In the play Antigone, how does Creon show he his loyal to his family?

Early in the play, Creon proclaims that he is the kind of ruler who has

"nothing but contempt for the kind of Governor who is afraid, for whatever reason, to follow the course that he knows is best for the State; and as for the man who sets private friendship above the public welfare,"

but his actions in attempting to spare Antigone, his niece, from execution for defying his order contradict his proclamation of being the upholder of law. Because she is his brother's daughter, he attempts to reason with her so that he doesn't have to have her executed. However, Antigone will not break; she stands by the morality of her decision to bury her brother even though in doing so, she has defied the state.
It takes the deaths of his niece, Antigone; his son, Haemon; and his wife, Eurydice, for Creon to finally understand that his loyalty should have been to his family. He laments his blindness when he says,

"I have been rash and foolish. I have killed my son and my wife. I look for comfort; my comfort lies here dead. Whatever my hands have touched has come to nothing,"

but it is too late; in his zeal to demonstrate his decisiveness as a ruler, he has betrayed his loyalty to his family.