Does culture define institutions or do institutions define culture?

To answer the question whether culture defines institutions or whether institutions define culture I have to go back to my previous post where I gave my definition of culture. I define culture as symbolic communication. Some of its symbols include a group's skills like knowledge, attitudes, values, and motives and many more. To my mind these meanings of the symbols are learned and internalized by a society through its institutions. Of course culture also had and still has a big impact on the institutions, since a culture can also change over time and so do the institutions as a result. Thus, it can be said that the institutions have been established through culture and are influenced by it. But for the following generations it can be assumed that the knowledge is being passed on to them through the institutions. For example schools carry and convey the values of a certain society and teach them so the culture lives on through them. To support my thesis that institutions define culture is that it is mostly the institutions which stand for and represent a certain culture. The following example should make my statement more comprehensible. For instance, in Nazi-German times, culture (the culture the Nazis wished to have) was “partly” conveyed through certain organizations/institutions like the “BDM” Bund Deutscher Mädel (League of German Girls). Every girl from 10 – 18 years was obliged to join the BDM. They learned how to serve their country and how to represent themselves as German females. So this institution was a big part of their life and shaped their characters and defined them as German. Today a good institution, which defines Germany, would be the Goethe-Institut and there are many more institutions which were prompted in the post-war years to account for the past and to create a new and better culture.