Finite Group DatabasebetaFinite Group DatabasebetaOrder 648.
80 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C81 | C2 | 1 | 7 | yes | maximal, direct product factor |
| 2 | C23×C27 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C81 | C22 | 1 | 7 | yes | direct product factor |
| 4 | C22×C27 | C2×C3 | 1 | 7 | yes | — |
| 5 | C81 | C23 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 6 | C23×C9 | C9 | 1 | 1 | yes | characteristic |
| 7 | C2×C27 | C22×C3 | 1 | 7 | yes | — |
| 8 | C22×C9 | C2×C9 | 1 | 7 | yes | — |
| 9 | C27 | C23×C3 | 1 | 1 | yes | characteristic |
| 10 | C23×C3 | C27 | 1 | 1 | yes | characteristic |
| 11 | C2×C9 | C22×C9 | 1 | 7 | yes | — |
| 12 | C22×C3 | C2×C27 | 1 | 7 | yes | — |
| 13 | C9 | C23×C9 | 1 | 1 | yes | characteristic |
| 14 | C23 | C81 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 15 | C2×C3 | C22×C27 | 1 | 7 | yes | — |
| 16 | C22 | C2×C81 | 1 | 7 | yes | direct product factor |
| 17 | C3 | C23×C27 | 1 | 1 | yes | characteristic |
| 18 | C2 | C22×C81 | 1 | 7 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C27 | C23×C3 |
| socle, prime radical | C23×C3 | C27 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C81 | C2 | direct (N × H) |
| 3. C2×C81 | C22 | direct (N × H) |
| 5. C81 | C23 | direct (N × H) |
| 14. C23 | C81 | direct (N × H) |
| 16. C22 | C2×C81 | direct (N × H) |
| 18. C2 | C22×C81 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C81 | 2 |
| C23×C27 | 3 |