What does the poem "The Last Duchess" tell us about the position of women in society in that age?

"My Last Duchess" suggests much about the limited social roles women had during the Renaissance period (and to a degree, the contemporary period in which the poem was written, the Victorian era). Mainly, the poem shows how women were viewed as the property of their husbands and how they had to make sure their reputation was never put under suspicion.
In the poem, the duchess is expected to be subordinate to her husband in all things down to the permission to smile at other men, even if said smile is only platonic in nature. While reputation and marital faithfulness are emphasized in such a culture, the duke exaggerates this, going to homicidal degrees to make sure his wife is obedient to him in every way. He knows he cannot ever fully control her mind or will, so the only way he can is by killing her.
The objectification of women in this setting is inherent in the central symbol of the poem: the portrait itself. Even though the duchess is no longer alive, her likeness is still under her husband's control. He claims a sense of ownership over her even after her death. The duke keeps a curtain over her portrait, controlling who does and does not get to look upon her.