Father Son
The father-son relationship is one of the major themes in the film. Brick's relationship to his father is strained as he desires to be truly loved by the man, and Big Daddy desires that his life live on through his factories, wealth and that his son would carry on his greatness through his life. Thus, this is one of the major points of contention for the pair throughout the film.
Overcoming Poverty
Maggie was raised in a family that had no financial wealth. She lived in poverty all of her life and her greatest need is to become rich, which she can achieve by having a child with Brick. When she does, Big Daddy will leave his land to Brick and her and they will never have to worry about money for as long as they live. This is why Maggie's nickname is The Cat, she will claw and fight in order to get what she needs no mater who is in her way.
The Old South
The Old South is a major theme of the film. Big Daddy built up his 28,000 acres of the riches land in the South all by himself. He grew up in a patriarchal culture that he desires to be carried on by his son, Brick. Big Daddy represents the Old South, that it is dying but clings to anything that it can in order that it will live on. Thus, Big Daddy creates a textile factory to spin his cotton off his land into products for sale, and he desires his son Brick to carry on his legacy.