I have a few observations about this question, but only time today to write down one of them. For this I will write $H\cap K$ for the intersection of two subgroups (just as everyone else does), and write $H+K$ for the join of the subgroups.
Theorem. Let $G$ be a group, and let $(P_1,Q_1)$ and $(P_2,Q_2)$ be two pairs of complementary normal subgroups (a.k.a. pairs of complementary direct factor subgroups of $G$). If $P = P_1\cap P_2$ and $Q = Q_1+Q_2$, then
- $[G,G]\subseteq P+Q$.
- $[P\cap Q,P+Q] = \{1\}$.
Therefore, if $G$ is any (finite) centerless, perfect group, then $G$ is a $\mathcal D$-group.
Proof: For the first item, $$ \begin{array}{rl} [G,G]&=[P_1+Q_1,P_2+Q_2]\\ &=[P_1,P_2]+[P_1,Q_2]+[Q_1,P_2]+[Q_1,Q_2]\\ &\leq [P_1,P_2]+Q\\ &=(P_1\cap P_2) + Q = P+Q. \end{array} $$ Here I am using the additivity of the commutator, the fact that $[H,K]\leq H\cap K$, and the fact that $Q_1, Q_2\leq Q$.
For the second item, $[P,Q_1] \leq P\cap Q_1 \leq P_1\cap Q_1 = \{1\}$. Similarly $[P,Q_2] = \{1\}$. By the additivity of the commutator, $[P,Q]=[P,Q_1+Q_2]=[P,Q_1]+[P,Q_2]=\{1\}$. Now let $Z=P\cap Q$, which is $\leq P$ or $Q$. From the last two sentences and the monotonicity of the commutator in each variable we deduce $[Z,Q]\leq [P,Q] = \{1\}$ and $[Z,P]\leq [Q,P]=[P,Q]=\{1\}$, so by additivity we get $$[P\cap Q,P+Q]=[Z,P+Q]=[Z,P]+[Z,Q]=\{1\}.$$ This is the assertion to be proved.
For the final sentence of the proof, let $G$ be a perfect group ($[G,G]=G$) that is also a centerless group ($[G,N]=\{1\}$ implies $N=\{1\}$). Using the perfectness of $G$, the first item of the theorem can be written $G\subseteq P+Q$. Using this (i.e. $G=P+Q$), the second item can be written $[P\cap Q,G]=\{1\}$, or $P\cap Q\leq Z(G)$. Using the centerlessness of $G$ we get $P\cap Q=\{1\}$. Altogether we obtain that $P=P_1\cap P_2$ and $Q=Q_1+Q_2$ are complementary normal subgroups of $G$. This shows that the collection of factor congruences is closed under $\cap$ and $+$, so $G$ is a $\mathcal D$-group \\\
[One can go a bit further and show that the lattice of factor subgroups of a perfect, centerless group is a complemented distributive sublattice of ${\mathcal N}(G)$.]