Algorithmic approach to creating all combinations based on a large truth table

Apologies if this was given elsewhere, but I haven't found it yet using my limited algorithmic terminology.;)

My situation is this: I have a variable number of data items, each of which is checked against each other against data items to determine compatibility. Compatibility is stored in the equivalent of a two-dimensional array (truth table?). My goal is to create every possible combination of these items, where every item in combination is compatible with every other item.

For example, if item 1 (of 4) was compatible with items 2 and 4, and item 2 was compatible with 1, 3, and 4, item 3 was compatible with 2, and item 4 was compatible with 1 and 2, my truth table would be look something like this:

1) {1,1,0,1}
2) {1,1,1,1}
3) {0,1,1,0}
4) {1,1,0,1}

The combinations I want from this would be:
1.2.4
1.2
1.4
1
2.3
2.4
2
3
4

My approach works well in a lot of situations, but sometimes it gets very cluttered when the number of items exceeds 5000, depending on the datasets. My secondary concern is to define a pattern that provides a runtime of 5 seconds to 3 hours ...

Just by looking at a boolean array, I just feel like there MUST be an easier solution - perhaps an algorithm named for someone. As you might deduce from the above, I don't necessarily know how to ask the question.;)

Thank you for your time!