The sphere centred on $(1,1,1)/2$ and radius $\sqrt{131}/2\approx 5.723$ contains $24+48+48=120$ points with integer coordinates, thanks to $$2\{\pm x,\pm y,\pm z\}-(1,1,1)=\{9,5,5\},\{9,7,1\},\{11,3,1\}.$$
Successive best $a,n$ where the sphere centred on $(1,1,1)/2$ and radius $\sqrt{a}/2$ contains $n$ points with integer coordinates:
(3,8),(11,24),(27,32),(35,48),(59,72),(131,120),(251,168),(299,192),(419,216),(611,240),(659,264),(731,288),(899, 336),(971,360),(1091,408),(1691,432),(1739,480),(1811,552),(2219,576),(2651,624),(2939,696),(3251,744),(4091,792),( 4259,840),(4619,864),(5099,936),(5771,1056),(6971,1080),(7619,1104),(8291,1128),(8531,1200),(9539,1320),(11051,1488),( 12011,1560),(13859,1608),(14339,1680),(15539,1728),(18851,1776),(19211,1800),(19379,1944),(20459,1968),(22571,2088),( 25091,2112),(25451,2160),(26171,2184),(26771,2352),(28019,2376),(31379,2472),(31979,2496),(33491,2592),(36539,2736),( 38099,2784),(38939,2832),(39731,3024),(42059,3120),(49139,3144),(51939,3168),(53819,3192),(55571,3360),(58211,3432),( 59219,3672),(65771,3696),(66491,3960),(74051,4200),(87779,4416)
GAP code:
a:=[];R:=89999;for i in [1..R] do a[i]:=0;od;
for x in [1,3..299] do
for y in [1,3..x] do
for z in [1,3..y] do
r:=x*x+y*y+z*z; if r<=R then
if x>y and y>z and z>0 then n:=48;
else if (x>z and z>0) or (x>y and z=0) then n:=24;
else if x=z then n:=8;
else if y=0 then n:=6;fi;fi;fi;fi;
a[r]:=a[r]+n;fi;od;od;od;
b:=0;for i in [1..R] do if a[i]>b then b:=a[i];Print("(",i,",",b,"),");fi;od;