Something that I think is important to consider when making a faction map as a play aid, is to decide what level of granularity do you need. A map that factions and relationships between those factions might miss that some factions contain significant and important sub-factions, the conflict between which is important to capture. However, there is a slippery slope that leads to a map that has a hundred different individuals and minute sub-factions listed and ends up being worthless at the table.
I'm currently working on some ideas for the format of faction maps, and I think that a lot of info can be conveyed using colour and size. For example, I made the following (very rough) faction map for an OSR town I wrote up for a Black Hack campaign (those of you who just saw this in a different thread will have to forgive me).
View attachment 261189
So all the blue boxes are parts of the Council faction, but just having one box wouldn't have captured the interplay between those parts. There is an additional layer of individual relationships that seemed like too much, so I left it out (it would be covered in the NPC sketches). The arrows show positive, negative, and conflicted relationships as well as directionality, when I thought it mattered. There are also some sub-faction elements in the mining consortium that I forgot about. Anyway, this is the sort of thing I like to use, although ideally it would look somewhat neater.