diff --git a/CHANGES.md b/CHANGES.md
index 50a0499..c2422f7 100644
--- a/CHANGES.md
+++ b/CHANGES.md
@@ -3,6 +3,10 @@ This file describes changes in the smallgrp package.
 # 1.6dev
 
   - Require GAP 4.12
+  - `NumberSmallGroups` now accepts a list of orders and the selection
+    criteria of `AllSmallGroups`, as in `NumberSmallGroups(96, IsAbelian)`.
+    It counts by listing the ids, so it costs what `IdsOfAllSmallGroups`
+    costs (issue #68)
   - Selection by `IsAbelian`, `IsNilpotentGroup`, `IsSupersolvableGroup`,
     `IsSolvableGroup`, `RankPGroup` and `PClassPGroup` now avoids
     constructing the groups for most orders which had no precomputed data,
diff --git a/gap/small.gd b/gap/small.gd
index 4ef887f..63b2698 100644
--- a/gap/small.gd
+++ b/gap/small.gd
@@ -703,6 +703,10 @@ DeclareGlobalFunction( "SelectSmallGroups" );
 ##  <P/>
 ##  <Ref Func="SmallGroupsInformation"/> reports which criteria are indexed
 ##  for a given order.
+##  <P/>
+##  <Ref Func="IdsOfAllSmallGroups"/> returns the ids of these groups instead
+##  of the groups themselves, and <Ref Func="NumberSmallGroups"/> just their
+##  number.
 ##  <Example><![CDATA[
 ##  gap> AllSmallGroups( 6 );
 ##  [ <pc group of size 6 with 2 generators>, 
@@ -790,11 +794,12 @@ DeclareSynonym( "IdsOfAllGroups", IdsOfAllSmallGroups );
 #############################################################################
 ##
 #F  NumberSmallGroups( <order> )
+#F  NumberSmallGroups( <arg> )
 ##
 ##  <#GAPDoc Label="NumberSmallGroups">
 ##  <ManSection>
-##  <Func Name="NumberSmallGroups" Arg='order'/>
-##  <Func Name="NrSmallGroups" Arg='order'/>
+##  <Func Name="NumberSmallGroups" Arg='order[, func1, val1, ...]'/>
+##  <Func Name="NrSmallGroups" Arg='order[, func1, val1, ...]'/>
 ##
 ##  <Description>
 ##  returns the number of groups of order <A>order</A>.
@@ -806,6 +811,21 @@ DeclareSynonym( "IdsOfAllGroups", IdsOfAllSmallGroups );
 ##  gap> NumberSmallGroups( 4096 );
 ##  Error, the library of groups of size 4096 is not available
 ##  ]]></Example>
+##  <P/>
+##  The function also accepts the arguments of
+##  <Ref Func="AllSmallGroups"/> and counts the groups that function would
+##  return.
+##  <Example><![CDATA[
+##  gap> NumberSmallGroups( [ 1 .. 100 ] );
+##  1048
+##  gap> NumberSmallGroups( 96, IsAbelian );
+##  7
+##  ]]></Example>
+##  <P/>
+##  Orders alone are answered from the numbers of groups. Otherwise the
+##  cost is that of <Ref Func="IdsOfAllSmallGroups"/>: the indexed criteria
+##  are applied first, whatever order the criteria are given in, and only
+##  the groups a remaining criterion must be tested on are constructed.
 ##  </Description>
 ##  </ManSection>
 ##  <#/GAPDoc>
diff --git a/gap/small.gi b/gap/small.gi
index a396f17..ccdae55 100644
--- a/gap/small.gi
+++ b/gap/small.gi
@@ -638,16 +638,12 @@ end );
 
 #############################################################################
 ##
-#F  NumberSmallGroups(<size>)
+#F  SMALL_GROUPS_NUMBER( <size> )
 ##
-##  returns the  number of groups of the order <size>.
-InstallGlobalFunction( NumberSmallGroups, function( size )
+SMALL_GROUPS_NUMBER := function( size )
     local inforec;
 
-    if not IsPosInt( size ) then 
-        Error( "usage: NumberSmallGroups( order )" ); 
-    fi;
-    if size = 1024 then 
+    if size = 1024 then
         return 49487367289;
     fi;
 
@@ -656,10 +652,51 @@ InstallGlobalFunction( NumberSmallGroups, function( size )
         Error( "the library of groups of size ", size, " is not available" );
     fi;
 
-    if IsBound( inforec.number ) then 
+    if IsBound( inforec.number ) then
         return inforec.number;
     fi;
     return NUMBER_SMALL_GROUPS_FUNCS[ inforec.func ]( size, inforec ).number;
+end;
+
+#############################################################################
+##
+#F  NumberSmallGroups(<size>)
+#F  NumberSmallGroups(<arg>)
+##
+##  returns the number of the groups 'AllSmallGroups' would return for <arg>.
+InstallGlobalFunction( NumberSmallGroups, function( arg )
+    local argl, sizes;
+
+    # the classical form, just an order
+    if Length( arg ) = 1 and IsPosInt( arg[ 1 ] ) then
+        return SMALL_GROUPS_NUMBER( arg[ 1 ] );
+    fi;
+
+    # 'Size' may precede the orders
+    argl := arg;
+    if Length( argl ) > 0 and argl[ 1 ] = Size then
+        argl := argl{ [ 2 .. Length( argl ) ] };
+    fi;
+
+    # without a criterion the numbers of the orders just add up. A single
+    # list argument is the orders; a second one would be group numbers and
+    # is left to 'SelectSmallGroups'.
+    sizes := fail;
+    if Length( argl ) = 1 and IsList( argl[ 1 ] ) then
+        sizes := argl[ 1 ];
+    elif Length( argl ) > 0 and ForAll( argl, IsPosInt ) then
+        sizes := argl;
+    fi;
+    if sizes <> fail and ForAll( sizes, IsPosInt ) then
+        return Sum( sizes, SMALL_GROUPS_NUMBER );
+    fi;
+
+    if Length( arg ) <= 1 then
+        Error( "usage: NumberSmallGroups( order )" );
+    fi;
+
+    # so far counting the groups is not cheaper than listing their ids
+    return Length( SelectSmallGroups( arg, true, true ) );
 end );
 
 #############################################################################
diff --git a/tst/small.tst b/tst/small.tst
index 95d9064..7ceb05d 100644
--- a/tst/small.tst
+++ b/tst/small.tst
@@ -104,10 +104,18 @@ gap> IdGroup(G);
 ################################################################################
 # NumberSmallGroups GlobalFunction
 ################################################################################
+gap> NumberSmallGroups();
+Error, usage: NumberSmallGroups( order )
 gap> NumberSmallGroups(fail);
 Error, usage: NumberSmallGroups( order )
 gap> NumberSmallGroups(0);
 Error, usage: NumberSmallGroups( order )
+gap> NumberSmallGroups(Size);
+Error, usage: NumberSmallGroups( order )
+gap> NumberSmallGroups("abc");
+Error, usage: NumberSmallGroups( order )
+gap> NumberSmallGroups([ 1, fail ]);
+Error, usage: NumberSmallGroups( order )
 gap> NumberSmallGroups(19);
 1
 gap> NumberSmallGroups(1024);
@@ -121,6 +129,50 @@ gap> NumberSmallGroups(5^6);
 gap> NumberSmallGroups(5*7*9*11*13);
 22
 
+# orders alone need no group at all
+gap> NumberSmallGroups([]);
+0
+gap> NumberSmallGroups([ 1 .. 100 ]);
+1048
+gap> NumberSmallGroups(Size, [ 1 .. 100 ]);
+1048
+gap> NumberSmallGroups(60, 1024);
+49487367302
+gap> NumberSmallGroups(Size, [ 1000, 3996 ]);
+Error, the library of groups of size 3996 is not available
+
+# with selection criteria
+gap> NumberSmallGroups(96, IsAbelian);
+7
+gap> NumberSmallGroups(96, IsAbelian, false);
+224
+gap> NrSmallGroups(96, IsAbelian) + NrSmallGroups(96, IsAbelian, false)
+>      = NrSmallGroups(96);
+true
+gap> NumberSmallGroups(Size, [ 1 .. 100 ], IsSolvableGroup, false);
+1
+gap> NumberSmallGroups(Size, 60, IsSupersolvableGroup, true);
+11
+gap> ForAll([ 1 .. 100 ], n -> NumberSmallGroups(n, IsNilpotentGroup)
+>          = Length(IdsOfAllSmallGroups(n, IsNilpotentGroup)));
+true
+
+# the indexed criteria come first, whatever order they are given in
+gap> NumberSmallGroups(512, IsAbelian, true, Exponent, 2);
+1
+gap> NumberSmallGroups(512, Exponent, 2, IsAbelian, true);
+1
+
+# a list of group numbers may narrow down the selection
+gap> NumberSmallGroups(60, [ 1 .. 10 ]);
+10
+gap> NumberSmallGroups(60, [ 1 .. 10 ], IsNilpotentGroup);
+1
+
+# selection criteria need the library, even where the number does not
+gap> NumberSmallGroups(1024, IsAbelian);
+Error, AllSmallGroups / OneSmallGroup: groups of order 1024 not available
+
 ################################################################################
 # SelectSmallGroups GlobalFunction (called by AllSmallGroups or OneSmallGroup)
 ################################################################################