Is there a classification of groups having the property that any set of $d$ elements (say including the identity) is contained in a proper subgroup?

It is appealing to call the maximum such integer (when finite) some sort of "dimension" or measure of being "not cyclic". As one example, we have elementary abelian groups of order $p^d$. I haven't thought much about nonabelian groups, but there are examples.

Disclaimers: Apologies if this is either a trivial classification or not of much research interest. It is perhaps well-studied already, but I don't know what to call this property. I thought I'd try to ask, since this is a group-theoretic analog of something else which interests me. Should I try group pub forum instead?

Thanks in advance.