Finite Group Databasebeta

← #16776789

Subgroups — #16776789

Order 3328.

10698 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13129649C212yesmaximal, complemented
2#13129622C212yesmaximal, complemented
3#13129620C2112yesmaximal, complemented
4#13129618C2116yesmaximal, complemented
5#13129605C216yesmaximal, complemented
6#13129604C218yesmaximal, complemented
7C2×#11719458C211yesmaximal, characteristic, complemented
8C2×#11719476C216yesmaximal, complemented
9D13×#578C218yesmaximal, complemented
10#13129547C211yesmaximal, characteristic, complemented
11C13×#3584C211yesmaximal, characteristic, complemented
12C13×#585C2212yescomplemented
13D13×2−1+4—24no—
14D13×2−1+4C22116yescomplemented
15#11719650—212no—
16#11719650C22116yescomplemented
17#11719648—28no—
18#11719648C2211yescharacteristic
19#11719648C22112yescomplemented
20C2×#88257—28no—
21C2×#88257C22112yescomplemented
22C2×#88257C2212yescomplemented
23C13×#578C2214yescomplemented
24C13×#577C2218yescomplemented
25C13×#576C2214yescomplemented
26C13×#575C2216yescomplemented
27C2×C13×#135C2216yescomplemented
28C2×C13×#131C2211yescharacteristic, complemented
29#11719543C22116yescomplemented
30#11719541C22116yescomplemented
31C2×#88205C22112yescomplemented
32#11719527C22112yescomplemented
33#11719524C2211yescharacteristic, complemented
34#11719484C22148yescomplemented
35D13×#137C22148yescomplemented
36#11719480C22148yescomplemented
37#11719478C22148yescomplemented
38#11719476C22124yescomplemented
39#11719476C22112yescomplemented
40#11719476C22124yescomplemented
41#11719476C22124yescomplemented
42#11719475C22124yescomplemented
43#11719474C2214yescomplemented
44#11719473C22124yescomplemented
45D13×#135C22124yescomplemented
46#11719471C22112yescomplemented
47C2×Q16×D13C22112yescomplemented
48#11719467C22124yescomplemented
49C2×#88169C22124yescomplemented
50#11719462C22112yescomplemented
51C2×#88166C22112yescomplemented
52#11719458C2218yescomplemented
53#11719458C2218yescomplemented
54#11719455C2218yescomplemented
55D13×#131C2218yescomplemented
56C2×#88163C2216yescomplemented
57#11719450C2216yescomplemented
58C22×Dic52C2213yescomplemented
59C2×#88158C2213yescomplemented
60C2×#88155C2216yescomplemented
61C13×2−1+4—22no—
62C13×2−1+4C2318yes—
63C13×H(5,𝔽2)—26no—
64C13×H(5,𝔽2)C2318yes—
65C2×C13×#41—28no—
66C2×C13×#41C23112yes—
67C2×C13×#41C2311yescharacteristic
68#88257—28no—
69#88257—216no—
70#88257—248no—
71#88257—224no—
72#88257—224no—
73#88257—212no—
74#88257D414yes—
75#88257C23124yes—
76#88257C23148yes—
77#88257C23116yes—
78#88256—212no—
79#88256D414yes—
80#88256C2318yes—
81#41×D13—216no—
82#41×D13—248no—
83#41×D13—224no—
84#41×D13C23148yes—
85#41×D13C2318yes—
86#88254—216no—
87#88254D412yes—
88#88254C23124yes—
89C2×Q8×D13—22no—
90C2×Q8×D13—216no—
91C2×Q8×D13C23124yes—
92#88251—224no—
93#88251—248no—
94#88251D416yes—
95#88251C23124yes—
96C2×#6129—26no—
97C2×#6129—224no—
98C2×#6129—248no—
99C2×#6129C23124yes—
100C2×#6127—224no—
101C2×#6127—212no—
102C2×#6127C2316yes—
103C2×#6127C2316yes—
104C22×Dic26—212no—
105C22×Dic26C2316yes—
106C13×#137C23124yes—
107C13×AGL(1,ℤ/8)C23124yes—
108C13×#135C23112yes—
109C13×#135C23112yes—
110C13×#135C2314yes—
111C13×#135C23112yes—
112C2×C13×Q16C2316yes—
113C2×C13×SD16C23112yes—
114C2×C13×D8C2316yes—
115C13×#131C2314yes—
116C13×#131C2314yes—
117C2×C13×M16C2313yes—
118C22×C8×C13C2313yes—
119#88205C23124yes—
120#88205C23148yes—
121#88204C23148yes—
122#88203C2318yes—
123#88202C2318yes—
124C2×#6107C23124yes—
125#88196C23124yes—
126C2×#6105C23124yes—
127#88186C23124yes—
128C2×#6099C2316yes—
129#88173C2318yes—
130#88172C23148yes—
131Q16×D13C23124yes—
132Q16×D13C23148yes—
133#88170C23148yes—
134#88169C23148yes—
135#88169C23196yes—
136#88168C23116yes—
137SD16×D13C23148yes—
138#88166C23124yes—
139#88166C23148yes—
140#88165C23148yes—
141D8×D13C2318yes—
142#88163C23148yes—
143#88161C23124yes—
144M16×D13C23124yes—
145C2×Dic52C2313yes—
146C2×Dic52C2316yes—
147C2×Dic52C23112yes—
148C2×Dic52C23112yes—
149#88158C23112yes—
150#88158C23112yes—
151C2×D104C2311yescharacteristic
152C2×#6095C2316yes—
153#88155C2314yes—
154#88155C23112yes—
155#88155C23112yes—
156#88155C23112yes—
157C2×#6094C23112yes—
158C2×C8×D13C23112yes—
159#63093—131nomaximal, Sylow
160C13×#41—28no—
161C13×#41—224no—
162C13×#41—212no—
163C13×#41—212no—
164C13×#41C2×D414yes—
165C13×#41C2418yes—
166C13×#41C24124yes—
167C13×#41C2414yes—
168C2×Q8×C13—28no—
169C2×Q8×C13C2×D411yescharacteristic
170C2×Q8×C13C24112yes—
171C2×D4×C13—212no—
172C2×D4×C13—224no—
173C2×D4×C13C2×D413yes—
174C2×D4×C13C24112yes—
175C22×C4×C13—212no—
176C22×C4×C13C2413yes—
177C2×C22⋊D13—26no—
178C2×C22⋊D13—224no—
179C2×C22⋊D13—248no—
180C2×C22⋊D13C2×D4112yes—
181C22×Dic13—26no—
182C22×Dic13—224no—
183#6131—224no—
184#6131C2×D418yes—
185#6131C24116yes—
186Q8×D13—216no—
187Q8×D13—248no—
188Q8×D13—224no—
189Q8×D13C2×D418yes—
190Q8×D13C24148yes—
191#6129—248no—
192#6129—296no—
193#6129—248no—
194#6129—248no—
195#6129—248no—
196#6129—224no—
197#6129C2×D4124yes—
198#6129C24148yes—
199D4×D13—248no—
200D4×D13—216no—
201D4×D13—224no—
202D4×D13C2×D4124yes—
203D4×D13C24116yes—
204#6127—212no—
205#6127—248no—
206#6127—224no—
207#6127—248no—
208#6127—248no—
209#6127—28no—
210#6127C2×D4112yes—
211#6127C2×D4112yes—
212#6127C2414yes—
213#6127C24112yes—
214#6127C24124yes—
215C2×D52—28no—
216C2×D52C2×D416yes—
217C2×D52C2412yes—
218C2×C4×D13—224no—
219C2×C4×D13—26no—
220C2×C4×D13—248no—
221C2×C4×D13C24112yes—
222C2×Dic26—212no—
223C2×Dic26—26no—
224C2×Dic26—224no—
225C2×Dic26—224no—
226C2×Dic26—248no—
227C2×Dic26C2×D416yes—
228C2×Dic26C2416yes—
229C2×Dic26C24124yes—
230C13×Q16C2414yes—
231C13×Q16C24112yes—
232C13×SD16C24124yes—
233C13×SD16C2418yes—
234C13×D8C2414yes—
235C13×D8C24112yes—
236C13×M16C24112yes—
237C2×C8×C13C2416yes—
238C2×C8×C13C2416yes—
239C2×C8×C13C2413yes—
240C2×C8×C13C2411yescharacteristic
241#6107C24148yes—
242#6106C24116yes—
243#6105C24148yes—
244#6104C24116yes—
245#6099C2414yes—
246#6099C24112yes—
247C2×#1055C24112yes—
248Dic52C2416yes—
249Dic52C2416yes—
250Dic52C24124yes—
251D104C2412yes—
252D104C2412yes—
253#6095C24124yes—
254#6094C2418yes—
255#6094C24124yes—
256C8×D13C2418yes—
257C8×D13C24124yes—
2582−1+6—132no—
259#3586—132no—
260#3586—1316no—
261#3586—1312no—
262#3584—138no—
263#3584—136no—
264#3584D1311yescharacteristic, complemented
265C2×#578—138no—
266C2×#578—136no—
267C2×#578—131no—
268#3573—131no—
269C23×C13—212no—
270C23×C13C22×D413yes—
271C22×D13—22no—
272C22×D13—216no—
273C22×D13C22×D4112yes—
274Q8×C13—212no—
275Q8×C13C22×D414yes—
276Q8×C13C2518yes—
277D4×C13—28no—
278D4×C13—224no—
279D4×C13—212no—
280D4×C13C22×D4112yes—
281D4×C13C2518yes—
282C2×C4×C13—212no—
283C2×C4×C13—28no—
284C2×C4×C13—224no—
285C2×C4×C13C22×D416yes—
286C2×C4×C13C22×D413yes—
287C2×C4×C13C2516yes—
288C2×C4×C13C2511yescharacteristic
289C22⋊D13—224no—
290C22⋊D13—248no—
291C22⋊D13—216no—
292C22⋊D13C22×D4124yes—
293C22⋊D13C22×D418yes—
294C2×Dic13—26no—
295C2×Dic13—224no—
296C2×Dic13—248no—
297C2×Dic13C22×D4112yes—
298D52—216no—
299D52—22no—
300D52C22×D414yes—
301D52C22×D4112yes—
302D52C2514yes—
303C4×D13—248no—
304C4×D13—224no—
305C4×D13—216no—
306C4×D13C22×D4124yes—
307C4×D13C2518yes—
308Dic26—26no—
309Dic26—224no—
310Dic26—248no—
311Dic26C22×D414yes—
312Dic26C22×D4112yes—
313Dic26C25112yes—
314C8×C13C2511yescharacteristic
315C8×C13C2516yes—
316C8×C13C2511yescharacteristic
317#1055C2518yes—
318#585—131no—
319#585—1316no—
320#585—1312no—
321#585—268no—
322#585—2612no—
323#585C2×D1312yescomplemented
324C2×2−1+4—1316no—
325C2×2−1+4—1312no—
326C2×2−1+4—132no—
327C2×2−1+4—268no—
328C2×2−1+4—264no—
329#578—1316no—
330#578—1348no—
331#578—1324no—
332#578—1312no—
333#578—1324no—
334#578—1324no—
335#578—1312no—
336#578—138no—
337#578—138no—
338#578C2×D1314yescomplemented
339#577—1316no—
340#577—1348no—
341#577—1324no—
342#577—1324no—
343#577C2×D1318yescomplemented
344#576—134no—
345#576—1312no—
346#576C2×D1314yescomplemented
347#575—1312no—
348#575—1348no—
349#575—1324no—
350#575C2×D1316yescomplemented
351C2×#137—1312no—
352C2×#137—1348no—
353C2×#137—1324no—
354C2×#137—136no—
355C2×#135—1324no—
356C2×#135—1312no—
357C2×#135—133no—
358C2×#135C2×D1316yescomplemented
359C22×Q16—1312no—
360C22×Q16—133no—
361#568—131no—
362#568—138no—
363#568—136no—
364C2×#131—138no—
365C2×#131—136no—
366C2×#131C2×D1311yescharacteristic, complemented
367C22×C13—28no—
368C22×C13—43no—
369C22×C13—812no—
370C22×C13C23×D411yescharacteristic
371C22×C13C23×D416yes—
372C2×D13—24no—
373C2×D13—412no—
374C2×D13—816no—
375C2×D13—82no—
376C2×D13C23×D418yes—
377C4×C13—28no—
378C4×C13C23×D411yescharacteristic
379C4×C13C23×D416yes—
380C4×C13C2611yescharacteristic
381Dic13—212no—
382Dic13C23×D418yes—
3832−1+4—22no—
3842−1+4—134no—
3852−1+4—132no—
3862−1+4—1348no—
3872−1+4—1324no—
3882−1+4—1316no—
3892−1+4—1324no—
3902−1+4—2616no—
3912−1+4—2616no—
3922−1+4—2612no—
3932−1+4—2648no—
3942−1+4—2624no—
3952−1+4—2624no—
3962−1+4—268no—
3972−1+4C22×D1318yes—
398H(5,𝔽2)—26no—
399H(5,𝔽2)—134no—
400H(5,𝔽2)—136no—
401H(5,𝔽2)—138no—
402H(5,𝔽2)—1324no—
403H(5,𝔽2)—2624no—
404H(5,𝔽2)—2648no—
405H(5,𝔽2)—2612no—
406H(5,𝔽2)C22×D1318yes—
407C2×#41—28no—
408C2×#41—138no—
409C2×#41—136no—
410C2×#41—1348no—
411C2×#41—1324no—
412C2×#41—136no—
413C2×#41—2612no—
414C2×#41—2624no—
415C2×#41—2624no—
416C2×#41—2648no—
417C2×#41—2648no—
418C2×#41—2616no—
419C2×#41—2624no—
420C2×#41—266no—
421C2×#41C22×D13112yes—
422C2×#41C22×D1311yescharacteristic
423C22×Q8—1324no—
424C22×Q8—136no—
425C22×Q8—2612no—
426C22×Q8—2616no—
427C22×Q8—262no—
428#137—1348no—
429#137—1324no—
430#137—1324no—
431#137—1348no—
432#137—1396no—
433#137—1348no—
434#137—1348no—
435#137C22×D13124yes—
436AGL(1,ℤ/8)—138no—
437AGL(1,ℤ/8)—1324no—
438AGL(1,ℤ/8)—1316no—
439AGL(1,ℤ/8)—1348no—
440AGL(1,ℤ/8)C22×D13124yes—
441#135—1348no—
442#135—138no—
443#135—1348no—
444#135—1324no—
445#135—1348no—
446#135—1312no—
447#135—1312no—
448#135C22×D13112yes—
449#135C22×D1314yes—
450#135C22×D13112yes—
451#135C22×D13112yes—
452C2×Q16—1324no—
453C2×Q16—1324no—
454C2×Q16—1348no—
455C2×Q16—1312no—
456C2×Q16—133no—
457C2×Q16—1312no—
458C2×Q16—136no—
459C2×Q16C22×D1316yes—
460C2×SD16—1324no—
461C2×SD16—1348no—
462C2×SD16—136no—
463C2×SD16C22×D13112yes—
464C2×D8—138no—
465C2×D8—131no—
466C2×D8C22×D1316yes—
467#131—138no—
468#131—1324no—
469#131—1312no—
470#131—134no—
471#131—1312no—
472#131—1312no—
473#131C22×D1314yes—
474#131C22×D1314yes—
475C2×M16—136no—
476C2×M16—1324no—
477C2×M16—1312no—
478C2×M16C22×D1313yes—
479C22×C8—1312no—
480C22×C8C22×D1313yes—
481C2×C13—26no—
482C2×C13—21no—
483C2×C13—48no—
484C2×C13C24×D411yescharacteristic
485D13—28no—
486D13—44no—
487#41—28no—
488#41—224no—
489#41—212no—
490#41—212no—
491#41—138no—
492#41—1324no—
493#41—1312no—
494#41—1312no—
495#41—1312no—
496#41—134no—
497#41—1316no—
498#41—1348no—
499#41—1324no—
500#41—2648no—
501#41—2648no—
502#41—2624no—
503#41—2648no—
504#41—268no—
505#41—2648no—
506#41—2648no—
507#41—2648no—
508#41—2696no—
509#41—2624no—
510#41—2624no—
511#41—2648no—
512#41—2612no—
513#41D4×D1314yes—
514#41C23×D13124yes—
515#41C23×D1318yes—
516#41C23×D1314yes—
517C2×Q8—28no—
518C2×Q8—138no—
519C2×Q8—136no—
520C2×Q8—1348no—
521C2×Q8—1324no—
522C2×Q8—136no—
523C2×Q8—2624no—
524C2×Q8—2624no—
525C2×Q8—2648no—
526C2×Q8—2648no—
527C2×Q8—2624no—
528C2×Q8—2616no—
529C2×Q8—2612no—
530C2×Q8—266no—
531C2×Q8D4×D1311yescharacteristic
532C2×Q8C23×D13112yes—
533C2×D4—212no—
534C2×D4—224no—
535C2×D4—1312no—
536C2×D4—1324no—
537C2×D4—1316no—
538C2×D4—136no—
539C2×D4—132no—
540C2×D4—268no—
541C2×D4—2616no—
542C2×D4—2624no—
543C2×D4—2648no—
544C2×D4—2648no—
545C2×D4—2624no—
546C2×D4—266no—
547C2×D4D4×D1313yes—
548C2×D4C23×D13112yes—
549C22×C4—212no—
550C22×C4—1312no—
551C22×C4—2648no—
552C22×C4—2624no—
553C22×C4—2624no—
554C22×C4—266no—
555C22×C4—266no—
556C22×C4C23×D1313yes—
557Q16—1348no—
558Q16—136no—
559Q16—1324no—
560Q16—136no—
561Q16C23×D1314yes—
562Q16C23×D13112yes—
563SD16—1316no—
564SD16—1348no—
565SD16—1324no—
566SD16C23×D13124yes—
567SD16C23×D1318yes—
568D8—1316no—
569D8—132no—
570D8—132no—
571D8C23×D1314yes—
572D8C23×D13112yes—
573M16—1312no—
574M16—134no—
575M16—1324no—
576M16—138no—
577M16C23×D13112yes—
578C2×C8—1312no—
579C2×C8—138no—
580C2×C8—1324no—
581C2×C8C23×D1316yes—
582C2×C8C23×D1311yescharacteristic
583C2×C8C23×D1316yes—
584C2×C8C23×D1313yes—
585C13#6309311yescharacteristic, complemented, Sylow
586C23—212no—
587C23—1312no—
588C23—2616no—
589C23—262no—
590C23C2×D4×D1313yes—
591Q8—212no—
592Q8—134no—
593Q8—1312no—
594Q8—1312no—
595Q8—2648no—
596Q8—266no—
597Q8—2624no—
598Q8C2×D4×D1314yes—
599Q8C24×D1318yes—
600D4—224no—
601D4—28no—
602D4—212no—
603D4—1324no—
604D4—138no—
605D4—134no—
606D4—1312no—
607D4—134no—
608D4—2616no—
609D4—2616no—
610D4—2648no—
611D4—2624no—
612D4—262no—
613D4C2×D4×D13112yes—
614D4C24×D1318yes—
615C2×C4—212no—
616C2×C4—28no—
617C2×C4—224no—
618C2×C4—1312no—
619C2×C4—1324no—
620C2×C4—138no—
621C2×C4—2616no—
622C2×C4—2648no—
623C2×C4—2648no—
624C2×C4—2624no—
625C2×C4—266no—
626C2×C4—2624no—
627C2×C4C2×D4×D1316yes—
628C2×C4C2×D4×D1313yes—
629C2×C4C24×D1316yes—
630C2×C4C24×D1311yescharacteristic
631C8—138no—
632C8C24×D1311yescharacteristic
633C8C24×D1316yes—
634C8C24×D1311yescharacteristic
635C22—28no—
636C22—43no—
637C22—812no—
638C22—138no—
639C22—264no—
640C22—5212no—
641C22—1042no—
642C22—10416no—
643C22C22×D4×D1316yes—
644C22C22×D4×D1311yescharacteristic
645C4—28no—
646C4—138no—
647C4—2612no—
648C4C22×D4×D1311yescharacteristic
649C4C22×D4×D1316yes—
650C4C25×D1311yescharacteristic
651C2—21no—
652C2—26no—
653C2—48no—
654C2—268no—
655C2—524no—
656C2C23×D4×D1311yescharacteristic
subgroupG/H
centerC2C23×D4×D13
commutator, cosocleC4×C13C26
FrattiniC4C25×D13
FittingC13×#3584C2
socleC2×C13C24×D4
hypercenter#3584D13
hypocenterC13#63093
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C2×#41 ◁ #3584

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #13129649C2semidirect
2. #13129622C2semidirect
3. #13129620C2semidirect
4. #13129618C2semidirect
5. #13129605C2semidirect
6. #13129604C2semidirect
7. C2×#11719458C2semidirect
8. C2×#11719476C2semidirect
9. D13×#578C2semidirect
10. #13129547C2semidirect
11. C13×#3584C2semidirect
12. C13×#585C22semidirect
14. D13×2−1+4C22semidirect
16. #11719650C22semidirect
19. #11719648C22semidirect
21. C2×#88257C22semidirect
22. C2×#88257C22semidirect
23. C13×#578C22semidirect
24. C13×#577C22semidirect
25. C13×#576C22semidirect
26. C13×#575C22semidirect
27. C2×C13×#135C22semidirect
28. C2×C13×#131C22semidirect
29. #11719543C22semidirect
30. #11719541C22semidirect
31. C2×#88205C22semidirect
32. #11719527C22semidirect
33. #11719524C22semidirect
34. #11719484C22semidirect
35. D13×#137C22semidirect
36. #11719480C22semidirect
37. #11719478C22semidirect
38. #11719476C22semidirect
39. #11719476C22semidirect
40. #11719476C22semidirect
41. #11719476C22semidirect
42. #11719475C22semidirect
43. #11719474C22semidirect
44. #11719473C22semidirect
45. D13×#135C22semidirect
46. #11719471C22semidirect
47. C2×Q16×D13C22semidirect
48. #11719467C22semidirect
49. C2×#88169C22semidirect
50. #11719462C22semidirect
51. C2×#88166C22semidirect
52. #11719458C22semidirect
53. #11719458C22semidirect
54. #11719455C22semidirect
55. D13×#131C22semidirect
56. C2×#88163C22semidirect
57. #11719450C22semidirect
58. C22×Dic52C22semidirect
59. C2×#88158C22semidirect
60. C2×#88155C22semidirect
264. #3584D13semidirect
323. #585C2×D13semidirect
338. #578C2×D13semidirect
343. #577C2×D13semidirect
346. #576C2×D13semidirect
350. #575C2×D13semidirect
358. C2×#135C2×D13semidirect
366. C2×#131C2×D13semidirect
585. C13#63093semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#131296492
#131296222
#131296202
#131296182
#131296052
#131296042
C2×#117194582
C2×#117194762
D13×#5782
#131295472
C13×#35842
#6309313