The external conflict in Sophocles' Antigone between the title character and her sister Ismene is pretty straightforward, but it evolves slightly by the end of the play. In the beginning Antigone wishes to bury her brother—Polynices—while her sister insists that it is against the king's decree:
ISMENE: We are in the grip of those stronger than ourselves, and must obey them in this and in things still more cruel. Therefore I will ask forgiveness of the gods and spirits who dwell below, for they will see that I yield to force, and I will hearken to our rulers. It is foolish to be too zealous even in a good cause. (84–85)
This initial conflict is derived from the two sisters' separate notions of devotion to family. Antigone feels she owes "a longer allegiance to the dead than to the living," meaning she believes it is her duty to bury Polynices. Ismene, on the other hand, is worried that if her sister breaks the law, she too will follow in the footsteps of all the rest of her family members and "perish, more miserably than all the rest." Antigone resents her sister's stance and opts to go it alone.
Later in the play when Antigone is caught burying Polynices, Ismene changes her tune. Afraid she will be left all alone, she takes it upon herself to try to share in her sister's punishment. Antigone, however, rejects this:
ISMENE: I am guilty if she is, and share in the blame.
ANTIGONE: No, no! Justice will not permit this. You did not consent to the deed, nor would I let you have part in it.
ISMENE: But now that danger threatens you, I am not ashamed to come to your side.
ANTIGONE: Who did the deed, the gods and the dead know; a friend in words is not the friend I love. (93)
While admitting that she still loves her sister, Antigone refuses to acknowledge Ismene's attempt to stand by her. She does not feel that Ismene deserves to share in the punishment (or the "honor" of her deed). "You chose to live, and I to die," she says shortly after (94). This shows the evolution of the conflict; whereas before it involved the debate about whether to bury Polynices, not it revolves around Ismene taking equal credit and punishment for Antigone's actions.
Note: For this response, the above quotations come from Ten Greek Plays in Contemporary Translations, edited by L.R. Lind and published by the Houghton Mifflin Company.