In their book Subgroup Growth, Lubotzky and Segal asked: What are the possible types of subgroup growth of the pro-
$p$
group? In this paper, we construct certain extensions of the Grigorchuk group and the Gupta–Sidki groups, which have all possible types of subgroup growth between
$n^{(\log n)^{2}}$
and
$e^{n}$
. Thus, we give an almost complete answer to Lubotzky and Segal’s question. In addition, we show that a class of pro-
$p$
branch groups, including the Grigorchuk group and the Gupta–Sidki groups, all have subgroup growth type
$n^{\log n}$
.