If subtype stands alone, then it isn't subordinate to type and if you want to relate many types with many othertypes, then yes there is a m-m relationship. But "subtype" would be assumed to belong to type and therefore NOT stand alone. For example, there are many cities in the US named Lincoln. is one Lincoln the same a another? NO, each Lincoln is different and exists in a different state. So State is "type" and city is "subtype". They have a 1-m relationship, not a m-m. We have a State (type) table. It has 51 entries (50 states plus DC or more entries if it includes dependencies like Puerto Rico). In the City table (subtype) there is a Foreign Key that points to the State in which the city exists.
Don't be confused by the subtypes having the same name.