Finite Group Databasebeta

Isoclinism class

Groups whose commutator structure agrees: there are isomorphisms G/Z(G) → H/Z(H) and G′ → H′ carrying one commutator map to the other. The class spans orders — G × A is isoclinic to G for every abelian A.

Stem order 64: every member's order is a multiple of it, and the members attaining it — the stem groups — are marked ◆.

4,890 groups of the catalog are in this class; the 500 smallest with no abelian direct factor are listed.

grouporder× stem
D4264×1◆ stem
#54664×1◆ stem
#54764×1◆ stem
#54864×1◆ stem
D4×Q864×1◆ stem
#55064×1◆ stem
#55164×1◆ stem
#55264×1◆ stem
#55364×1◆ stem
#55464×1◆ stem
#55564×1◆ stem
#55664×1◆ stem
#55764×1◆ stem
Q8264×1◆ stem
#55964×1◆ stem
D4×C22⋊C4128×2
#2339128×2
Q8×C22⋊C4128×2
#2341128×2
#2342128×2
#2343128×2
#2344128×2
#2345128×2
#2346128×2
#2347128×2
D4×C4⋊C4128×2
#2349128×2
Q8×C4⋊C4128×2
#2351128×2
#2352128×2
#2353128×2
#2354128×2
#2355128×2
#2356128×2
#2357128×2
#2358128×2
#2359128×2
#2360128×2
#2361128×2
#2362128×2
#2363128×2
#2364128×2
#2365128×2
#2366128×2
#2367128×2
#2368128×2
#2369128×2
#2370128×2
#2371128×2
#2372128×2
#2373128×2
#2374128×2
#2880128×2
#2906128×2
#2924128×2
#2925128×2
#2933128×2
D4×M16128×2
#2935128×2
#2962128×2
Q8×M16128×2
#2968128×2
#2969128×2
#2970128×2
#2971128×2
#2974128×2
#2977128×2
#2979128×2
#2984128×2
#2986128×2
#2990128×2
#2991128×2
#2994128×2
D4×#41128×2
#3475128×2
Q8×#41128×2
#3480128×2
#3481128×2
#3482128×2
#3483128×2
C22⋊C42256×4
#13799256×4
C22⋊C4×C4⋊C4256×4
#13801256×4
#13802256×4
C4⋊C42256×4
#13804256×4
#13805256×4
#13806256×4
#13807256×4
D4×#95256×4
#13809256×4
#13810256×4
Q8×#95256×4
#13812256×4
#13813256×4
#13814256×4
#13815256×4
#13816256×4
#13817256×4
#13818256×4
#13819256×4
#13820256×4
#13821256×4
#13822256×4
#13823256×4
#13824256×4
#13825256×4
#13826256×4
#17320256×4
#17325256×4
#17326256×4
#17327256×4
#17330256×4
#17331256×4
M162256×4
#17337256×4
#17341256×4
#20336256×4
#20368256×4
#20378256×4
#20379256×4
#20387256×4
C22⋊C4×M16256×4
#20389256×4
#20430256×4
#20431256×4
#20439256×4
C4⋊C4×M16256×4
#20441256×4
#20468256×4
#20469256×4
#20470256×4
#20478256×4
#20479256×4
#20480256×4
#20481256×4
#20482256×4
#20490256×4
#20491256×4
#20492256×4
#20493256×4
#20494256×4
#20495256×4
D4×#97256×4
#20519256×4
#20520256×4
#20521256×4
Q8×#97256×4
#20525256×4
#20528256×4
#20530256×4
#20535256×4
#20538256×4
#20540256×4
#20545256×4
#20546256×4
#20550256×4
#20552256×4
#20557256×4
#20558256×4
#20559256×4
#20562256×4
#20567256×4
#20571256×4
#20572256×4
#20573256×4
D4×C22⋊C8256×4
#22469256×4
Q8×C22⋊C8256×4
#22530256×4
#22552256×4
#22589256×4
#22599256×4
#22625256×4
#22632256×4
#22639256×4
D4×C4⋊C8256×4
#22696256×4
Q8×C4⋊C8256×4
#22734256×4
#22748256×4
#22785256×4
#22795256×4
#22821256×4
#22828256×4
#22835256×4
#22842256×4
#22900256×4
#22952256×4
#22985256×4
#23050256×4
#23083256×4
#23116256×4
#23149256×4
#23173256×4
#23204256×4
#23237256×4
#23247256×4
#23253256×4
#23263256×4
#23269256×4
#23278256×4
#23294256×4
#33334256×4
#33398256×4
#33422256×4
#33423256×4
#33436256×4
D4×M32256×4
#33438256×4
#33476256×4
Q8×M32256×4
#33482256×4
#33483256×4
#33484256×4
#33485256×4
#33488256×4
#33491256×4
#33493256×4
#33498256×4
#33500256×4
#33504256×4
#33505256×4
#33508256×4
C22⋊C4×#41256×4
#34109256×4
C4⋊C4×#41256×4
#34121256×4
D4×#117256×4
#34123256×4
Q8×#117256×4
#34125256×4
#34126256×4
#34127256×4
#34128256×4
#34129256×4
#34136256×4
#34141256×4
#34145256×4
#34149256×4
#34151256×4
#34152256×4
#34153256×4
#34154256×4
#34155256×4
#60127256×4
D4×#131256×4
Q8×#131256×4
#60210256×4
M16×#41256×4
#60215256×4
#60218256×4
#60278256×4
#60280256×4
#60305256×4
#60308256×4
#60331256×4
#412256×4
C22⋊C4×#95512×8
C4⋊C4×#95512×8
#153725512×8
#153726512×8
#153727512×8
#153728512×8
#153729512×8
#180797512×8
#180802512×8
#180803512×8
#180804512×8
#180805512×8
#180808512×8
#180809512×8
M16×#97512×8
#180812512×8
#180817512×8
#180820512×8
#180821512×8
#180824512×8
#180827512×8
#180828512×8
#180834512×8
#180837512×8
#180841512×8
#180844512×8
#180849512×8
#181806512×8
#181889512×8
#181934512×8
M16×C22⋊C8512×8
#183054512×8
#183063512×8
#183285512×8
#183330512×8
M16×C4⋊C8512×8
#183548512×8
#183557512×8
#183728512×8
#183729512×8
#183730512×8
#183793512×8
#183809512×8
#183818512×8
#183823512×8
#183833512×8
#183841512×8
#183842512×8
D4×#322512×8
#183844512×8
#184052512×8
#184053512×8
#184054512×8
#184372512×8
#184373512×8
Q8×#322512×8
#184381512×8
#184427512×8
#184428512×8
#184432512×8
#184472512×8
#184476512×8
#184485512×8
#184486512×8
#184490512×8
#184492512×8
#184508512×8
#184579512×8
#184580512×8
#184582512×8
#184607512×8
#184608512×8
#184690512×8
#184691512×8
#184692512×8
#353115512×8
#353120512×8
M16×#95512×8
#353165512×8
#353170512×8
#353171512×8
#353178512×8
#353179512×8
#353200512×8
C22⋊C4×#97512×8
#353204512×8
#353205512×8
#353206512×8
#353218512×8
#353219512×8
#353235512×8
C4⋊C4×#97512×8
#353239512×8
#353240512×8
#353241512×8
#353250512×8
#353252512×8
#353256512×8
#353257512×8
#353261512×8
#353263512×8
#353268512×8
#353270512×8
#353274512×8
#353275512×8
#353279512×8
#353281512×8
#356050512×8
C22⋊C4×C22⋊C8512×8
#356111512×8
#356118512×8
#356125512×8
C22⋊C4×C4⋊C8512×8
C4⋊C4×C22⋊C8512×8
#356175512×8
#356246512×8
#356253512×8
#356260512×8
#356296512×8
#356303512×8
#356310512×8
#356336512×8
C4⋊C4×C4⋊C8512×8
#356376512×8
#356383512×8
#356390512×8
#356416512×8
#356421512×8
#356429512×8
#356436512×8
#356443512×8
#356450512×8
#356455512×8
D4×#336512×8
#356528512×8
#356541512×8
Q8×#336512×8
#356582512×8
#356589512×8
#356624512×8
#356684512×8
#356728512×8
#356754512×8
#356779512×8
#356799512×8
#356824512×8
#356844512×8
#356856512×8
#356886512×8
#356900512×8
#356916512×8
#356926512×8
#356942512×8
#356971512×8
#357018512×8
#357049512×8
#357071512×8
#357096512×8
#357116512×8
#357134512×8
#357180512×8
#357210512×8
#357228512×8
#357251512×8
#357271512×8
#479903512×8
#479918512×8
M16×M32512×8
#479920512×8
#479928512×8
#479929512×8
#479933512×8
#479934512×8
#479938512×8
#479943512×8
#479944512×8
#479945512×8
#479947512×8
#479949512×8
#479951512×8
#479953512×8
#492839512×8
#492916512×8
#492938512×8
#492947512×8
#492968512×8
C22⋊C4×M32512×8
#492977512×8
#493096512×8
#493101512×8
#493120512×8
C4⋊C4×M32512×8
#493126512×8
#493170512×8
#493171512×8
#493172512×8
#493194512×8
#493195512×8
#493196512×8
#493197512×8
#493198512×8
#493208512×8
#493209512×8
#493210512×8
D4×#346512×8
#493221512×8
#493222512×8
#493223512×8
#493255512×8
#493256512×8
Q8×#346512×8
#493260512×8
#493266512×8
#493268512×8
#493271512×8
#493279512×8
#493281512×8
#493284512×8
#493291512×8
#493292512×8
#493296512×8
#493298512×8
#493305512×8
#493306512×8
#493310512×8
#493313512×8
#493314512×8
#493323512×8
#493324512×8
#493325512×8
D4×C22⋊C16512×8
#497465512×8
Q8×C22⋊C16512×8
#497626512×8
#497682512×8
#497736512×8
#497750512×8
#497788512×8
#497798512×8
#497810512×8
D4×C4⋊C16512×8

The class itself is infinite — G × A is isoclinic to G for every abelian A, and there are abelian groups of every order — so a table shows the part of it the catalog names, never the whole.