Congruence condition fails for number of subrings of given prime power order

Statement
It is possible to have a finite Lie ring $$L$$ (in fact, we can choose $$L$$ to have prime power order $$p^k$$) and a value $$p^r, 0 \le r \le k$$ such that the number of Lie subrings of $$L$$ of order $$p^r$$ is nonzero and not congruent to 1 mod $$p$$.

Related facts

 * Congruence condition on number of subgroups of given prime power order
 * Congruence condition on number of subrings of given prime power order in nilpotent Lie ring

For an odd prime
Let $$p$$ be an odd prime. Consider the special linear Lie ring of degree two over the prime field $$\mathbb{F}_p$$. As a set, this is the set of $$2 \times 2$$ trace zero matrices over the field of $$p$$ elements. The addition is given by matrix addition and the commutator is defined as $$[A,B] := AB - BA$$ with $$AB,BA$$ denoting usual matrix products.

This is a Lie ring of order $$p^3$$, with the additive group an elementary abelian group of prime-cube order.

The Lie ring has an additive basis of elements:

$$e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \\\end{pmatrix},\qquad f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \\\end{pmatrix}, \qquad h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}$$

and these satisfy the relations:

$$[e,f] = h, \qquad [h,e] = 2e, \qquad [h,f] = -2f$$

We consider subrings of order $$p^2$$:


 * 1) Any subring of order $$p^2$$ must contain the element $$h$$, because if an additive subgroup of order $$p^2$$ does not contain $$h$$, then the Lie bracket of any two basis vectors is a scalar multiple of $$h$$.
 * 2) We can choose a basis $$h, \alpha e + \beta f$$ for the subring. In that case, the Lie bracket of these two elements is $$[h,\alpha e + \beta f] = 2\alpha e - 2\beta f$$. For the latter element to also lie in the subring, we need that either $$\alpha = 0$$ or $$\beta = 0$$.
 * 3) Thus, the only possible subrings of order $$p^2$$ are $$\langle h,e \rangle$$ and $$\langle h,f \rangle$$. There are exactly two such subrings, which contradicts the congruence condition.