Finite Group DatabasebetaFinite Group DatabasebetaOrder 6561.
127902864 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C37 | C3 | 1 | 3280 | yes | maximal, direct product factor |
| 2 | C36 | C32 | 1 | 896260 | yes | direct product factor |
| 3 | C35 | C33 | 1 | 25095280 | yes | direct product factor |
| 4 | C34 | C34 | 1 | 75913222 | yes | direct product factor |
| 5 | C33 | C35 | 1 | 25095280 | yes | direct product factor |
| 6 | C32 | C36 | 1 | 896260 | yes | direct product factor |
| 7 | C3 | C37 | 1 | 3280 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C37 | C3 | direct (N × H) |
| 2. C36 | C32 | direct (N × H) |
| 3. C35 | C33 | direct (N × H) |
| 4. C34 | C34 | direct (N × H) |
| 5. C33 | C35 | direct (N × H) |
| 6. C32 | C36 | direct (N × H) |
| 7. C3 | C37 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C37 | 3 |