Heptiamonds and heptahexes
The heptiamonds can fill heptahexes when a hex is composed of 24 triangles, i. e. with a side two. Many examples with symmetry can be seen on David Goodger's site. Here are some other ones, namely the two holey heptahexes:
There are exactly 333 heptahexes. Since a side 2 hexagon has 24 triangles, it is possible to tile a side 2 heptahex with the heptiamonds. This is possible for all of the shapes: a full list is available on a separate page.
Patrick M. Hamlyn has counted the total number of solutions for the stick heptahex. There are millions of them, which implies that the situation is no different for more complex heptahexes.