There's a brief broad outline - which is what I most like about it - at the Wikipedia article. To get the full classification, you need the books, but the full LCC isn't on the public web either (it can be subscribed to).

Anyway, I copied the abridged version, with some additions from the full original classification where my collection needed more detail, and use that. I don't use the revised edition, which is a faceted classification, and I don't like those.