In the USA, these days, if a person has a last name made up of 2 surnames, often there is a dash (also called a hyphen) between the 2 surnames to keep them together so computer programs don't sort/list them properly. Example: Connor-Mackintosh; without the hyphen, a computer sorting program could list Mackintosh as just the last name and list Connor as a first name or a middle name.
Also, I have known people who have 2 first names with a hyphen between the 2 given names. Examples: Jean-Luc, Mary-Kay. Again, often the reason for that is to keep the 2 names together as a "name unit," if you will.
Computer sorting programs are notorious for not recognizing surnames with spaces or other kinds of punctuation. Examples: O'Brien - I have seen listed as last name: Brien, first name initial O; De La Torre listed as last name: Torre, first name De, middle name: La. It's annoying to say the least.
As for trying to find someone on Facebook, she may not have used her actual name, or if she is married she may have just used her maiden name for FB. Example: Elizabeth Connor or Elizabeth Mackintosh.
Suggestion: The next time you meet, be friendly and say "hello." If you did not introduce yourself before, do that this next time. Example: "My name is Joe Smith. Just call me J.S." She will probably introduce herself. Example: "Hello J.S. I'm Liz Connor-Mackintosh." (Or, "I'm Liz Mackintosh," or whatever her name is.)
Unless she suggests you two become FB friends when you are chatting face-to-face, you may reconsider trying to find her on FB. That could be considered stalking. :-) That is more of an issue of etiquette, I think.
Did you also have a genealogy question? Genealogy, where your question landed, is the study/research of a person's family tree/dead ancestors.
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Best wishes