Recall that an H-group is a space $X$ (in the sense of homotopy theory, so say CW complex) that is a group object in the homotopy category. I.e., there's a multiplication map $X \times X \rightarrow X$ which is associative up to homotopy and with an inverse map again up to homotopy. The standard example of an H-group is a loop space $\Omega X$. What is a simple example of an H-group that is not a loop space?
If I understand the language right, I'm asking for a group-like $A_3$ algebra that's not $A_\infty$. If I were asking for $A_2$ but not $A_3$ then I know $S^7$ is a good example.
My motivation is illustrating some of the subtleties in defining $\infty$-groups for a HoTT seminar.