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Did the Nazis fund Lufthansa?

Firstly, I apologise for the nature of the question; I don't wish to upset anybody, it's just something that interests me.

During WWII, a number of German companies received financial backing and slave labourers from Hitler. Companies such as Hugo Boss, Volkswagen and Lufthansa.

A thread on Airliners.net from 1999 mentions a documentary showing Lufthansa planes flying with the Swastika on their tail. I found this image online:

What is the story behind Lufthansa's involvement with the Nazis?

How were they allowed to continue flying after WWII?