Socrates final conclusion has it's own premises, and I will in fact list these, as the prompt request. However, I do not believe that this is where Socrates logic is flawed. I do believe that his premises and conclusion are valid, and perhaps his any parts of his final conclusion sound, nevertheless I feel that he demonstrated inappropriate logic along the path to this final conclusion.
If we are good, then we are profitable. Virtue is good. Therefore virtue is profitable. If, virtue is profitable and virtue is a quality of the soul and none of the things of the soul are either profitable or hurtful in themselves, then there must exist another factor that controls the souls outcomes. The soul, under the guidance of wisdom results in happiness, and under irrationality results in unhappiness. Then it is only reasonable to conclude that virtue that exist within the soul, has predictable outcomes based on some sort of wisdom or practicality. If the soul's rightfulness is universally true of all human nature, and all other things within the soul rely on wisdom for goodness, and wisdom is inferred to be that which profits, then we can conclude that virtue is either wholly or partly wisdom.
"all other things hang upon the soul, and the things of the soul herself hang upon wisdom... and so wisdom is inferred to be that which profits- and virtue, as we say is profitable... (we) thus arrive at the conclusion that virtue is either wholly or partly wisdom" (Plato 80)