A nice graphical map can be found in the article "Relationships Among Common Univariate Distributions", American Statistician 40:143-146.
This link provides a detailed chart by Leemis in 2008.
See also the jpg file 'distribution_relationships' at the bottom of this page.
The site below has a nice tool for showing graphically the distributions in the R package, and how changes to their parameters affects the distribution.
If the site doesn't show in the frame below, use this link: http://shiny.unterrichtsmedien.ch/distributions/
One of the more interesting aspects is the effect of the noncentrality parameter.