The primary conflict is how parents effectively justify their children fighting and dying in the nature of war. The main characters are a husband and wife who are in a railway car. The husband is explaining to the other passengers in the car that his wife is distraught at their son fighting in the war. The other primary character is a fat man who enters the train and offers his own analysis which is that parents have a responsibility to encourage their kids to fight for the nation and embrace the consequences of dying for national glory. He pontificates quite a great deal and is only disarmed by the wife's emotionally and rhetorically jarring question at the end of the story. The fundamental conflict of being able to stand by a nation while there is obvious conflict on an personal level of what is being done in the name of said nation becomes the overarching dilemma in the story.
To me, the main conflict in this story is between the various people on the train. They are arguing about how they should think of the war. They are also arguing about who should be saddest and whose situation (in terms of sons going off to war) is the worst.
I think this symbolizes a deeper conflict, though. It is a conflict between the people and society. The characters (they are all just anonymous people who have sons who have been called to fight or who have already been in the war) are showing the conflict between individuals' desires to have their own happiness and society's need to have its citizens fight for it in wars.