A visual atlas of all groups up to order 8 with their diagrams and properties
How many groups are there? For each order n, there are finitely many groups (up to isomorphism). Order 1 has one group, order 2 has one, order 4 has two (Z4 and V4), and order 8 has five! Explore the complete classification of small groups.
Browse the major families: cyclic groups (rotations), dihedral groups (symmetries of polygons), and symmetric groups (all permutations). See how each family grows.
Generated by a single element. Z_n = rotations of a regular n-gon.
Key insight: Cyclic groups are the simplest -- generated by a single element. Symmetric groups are the most complex -- S_n has n! elements and contains copies of every group of order up to n.
All 14 groups up to order 8, organized by order. Click any group to see its properties, description, and multiplication table.
All 14 groups up to order 8. Click one to see its properties and multiplication table.
Key insight: Groups of prime order are always cyclic (and simple). The first non-trivial choice appears at order 4: cyclic Z4 or Klein four-group V4. Order 8 has five distinct groups, including the famous quaternion group Q8.
Select any group from the zoo and see its computed properties: abelian, cyclic, simple, center size, number of subgroups, and element order distribution.
Key insight: The element order distribution is a powerful invariant. Z4 has one element of order 4; V4 has three elements of order 2. D4 and Q8 both have order 8 but different order distributions.