Yes lipids is another name for fats

Here is the taxonomic hierarchy:

Domain

Kingdom

Subkingdom

Infrakingdom

Superphylum/Superdivision

Phylum/Division

Subphylum/Subdivision

Infraphylum/Infradivision

Superclass

Class

Subclass

Infraclass

Superorder

Order

Suborder

Infraorder

Superfamily

Family

Subfamily

Infrafamily

Genus

Species

Subspecies/Variety

Breed/Subvariety

Therefor, Breed and Subvariety are the most complex levels of biological organization.

species, genus, family, order, class, phylum, kingdom

Genus and species. If further ordering is required to distinguish an organism, such as the case of the gray wolf (Canis lupus) and the domestic dog (Canis lupus familiaris), subspecies/variety and breed/subvariety may be used (formatted as Genus species subspecies "Breed").

The scientific name of a species typically consists of two terms: the genus name and the species name. This binomial nomenclature system was introduced by Carl Linnaeus to uniquely identify and classify different species.