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I have to determine all Hasse diagrams of all non-isomorphic lattices with less than or equal to 6 elements. In order to determine the diagrams I systematically added characteristics to the diagrams and I think I have found all of them now. But I also need to prove the that my list is complete.

Does anyone have an idea to derive how many Hasse diagrams exist? (I have already proven that they are not isomorphic to each other)

I found 1 diagram for the 1-element lattices, 1 for the 2-element lattices, 1 for the 3-element lattices, 2 for the 4-element lattices, 5 for the 5-element lattices and 15 for the 6-element latices.

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According to this paper (there is a list in the end, and there are results to reach it in the text), the numbers you found are correct, for unlabelled lattices.
The list goes on up to $n=18$ and contains also the numbers for labelled ones.

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