Finite Group DatabasebetaFinite Group DatabasebetaOrder 648.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22⋊C81 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #6171 | — | 3 | 1 | no | maximal |
| 3 | D81 | — | 4 | 1 | no | maximal |
| 4 | C22×C27 | D3 | 1 | 1 | yes | characteristic |
| 5 | C2×D27 | — | 3 | 1 | no | — |
| 6 | Dic27 | — | 3 | 1 | no | — |
| 7 | C81 | — | 4 | 1 | no | Sylow |
| 8 | #611 | — | 9 | 1 | no | — |
| 9 | C2×C27 | — | 3 | 1 | no | — |
| 10 | D27 | — | 6 | 1 | no | — |
| 11 | C22×C9 | D9 | 1 | 1 | yes | characteristic |
| 12 | C2×D9 | — | 9 | 1 | no | — |
| 13 | Dic9 | — | 9 | 1 | no | — |
| 14 | C27 | S4 | 1 | 1 | yes | characteristic |
| 15 | #67 | — | 27 | 1 | no | — |
| 16 | C2×C9 | — | 3 | 1 | no | — |
| 17 | D9 | — | 18 | 1 | no | — |
| 18 | C22×C3 | D27 | 1 | 1 | yes | characteristic |
| 19 | C2×D3 | — | 27 | 1 | no | — |
| 20 | Dic3 | — | 27 | 1 | no | — |
| 21 | C9 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 22 | D4 | — | 81 | 1 | no | Sylow |
| 23 | C2×C3 | — | 3 | 1 | no | — |
| 24 | D3 | — | 54 | 1 | no | — |
| 25 | C22 | — | 81 | 1 | no | — |
| 26 | C22 | D81 | 1 | 1 | yes | characteristic, complemented |
| 27 | C4 | — | 81 | 1 | no | — |
| 28 | C3 | (C22⋊C27)⋊C2 | 1 | 1 | yes | characteristic |
| 29 | C2 | — | 3 | 1 | no | — |
| 30 | C2 | — | 162 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊C81 | C2 |
| Frattini | C27 | S4 |
| Fitting | C22×C27 | D3 |
| socle | C22×C3 | D27 |
| LCS | G ▷ C22⋊C81 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22⋊C81 | C2 | semidirect |
| 26. C22 | D81 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22⋊C81 | 2 |
| #6171 | 3 |
| D81 | 4 |