Congruence condition summary for groups of prime-cube order

This article gives specific information, namely, congruence condition summary, about a family of groups, namely: groups of prime-cube order.
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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:

Collection	Does it satisfy a universal congruence condition?	Restricted class of groups in which it satisfies a congruence condition	Explanation
elementary abelian group:E8 only	No	abelian groups	congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group explains why it is true in abelian groups. The smallest failure among non-abelian groups occurs in direct product of D8 and Z2
direct product of Z4 and Z2, elementary abelian group:E8	Yes	all groups	congruence condition on number of abelian subgroups of order eight and exponent dividing four
cyclic group:Z8, direct product of Z4 and Z2, elementary abelian group:E8 (i.e., all the abelian groups of order 8)	Yes	all groups	congruence condition on number of abelian subgroups of prime-cube order
cyclic group:Z8, direct product of Z4 and Z2, dihedral group:D8, quaternion group, and elementary abelian group:E8	Yes	all groups	congruence condition on number of subgroups of given prime power order

Case of the prime 3

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

Collection	Does it satisfy a universal congruence condition?	Restricted class of groups in which it satisfies a congruence condition	Explanation
elementary abelian group:E27 only	Yes	all groups	congruence condition on number of elementary abelian subgroups of prime-cube and prime-fourth order
direct product of Z9 and Z3 and elementary abelian group:E27	Yes	all groups	Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime
cyclic group:Z27, direct product of Z9 and Z3, elementary abelian group:E27	Yes	all groups	Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
direct product of Z9 and Z3, prime-cube order group:U(3,3), M27, elementary abelian group:E27	Yes	all groups	congruence condition on number of non-cyclic subgroups of prime-cube order for odd prime
all groups of order 27	Yes	all groups	congruence condition on number of subgroups of given prime power order
prime-cube order group:U(3,3) and elementary abelian group:E27	No	groups of nilpotency class two	congruence condition on number of subgroups of given order and bounded exponent in class two group for odd prime Why it's not true in general: Congruence condition fails for subgroups of order p^p and exponent p. In general, we can get a failure inside wreath product of Z3 and Z3 (order 81).

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 $p^2$).

Collection	Does it satisfy a universal congruence condition?	Restricted class of groups in which it satisfies a congruence condition	Explanation
elementary abelian group of prime-cube order only	Yes	all groups	congruence condition on number of elementary abelian subgroups of prime-cube and prime-fourth order
direct product of cyclic group of prime-square order and cyclic group of prime order and elementary abelian group of prime-cube order	Yes	all groups	Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime
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	Yes	all groups	Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
direct product of cyclic group of prime-square order and cyclic group of prime orderelementary abelian group of prime-cube order, 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 $p^2$)	Yes	all groups	congruence condition on number of non-cyclic subgroups of prime-cube order for odd prime
all groups of order $p^3$	Yes	all groups	congruence condition on number of subgroups of given prime power order
prime-cube order group:U(3,p) (this has exponent $p$) and elementary abelian group of prime-cube order	Yes	all groups	Mann's replacement theorem for subgroups of prime exponent