There is a standard convention that the icosahedral groups are classified separately from the cubic groups, but these two symmetry types have been conflated as "cubic" in some chemistry textbooks. In this note, the connection between cubic and icosahedral symmetries is examined, using a simple pictorial model. It is shown that octahedral and icosahedral groups have a tetrahedral subgroup in common; in the full groups, this is the group "T[subscript h]", or, using just the rotation groups, the group "T". In the analysis, various points emerge that may be of wider use in the teaching of symmetry. The misassignment of icosahedral symmetry as "cubic" is traced to a significant article in virology, whose authority has been great enough to allow the error to spread widely, and the nature of this mistake is analyzed. Reasons to maintain the conventional separation of cubic and icosahedral groups are given. (Contains 11 notes, 1 table and 2 figures.)