Enumeration of winning coalitions
You are asking two questions:


How to count "winning groups" up to isomorphism?
Which winning groups are realizable as winning coalitions?


From the point of view of winning groups, we can assume that each voter always casts all her votes, and the resulting games is known as a weighted voting game.

Counting winning groups

Your concept of "winning group" is a same as a monotone Boolean function or an upset. The number of monotone Boolean functions up to permutation is enumerated in the sequence A003182, and grows pretty quickly. For $n=7$ the number is already pretty large: 490,013,148. It's probably not too helpful to list all of them. It is even non-trivial just to calculate the number, see this paper.

You can find all the 16351 non-constant monotone Boolean functions for $n=6$ listed here as sets of minterms, which are (in your parlance) minimal winning coalitions. The minterms form what is known as an antichain, a collection of sets none of which is a subset of another.

Which winning groups are achievable?

Let $S_1,S_2$ be two sets of voters. A swap consists of taking some $x \in S_1 \setminus S_2$ and some $y \in S_2 \setminus S_1$, and forming the new sets $S_1 - x + y, S_2 - y + x$ (that is, we exchange two voters, taking care not to have the same voter twice in any of the sets). Given several sets $S_1,\ldots,S_k$ of voters, a trade is the result of applying arbitrarily many swaps, resulting in a new $k$-tuple $T_1,\ldots,T_k$.

(Alternatively, $T_1,\ldots,T_k$ is a trade of $S_1,\ldots,S_k$ if the multisets $S_1 \cup \cdots \cup S_k$ and $T_1 \cup \cdots \cup T_k$ are the same.)

Let $W$ be a set of "winning coalitions". We say that $W$ is trade robust if whenever $S_1,\ldots,S_k \in W$, in any trade $T_1,\ldots,T_k$ obtained from $S_1,\ldots,S_k$ at least one of the $T_i$ is in $W$. It is easy to see that every set of winning coalitions arising from weighted voting games is trade robust (exercise). Amazingly, Taylor and Zwicker showed that the converse holds as well.