Finite Group Databasebeta

← #16776788

Subgroups — #16776788

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#13129648C212yesmaximal, complemented
2#13129623C212yesmaximal, complemented
3#13129621C2116yesmaximal, complemented
4#13129619C2112yesmaximal, complemented
5#13129610C211yesmaximal, characteristic, complemented
6#13129606C218yesmaximal, complemented
7#13129605C216yesmaximal, complemented
8C2×#11719471C216yesmaximal, complemented
9D13×#578C218yesmaximal, complemented
10#13129547C211yesmaximal, characteristic, complemented
11C2×C13×#578C211yesmaximal, characteristic, complemented
12C2×C13×2−1+4C2212yescomplemented
13#11719652—28no—
14#11719652C22116yescomplemented
15D13×2−1+4—28no—
16D13×2−1+4C22116yescomplemented
17#11719648—212no—
18#11719648C2211yescharacteristic
19#11719648C22112yescomplemented
20#11719648C2212yescomplemented
21C2×#88254—24no—
22C2×#88254C22112yescomplemented
23C13×#578C2218yescomplemented
24C13×#578C2218yescomplemented
25C2×C13×#137C2216yescomplemented
26C2×C13×#135C2213yescomplemented
27C22×C13×Q16C2213yescomplemented
28C2×C13×#131C2211yescharacteristic, complemented
29#11719543C22116yescomplemented
30#11719542C22116yescomplemented
31#11719527C22112yescomplemented
32#11719524C2211yescharacteristic, complemented
33C2×#88196C22112yescomplemented
34#11719484C22148yescomplemented
35#11719483C22148yescomplemented
36#11719482C22148yescomplemented
37D13×#137C22148yescomplemented
38#11719476C22112yescomplemented
39#11719475C22124yescomplemented
40#11719474C22112yescomplemented
41#11719473C22124yescomplemented
42D13×#135C22124yescomplemented
43#11719471C22124yescomplemented
44#11719471C22112yescomplemented
45#11719471C22124yescomplemented
46#11719471C22124yescomplemented
47C2×#88173C22112yescomplemented
48C2×#88172C22124yescomplemented
49C2×Q16×D13C22112yescomplemented
50#11719467C22124yescomplemented
51#11719462C2214yescomplemented
52#11719458C2214yescomplemented
53#11719457C2214yescomplemented
54#11719456C2218yescomplemented
55#11719455C2218yescomplemented
56D13×#131C2218yescomplemented
57#11719453C2216yescomplemented
58#11719450C2216yescomplemented
59C2×#88158C2216yescomplemented
60C2×#88155C2216yescomplemented
61C13×2−1+4—28no—
62C13×2−1+4C23116yes—
63C2×C13×#41—26no—
64C2×C13×#41C2316yes—
65C2×C13×#41C2311yescharacteristic
66C22×Q8×C13—22no—
67C22×Q8×C13C2316yes—
68#88257—248no—
69#88257—212no—
70#88257D414yes—
71#88257C23124yes—
72#88257C2318yes—
73#88256—248no—
74#88256—212no—
75#88256D414yes—
76#88256C23124yes—
77#88256C2318yes—
78#41×D13—28no—
79#41×D13—248no—
80#41×D13—248no—
81#41×D13—224no—
82#41×D13C23148yes—
83#41×D13C2318yes—
84#88254—22no—
85#88254—216no—
86#88254—216no—
87#88254—224no—
88#88254—224no—
89#88254D412yes—
90#88254C23124yes—
91#88254C23148yes—
92C2×#6131—216no—
93C2×#6131—212no—
94C2×#6131C23124yes—
95C2×Q8×D13—216no—
96C2×Q8×D13—212no—
97C2×Q8×D13C23124yes—
98#88251—26no—
99#88251—224no—
100#88251D416yes—
101#88251C2318yes—
102C2×#6127—224no—
103C2×#6127—224no—
104C2×#6127C2316yes—
105C2×#6127C23112yes—
106C13×#137C23148yes—
107C13×#135C23112yes—
108C13×#135C23112yes—
109C2×C13×Q16C23112yes—
110C2×C13×Q16C23112yes—
111C2×C13×Q16C2313yes—
112C2×C13×Q16C2316yes—
113C2×C13×SD16C2316yes—
114C2×C13×D8C2311yescharacteristic
115C13×#131C2314yes—
116C13×#131C2314yes—
117C2×C13×M16C2313yes—
118C22×C8×C13C2313yes—
119#88205C23124yes—
120#88204C23148yes—
121#88203C23124yes—
122#88202C2318yes—
123C2×#6107C23124yes—
124#88196C23124yes—
125#88196C23148yes—
126C2×#6106C23124yes—
127#88186C2318yes—
128C2×#6099C2316yes—
129#88173C23124yes—
130#88173C23148yes—
131#88172C23148yes—
132#88172C23196yes—
133Q16×D13C23124yes—
134Q16×D13C23148yes—
135#88170C23148yes—
136#88169C23148yes—
137#88168C23148yes—
138SD16×D13C23148yes—
139#88166C2318yes—
140#88165C23116yes—
141D8×D13C2318yes—
142#88163C23124yes—
143#88162C23124yes—
144#88161C23124yes—
145M16×D13C23124yes—
146C2×Dic52C2316yes—
147#88158C2314yes—
148#88158C23112yes—
149#88158C23112yes—
150#88158C23112yes—
151C2×D104C2316yes—
152C2×#6095C23112yes—
153#88155C2314yes—
154#88155C23112yes—
155#88155C23112yes—
156#88155C23112yes—
157C2×#6094C23112yes—
158C2×C8×D13C23112yes—
159#63093—131nomaximal, Sylow
160C13×#41—248no—
161C13×#41—212no—
162C13×#41C2×D414yes—
163C13×#41C24124yes—
164C13×#41C2414yes—
165C2×Q8×C13—216no—
166C2×Q8×C13—212no—
167C2×Q8×C13—26no—
168C2×Q8×C13C2×D411yescharacteristic
169C2×Q8×C13C24124yes—
170C2×Q8×C13C2416yes—
171C2×D4×C13—26no—
172C2×D4×C13C2×D413yes—
173C2×D4×C13C2412yes—
174C22×C4×C13—26no—
175C22×C4×C13—26no—
176C22×C4×C13C2413yes—
177C2×C22⋊D13—212no—
178C2×C22⋊D13—216no—
179C2×C22⋊D13C2×D4112yes—
180#6131—28no—
181#6131—248no—
182#6131—224no—
183#6131—248no—
184#6131C2×D418yes—
185#6131C24148yes—
186Q8×D13—28no—
187Q8×D13—248no—
188Q8×D13—248no—
189Q8×D13—224no—
190Q8×D13C2×D418yes—
191Q8×D13C24148yes—
192#6129—224no—
193#6129—248no—
194#6129—248no—
195#6129—28no—
196#6129C2×D4124yes—
197#6129C24116yes—
198D4×D13—248no—
199D4×D13—224no—
200D4×D13—248no—
201D4×D13—28no—
202D4×D13C2×D4124yes—
203D4×D13C24116yes—
204#6127—212no—
205#6127—212no—
206#6127—248no—
207#6127—296no—
208#6127—224no—
209#6127—248no—
210#6127—224no—
211#6127C2×D4112yes—
212#6127C2×D4112yes—
213#6127C2414yes—
214#6127C24112yes—
215#6127C2418yes—
216#6127C24124yes—
217C2×D52—224no—
218C2×D52—224no—
219C2×D52C2×D416yes—
220C2×D52C24112yes—
221C2×C4×D13—224no—
222C2×C4×D13—212no—
223C2×C4×D13—224no—
224C2×C4×D13—248no—
225C2×C4×D13C24112yes—
226C2×Dic26—224no—
227C2×Dic26—224no—
228C2×Dic26C2×D416yes—
229C2×Dic26C24112yes—
230C13×Q16C2416yes—
231C13×Q16C24124yes—
232C13×Q16C2416yes—
233C13×SD16C24124yes—
234C13×D8C2412yes—
235C13×D8C2412yes—
236C13×M16C24112yes—
237C2×C8×C13C2416yes—
238C2×C8×C13C2416yes—
239C2×C8×C13C2411yescharacteristic
240C2×C8×C13C2413yes—
241#6107C24148yes—
242#6106C24148yes—
243#6105C24116yes—
244#6104C24116yes—
245#6099C2414yes—
246#6099C24112yes—
247C2×#1055C24112yes—
248Dic52C2414yes—
249Dic52C24112yes—
250D104C2414yes—
251D104C24112yes—
252#6095C2418yes—
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#3584—131no—
265C2×#578—138no—
266C2×#578—136no—
267C2×#578D1311yescharacteristic, complemented
268#3573—131no—
269C23×C13—22no—
270C23×C13C22×D413yes—
271C22×D13—212no—
272C22×D13—216no—
273C22×D13C22×D4112yes—
274Q8×C13—224no—
275Q8×C13—26no—
276Q8×C13C22×D414yes—
277Q8×C13C25112yes—
278D4×C13—224no—
279D4×C13—22no—
280D4×C13C22×D4112yes—
281D4×C13C2514yes—
282C2×C4×C13—224no—
283C2×C4×C13—26no—
284C2×C4×C13—224no—
285C2×C4×C13C22×D416yes—
286C2×C4×C13C22×D413yes—
287C2×C4×C13C2516yes—
288C2×C4×C13C2511yescharacteristic
289C22⋊D13—28no—
290C22⋊D13—224no—
291C22⋊D13—216no—
292C22⋊D13—216no—
293C22⋊D13C22×D418yes—
294C22⋊D13C22×D4124yes—
295C2×Dic13—212no—
296C2×Dic13—216no—
297C2×Dic13C22×D4112yes—
298D52—248no—
299D52—212no—
300D52C22×D4112yes—
301D52C22×D414yes—
302D52C2518yes—
303C4×D13—248no—
304C4×D13—224no—
305C4×D13—28no—
306C4×D13—248no—
307C4×D13C22×D4124yes—
308C4×D13C2518yes—
309Dic26—212no—
310Dic26—248no—
311Dic26C22×D4112yes—
312Dic26C22×D414yes—
313Dic26C2518yes—
314C8×C13C2511yescharacteristic
315C8×C13C2516yes—
316C8×C13C2511yescharacteristic
317#1055C2518yes—
318#585—131no—
319#585—1316no—
320#585—1312no—
321#585—132no—
322#585—2612no—
323#585—268no—
324C2×2−1+4—1316no—
325C2×2−1+4—1312no—
326C2×2−1+4—264no—
327C2×2−1+4—268no—
328C2×2−1+4C2×D1312yescomplemented
329#578—1316no—
330#578—1348no—
331#578—1312no—
332#578—1324no—
333#578—1312no—
334#578—1324no—
335#578—1324no—
336#578—134no—
337#578C2×D1318yescomplemented
338#578C2×D1318yescomplemented
339#577—1316no—
340#577—1348no—
341#577—1324no—
342#577—1324no—
343#577—138no—
344#576—1312no—
345#576—134no—
346#576—134no—
347#575—1312no—
348#575—1348no—
349#575—1324no—
350#575—136no—
351C2×#137—1312no—
352C2×#137—1348no—
353C2×#137—1324no—
354C2×#137C2×D1316yescomplemented
355C2×#135—1324no—
356C2×#135—1312no—
357C2×#135—136no—
358C2×#135C2×D1313yescomplemented
359C22×Q16—1312no—
360C22×Q16C2×D1313yescomplemented
361#568—131no—
362#568—138no—
363#568—136no—
364C2×#131—138no—
365C2×#131—136no—
366C2×#131C2×D1311yescharacteristic, complemented
367C22×C13—24no—
368C22×C13—43no—
369C22×C13—82no—
370C22×C13C23×D416yes—
371C22×C13C23×D411yescharacteristic
372C2×D13—28no—
373C2×D13—412no—
374C2×D13—816no—
375C2×D13—812no—
376C2×D13C23×D418yes—
377C4×C13—212no—
378C4×C13C23×D416yes—
379C4×C13C23×D411yescharacteristic
380C4×C13C2611yescharacteristic
381Dic13—28no—
382Dic13C23×D418yes—
3832−1+4—28no—
3842−1+4—134no—
3852−1+4—132no—
3862−1+4—1324no—
3872−1+4—138no—
3882−1+4—1348no—
3892−1+4—1324no—
3902−1+4—2616no—
3912−1+4—2616no—
3922−1+4—2624no—
3932−1+4—2624no—
3942−1+4—2612no—
3952−1+4—2648no—
3962−1+4—262no—
3972−1+4C22×D13116yes—
398H(5,𝔽2)—134no—
399H(5,𝔽2)—136no—
400H(5,𝔽2)—1324no—
401H(5,𝔽2)—138no—
402H(5,𝔽2)—138no—
403H(5,𝔽2)—2624no—
404H(5,𝔽2)—2648no—
405H(5,𝔽2)—2612no—
406H(5,𝔽2)—266no—
407C2×#41—26no—
408C2×#41—138no—
409C2×#41—136no—
410C2×#41—1348no—
411C2×#41—1324no—
412C2×#41—1312no—
413C2×#41—2624no—
414C2×#41—2624no—
415C2×#41—2612no—
416C2×#41—2648no—
417C2×#41—2616no—
418C2×#41—2648no—
419C2×#41—2624no—
420C2×#41—268no—
421C2×#41C22×D1316yes—
422C2×#41C22×D1311yescharacteristic
423C22×Q8—22no—
424C22×Q8—1324no—
425C22×Q8—2616no—
426C22×Q8—2612no—
427C22×Q8C22×D1316yes—
428#137—1324no—
429#137—1348no—
430#137—1324no—
431#137—1396no—
432#137—1348no—
433#137—1348no—
434#137—1324no—
435#137C22×D13148yes—
436AGL(1,ℤ/8)—1324no—
437AGL(1,ℤ/8)—138no—
438AGL(1,ℤ/8)—1348no—
439AGL(1,ℤ/8)—1316no—
440AGL(1,ℤ/8)—1324no—
441#135—1348no—
442#135—1324no—
443#135—1348no—
444#135—1348no—
445#135—138no—
446#135—1312no—
447#135—134no—
448#135—1312no—
449#135—1312no—
450#135C22×D13112yes—
451#135C22×D13112yes—
452C2×Q16—1324no—
453C2×Q16—1324no—
454C2×Q16—1348no—
455C2×Q16—136no—
456C2×Q16C22×D13112yes—
457C2×Q16C22×D1313yes—
458C2×Q16C22×D13112yes—
459C2×Q16C22×D1316yes—
460C2×SD16—1324no—
461C2×SD16—1348no—
462C2×SD16—1312no—
463C2×SD16C22×D1316yes—
464C2×D8—138no—
465C2×D8—136no—
466C2×D8C22×D1311yescharacteristic
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—44no—
484C2×C13C24×D411yescharacteristic
485D13—28no—
486D13—48no—
487#41—248no—
488#41—212no—
489#41—138no—
490#41—1324no—
491#41—1312no—
492#41—1312no—
493#41—1312no—
494#41—134no—
495#41—1348no—
496#41—1316no—
497#41—1324no—
498#41—138no—
499#41—2648no—
500#41—2648no—
501#41—2696no—
502#41—2624no—
503#41—2624no—
504#41—2648no—
505#41—2648no—
506#41—2648no—
507#41—268no—
508#41—2624no—
509#41—2648no—
510#41—268no—
511#41—2624no—
512#41—2612no—
513#41—2612no—
514#41D4×D1314yes—
515#41C23×D13124yes—
516#41C23×D1314yes—
517C2×Q8—216no—
518C2×Q8—26no—
519C2×Q8—212no—
520C2×Q8—138no—
521C2×Q8—136no—
522C2×Q8—1348no—
523C2×Q8—1312no—
524C2×Q8—2624no—
525C2×Q8—2624no—
526C2×Q8—2648no—
527C2×Q8—2648no—
528C2×Q8—2624no—
529C2×Q8—268no—
530C2×Q8D4×D1311yescharacteristic
531C2×Q8C23×D13124yes—
532C2×Q8C23×D1316yes—
533C2×D4—26no—
534C2×D4—1312no—
535C2×D4—1324no—
536C2×D4—1316no—
537C2×D4—136no—
538C2×D4—1312no—
539C2×D4—2624no—
540C2×D4—2624no—
541C2×D4—2648no—
542C2×D4—268no—
543C2×D4—2648no—
544C2×D4—2616no—
545C2×D4—2612no—
546C2×D4—2624no—
547C2×D4D4×D1313yes—
548C2×D4C23×D1312yes—
549C22×C4—26no—
550C22×C4—26no—
551C22×C4—1312no—
552C22×C4—2624no—
553C22×C4—2648no—
554C22×C4—2624no—
555C22×C4—2612no—
556C22×C4C23×D1313yes—
557Q16—1348no—
558Q16—134no—
559Q16—1312no—
560Q16C23×D1316yes—
561Q16C23×D13124yes—
562Q16C23×D1316yes—
563SD16—1348no—
564SD16—1316no—
565SD16—1324no—
566SD16—138no—
567SD16C23×D13124yes—
568D8—1316no—
569D8—134no—
570D8—1312no—
571D8C23×D1312yes—
572D8C23×D1312yes—
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—22no—
587C23—1312no—
588C23—2616no—
589C23—2612no—
590C23C2×D4×D1313yes—
591Q8—224no—
592Q8—26no—
593Q8—1312no—
594Q8—134no—
595Q8—138no—
596Q8—2648no—
597Q8—2612no—
598Q8C2×D4×D1314yes—
599Q8C24×D13112yes—
600D4—224no—
601D4—22no—
602D4—138no—
603D4—1324no—
604D4—1312no—
605D4—134no—
606D4—138no—
607D4—2648no—
608D4—2616no—
609D4—2616no—
610D4—268no—
611D4—2624no—
612D4—2612no—
613D4C2×D4×D13112yes—
614D4C24×D1314yes—
615C2×C4—26no—
616C2×C4—224no—
617C2×C4—224no—
618C2×C4—1312no—
619C2×C4—1324no—
620C2×C4—138no—
621C2×C4—2648no—
622C2×C4—2648no—
623C2×C4—2616no—
624C2×C4—2612no—
625C2×C4—2624no—
626C2×C4—268no—
627C2×C4C2×D4×D1313yes—
628C2×C4C2×D4×D1316yes—
629C2×C4C24×D1316yes—
630C2×C4C24×D1311yescharacteristic
631C8—138no—
632C8C24×D1311yescharacteristic
633C8C24×D1316yes—
634C8C24×D1311yescharacteristic
635C22—24no—
636C22—43no—
637C22—82no—
638C22—138no—
639C22—268no—
640C22—5212no—
641C22—10412no—
642C22—10416no—
643C22C22×D4×D1311yescharacteristic
644C22C22×D4×D1316yes—
645C4—212no—
646C4—138no—
647C4—268no—
648C4C22×D4×D1316yes—
649C4C22×D4×D1311yescharacteristic
650C4C25×D1311yescharacteristic
651C2—21no—
652C2—26no—
653C2—44no—
654C2—268no—
655C2—528no—
656C2C23×D4×D1311yescharacteristic
subgroupG/H
centerC2C23×D4×D13
commutator, cosocleC4×C13C26
FrattiniC4C25×D13
FittingC2×C13×#578C2
socleC2×C13C24×D4
hypercenterC2×#578D13
hypocenterC13#63093
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. #13129648C2semidirect
2. #13129623C2semidirect
3. #13129621C2semidirect
4. #13129619C2semidirect
5. #13129610C2semidirect
6. #13129606C2semidirect
7. #13129605C2semidirect
8. C2×#11719471C2semidirect
9. D13×#578C2semidirect
10. #13129547C2semidirect
11. C2×C13×#578C2semidirect
12. C2×C13×2−1+4C22semidirect
14. #11719652C22semidirect
16. D13×2−1+4C22semidirect
19. #11719648C22semidirect
20. #11719648C22semidirect
22. C2×#88254C22semidirect
23. C13×#578C22semidirect
24. C13×#578C22semidirect
25. C2×C13×#137C22semidirect
26. C2×C13×#135C22semidirect
27. C22×C13×Q16C22semidirect
28. C2×C13×#131C22semidirect
29. #11719543C22semidirect
30. #11719542C22semidirect
31. #11719527C22semidirect
32. #11719524C22semidirect
33. C2×#88196C22semidirect
34. #11719484C22semidirect
35. #11719483C22semidirect
36. #11719482C22semidirect
37. D13×#137C22semidirect
38. #11719476C22semidirect
39. #11719475C22semidirect
40. #11719474C22semidirect
41. #11719473C22semidirect
42. D13×#135C22semidirect
43. #11719471C22semidirect
44. #11719471C22semidirect
45. #11719471C22semidirect
46. #11719471C22semidirect
47. C2×#88173C22semidirect
48. C2×#88172C22semidirect
49. C2×Q16×D13C22semidirect
50. #11719467C22semidirect
51. #11719462C22semidirect
52. #11719458C22semidirect
53. #11719457C22semidirect
54. #11719456C22semidirect
55. #11719455C22semidirect
56. D13×#131C22semidirect
57. #11719453C22semidirect
58. #11719450C22semidirect
59. C2×#88158C22semidirect
60. C2×#88155C22semidirect
267. C2×#578D13semidirect
328. C2×2−1+4C2×D13semidirect
337. #578C2×D13semidirect
338. #578C2×D13semidirect
354. C2×#137C2×D13semidirect
358. C2×#135C2×D13semidirect
360. C22×Q16C2×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
#131296482
#131296232
#131296212
#131296192
#131296102
#131296062
#131296052
C2×#117194712
D13×#5782
#131295472
C2×C13×#5782
#6309313