 | C34 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C34 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C93 | C9 | direct (N × H) |
| 8. C92 | C92 | direct (N × H) |
| 12. C9 | C93 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C93 | 3 |