Congruence condition summary for groups of prime-cube order

Case of the prime 2
We first provide the congruence condition summary for groups of order 8, i.e., the case where $$p = 2$$.

There are five groups of order 8: cyclic group:Z8, direct product of Z4 and Z2, dihedral group:D8, quaternion group, and elementary abelian group:E8. There are thus $$2^5 - 1 = 31$$ possible collections of groups. Instead of listing all 31, we simply note the ones that do satisfy a universal congruence condition or a congruence condition to an interesting restricted class:

Case of the prime 3
We now provide the congruence condition summary for groups of order 27, i.e., the case $$p = 3$$.

Case of primes greater than 3
For any prime $$p \ge 5$$, the congruence condition summary for groups of order $$p^3$$ looks the same. It is given below.

First, note that there are five groups of order $$p^3$$. The three abelian groups are cyclic group of prime-cube order, direct product of cyclic group of prime-square order and cyclic group of prime order, and elementary abelian group of prime-cube order. The two non-abelian groups are prime-cube order group:U(3,p) (this has exponent $$p$$) and semidirect product of cyclic group of prime-square order and cyclic group of prime order (this has exponent $$p^2$$).