Finite Group Databasebeta

← #30593883

Subgroups — #30593883

Order 15616.

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#21460632C212yesmaximal, complemented
2#21460605C212yesmaximal, complemented
3#21460603C2112yesmaximal, complemented
4#21460601C2116yesmaximal, complemented
5#21460588C216yesmaximal, complemented
6#21460587C218yesmaximal, complemented
7C2×#16840579C211yesmaximal, characteristic, complemented
8C2×#16840576C216yesmaximal, complemented
9#578×D61C218yesmaximal, complemented
10#21460530C211yesmaximal, characteristic, complemented
11C61×#3584C211yesmaximal, characteristic, complemented
12#16840598—28no—
13#16840598C2211yescharacteristic
14#16840598C22112yescomplemented
15#16840597—212no—
16#16840597C22116yescomplemented
17C2×#14519824—28no—
18C2×#14519824C2212yescomplemented
19C2×#14519824C22112yescomplemented
202−1+4×D61—24no—
212−1+4×D61C22116yescomplemented
22#16840579C2218yescomplemented
23#16840579C2218yescomplemented
24#16840578C22148yescomplemented
25#16840577C22116yescomplemented
26#16840576C22124yescomplemented
27#16840576C22112yescomplemented
28#16840576C22124yescomplemented
29#16840576C22124yescomplemented
30#16840575C22112yescomplemented
31#16840573C22116yescomplemented
32#16840572C22148yescomplemented
33#16840569C22124yescomplemented
34#16840568C22124yescomplemented
35#16840564C2214yescomplemented
36#16840563C22112yescomplemented
37#16840560C22112yescomplemented
38#16840559C22124yescomplemented
39#16840558C22148yescomplemented
40C2×#14519810C2216yescomplemented
41C2×#14519807C22112yescomplemented
42C2×#14519806C22124yescomplemented
43#137×D61C22148yescomplemented
44C2×#14519799C2213yescomplemented
45C2×#14519796C22112yescomplemented
46#135×D61C22124yescomplemented
47C22×Dic244C2213yescomplemented
48C2×Q16×D61C22112yescomplemented
49#16840529C2211yescharacteristic, complemented
50#16840528C2216yescomplemented
51#16840527C2218yescomplemented
52C2×#14519783C2216yescomplemented
53#131×D61C2218yescomplemented
54C61×#585C2212yescomplemented
55C61×#578C2214yescomplemented
56C61×#577C2218yescomplemented
57C61×#576C2214yescomplemented
58C61×#575C2216yescomplemented
59C2×#135×C61C2216yescomplemented
60C2×#131×C61C2211yescharacteristic, complemented
61#14519825—216no—
62#14519825D412yes—
63#14519825C23124yes—
64#14519824—28no—
65#14519824—216no—
66#14519824—248no—
67#14519824—224no—
68#14519824—224no—
69#14519824—212no—
70#14519824D414yes—
71#14519824C23124yes—
72#14519824C23148yes—
73#14519824C23116yes—
74#14519823—212no—
75#14519823D414yes—
76#14519823C2318yes—
77#14519822—224no—
78#14519822—248no—
79#14519822D416yes—
80#14519822C23124yes—
81C2×#11758222—224no—
82C2×#11758222—212no—
83C2×#11758222C2316yes—
84C2×#11758222C2316yes—
85C2×#11758224—26no—
86C2×#11758224—224no—
87C2×#11758224—248no—
88C2×#11758224C23124yes—
89#41×D61—216no—
90#41×D61—248no—
91#41×D61—224no—
92#41×D61C23148yes—
93#41×D61C2318yes—
94C22×Dic122—212no—
95C22×Dic122C2316yes—
96C2×Q8×D61—22no—
97C2×Q8×D61—216no—
98C2×Q8×D61C23124yes—
99#14519810C23148yes—
100#14519809C23148yes—
101#14519808C23124yes—
102#14519807C23148yes—
103#14519807C23124yes—
104#14519806C23196yes—
105#14519806C23148yes—
106#14519804C23116yes—
107#14519803C23124yes—
108#14519802C2318yes—
109#14519801C23148yes—
110#14519800C2318yes—
111#14519799C23112yes—
112#14519799C23112yes—
113#14519798C23148yes—
114#14519797C23148yes—
115#14519796C23124yes—
116#14519796C23148yes—
117C2×Dic244C23112yes—
118C2×Dic244C2313yes—
119C2×Dic244C23112yes—
120C2×Dic244C2316yes—
121C2×#11758202C23124yes—
122Q16×D61C23124yes—
123Q16×D61C23148yes—
124C2×#11758190C2316yes—
125C2×#11758200C23124yes—
126SD16×D61C23148yes—
127C2×D488C2311yescharacteristic
128D8×D61C2318yes—
129#14519785C23124yes—
130#14519784C2318yes—
131#14519783C23112yes—
132#14519783C2314yes—
133#14519783C23112yes—
134#14519783C23112yes—
135C2×#11758194C2316yes—
136C2×#11758189C23112yes—
137M16×D61C23124yes—
138C2×C8×D61C23112yes—
1392−1+4×C61—22no—
1402−1+4×C61C2318yes—
141H(5,𝔽2)×C61—26no—
142H(5,𝔽2)×C61C2318yes—
143C2×#41×C61—28no—
144C2×#41×C61C23112yes—
145C2×#41×C61C2311yescharacteristic
146#137×C61C23124yes—
147AGL(1,ℤ/8)×C61C23124yes—
148#135×C61C2314yes—
149#135×C61C23112yes—
150#135×C61C23112yes—
151#135×C61C23112yes—
152C2×Q16×C61C2316yes—
153C2×SD16×C61C23112yes—
154C2×D8×C61C2316yes—
155#131×C61C2314yes—
156#131×C61C2314yes—
157C2×M16×C61C2313yes—
158C22×C8×C61C2313yes—
159#41×C61—28no—
160#41×C61—224no—
161#41×C61—212no—
162#41×C61—212no—
163#41×C61C2×D414yes—
164#41×C61C24124yes—
165#41×C61C2418yes—
166#41×C61C2414yes—
167C2×Q8×C61—28no—
168C2×Q8×C61C2×D411yescharacteristic
169C2×Q8×C61C24112yes—
170C2×D4×C61—212no—
171C2×D4×C61—224no—
172C2×D4×C61C2×D413yes—
173C2×D4×C61C24112yes—
174C22×C4×C61—212no—
175C22×C4×C61C2413yes—
176C2×C22⋊D61—26no—
177C2×C22⋊D61—224no—
178C2×C22⋊D61—248no—
179C2×C22⋊D61C2×D4112yes—
180C22×Dic61—26no—
181C22×Dic61—224no—
182#11758226—224no—
183#11758226C2×D418yes—
184#11758226C24116yes—
185Q8×D61—216no—
186Q8×D61—248no—
187Q8×D61—224no—
188Q8×D61C2×D418yes—
189Q8×D61C24148yes—
190#11758224—248no—
191#11758224—296no—
192#11758224—248no—
193#11758224—248no—
194#11758224—248no—
195#11758224—224no—
196#11758224C2×D4124yes—
197#11758224C24148yes—
198D4×D61—248no—
199D4×D61—216no—
200D4×D61—224no—
201D4×D61C2×D4124yes—
202D4×D61C24116yes—
203#11758222—212no—
204#11758222—248no—
205#11758222—248no—
206#11758222—224no—
207#11758222—248no—
208#11758222—28no—
209#11758222C2×D4112yes—
210#11758222C2×D4112yes—
211#11758222C24112yes—
212#11758222C2414yes—
213#11758222C24124yes—
214C2×D244—28no—
215C2×D244C2×D416yes—
216C2×D244C2412yes—
217C2×C4×D61—224no—
218C2×C4×D61—26no—
219C2×C4×D61—248no—
220C2×C4×D61C24112yes—
221C2×Dic122—26no—
222C2×Dic122—212no—
223C2×Dic122—224no—
224C2×Dic122—248no—
225C2×Dic122—224no—
226C2×Dic122C2×D416yes—
227C2×Dic122C24124yes—
228C2×Dic122C2416yes—
229Q16×C61C24112yes—
230Q16×C61C2414yes—
231SD16×C61C2418yes—
232SD16×C61C24124yes—
233D8×C61C2414yes—
234D8×C61C24112yes—
235M16×C61C24112yes—
236C2×C8×C61C2411yescharacteristic
237C2×C8×C61C2413yes—
238C2×C8×C61C2416yes—
239C2×C8×C61C2416yes—
240#11758202C24148yes—
241#11758201C24116yes—
242#11758200C24148yes—
243#11758199C24116yes—
244#11758194C24112yes—
245#11758194C2414yes—
246C2×#92418C24112yes—
247Dic244C24124yes—
248Dic244C2416yes—
249Dic244C2416yes—
250D488C2412yes—
251D488C2412yes—
252#11758190C24124yes—
253#11758189C24124yes—
254#11758189C2418yes—
255C8×D61C2418yes—
256C8×D61C24124yes—
257C23×C61—212no—
258C23×C61C22×D413yes—
259C22×D61—22no—
260C22×D61—216no—
261C22×D61C22×D4112yes—
262Q8×C61—212no—
263Q8×C61C22×D414yes—
264Q8×C61C2518yes—
265D4×C61—28no—
266D4×C61—224no—
267D4×C61—212no—
268D4×C61C22×D4112yes—
269D4×C61C2518yes—
270C2×C4×C61—212no—
271C2×C4×C61—28no—
272C2×C4×C61—224no—
273C2×C4×C61C22×D416yes—
274C2×C4×C61C22×D413yes—
275C2×C4×C61C2516yes—
276C2×C4×C61C2511yescharacteristic
277C22⋊D61—224no—
278C22⋊D61—248no—
279C22⋊D61—216no—
280C22⋊D61C22×D4124yes—
281C22⋊D61C22×D418yes—
282C2×Dic61—26no—
283C2×Dic61—224no—
284C2×Dic61—248no—
285C2×Dic61C22×D4112yes—
286D244—216no—
287D244—22no—
288D244C22×D414yes—
289D244C22×D4112yes—
290D244C2514yes—
291C4×D61—248no—
292C4×D61—224no—
293C4×D61—216no—
294C4×D61C22×D4124yes—
295C4×D61C2518yes—
296Dic122—26no—
297Dic122—224no—
298Dic122—248no—
299Dic122C22×D414yes—
300Dic122C22×D4112yes—
301Dic122C25112yes—
302C8×C61C2511yescharacteristic
303C8×C61C2516yes—
304C8×C61C2511yescharacteristic
305#92418C2518yes—
306#63093—611nomaximal, Sylow
307C22×C61—28no—
308C22×C61—43no—
309C22×C61—812no—
310C22×C61C23×D411yescharacteristic
311C22×C61C23×D416yes—
312C2×D61—24no—
313C2×D61—412no—
314C2×D61—816no—
315C2×D61—82no—
316C2×D61C23×D418yes—
317C4×C61—28no—
318C4×C61C23×D411yescharacteristic
319C4×C61C23×D416yes—
320C4×C61C2611yescharacteristic
321Dic61—212no—
322Dic61C23×D418yes—
3232−1+6—612no—
324#3586—612no—
325#3586—6116no—
326#3586—6112no—
327#3584—618no—
328#3584—616no—
329#3584D6111yescharacteristic, complemented
330C2×#578—618no—
331C2×#578—616no—
332C2×#578—611no—
333#3573—611no—
334C2×C61—26no—
335C2×C61—21no—
336C2×C61—48no—
337C2×C61C24×D411yescharacteristic
338D61—28no—
339D61—44no—
340#585—611no—
341#585—6116no—
342#585—6112no—
343#585—1228no—
344#585—12212no—
345#585C2×D6112yescomplemented
346C2×2−1+4—6116no—
347C2×2−1+4—6112no—
348C2×2−1+4—612no—
349C2×2−1+4—1228no—
350C2×2−1+4—1224no—
351#578—6116no—
352#578—6148no—
353#578—6124no—
354#578—6112no—
355#578—6124no—
356#578—6124no—
357#578—6112no—
358#578—618no—
359#578—618no—
360#578C2×D6114yescomplemented
361#577—6116no—
362#577—6148no—
363#577—6124no—
364#577—6124no—
365#577C2×D6118yescomplemented
366#576—614no—
367#576—6112no—
368#576C2×D6114yescomplemented
369#575—6112no—
370#575—6148no—
371#575—6124no—
372#575C2×D6116yescomplemented
373C2×#137—6112no—
374C2×#137—6148no—
375C2×#137—6124no—
376C2×#137—616no—
377C2×#135—6124no—
378C2×#135—6112no—
379C2×#135—613no—
380C2×#135C2×D6116yescomplemented
381C22×Q16—6112no—
382C22×Q16—613no—
383#568—611no—
384#568—618no—
385#568—616no—
386C2×#131—618no—
387C2×#131—616no—
388C2×#131C2×D6111yescharacteristic, complemented
389C61#6309311yescharacteristic, complemented, Sylow
3902−1+4—22no—
3912−1+4—614no—
3922−1+4—612no—
3932−1+4—6148no—
3942−1+4—6124no—
3952−1+4—6116no—
3962−1+4—6124no—
3972−1+4—12216no—
3982−1+4—12216no—
3992−1+4—12212no—
4002−1+4—12248no—
4012−1+4—12224no—
4022−1+4—12224no—
4032−1+4—1228no—
4042−1+4C22×D6118yes—
405H(5,𝔽2)—26no—
406H(5,𝔽2)—614no—
407H(5,𝔽2)—616no—
408H(5,𝔽2)—618no—
409H(5,𝔽2)—6124no—
410H(5,𝔽2)—12224no—
411H(5,𝔽2)—12248no—
412H(5,𝔽2)—12212no—
413H(5,𝔽2)C22×D6118yes—
414C2×#41—28no—
415C2×#41—618no—
416C2×#41—616no—
417C2×#41—6148no—
418C2×#41—6124no—
419C2×#41—616no—
420C2×#41—12212no—
421C2×#41—12224no—
422C2×#41—12224no—
423C2×#41—12248no—
424C2×#41—12248no—
425C2×#41—12216no—
426C2×#41—12224no—
427C2×#41—1226no—
428C2×#41C22×D61112yes—
429C2×#41C22×D6111yescharacteristic
430C22×Q8—6124no—
431C22×Q8—616no—
432C22×Q8—12212no—
433C22×Q8—12216no—
434C22×Q8—1222no—
435#137—6148no—
436#137—6124no—
437#137—6124no—
438#137—6148no—
439#137—6196no—
440#137—6148no—
441#137—6148no—
442#137C22×D61124yes—
443AGL(1,ℤ/8)—618no—
444AGL(1,ℤ/8)—6124no—
445AGL(1,ℤ/8)—6116no—
446AGL(1,ℤ/8)—6148no—
447AGL(1,ℤ/8)C22×D61124yes—
448#135—6148no—
449#135—618no—
450#135—6148no—
451#135—6124no—
452#135—6148no—
453#135—6112no—
454#135—6112no—
455#135C22×D61112yes—
456#135C22×D6114yes—
457#135C22×D61112yes—
458#135C22×D61112yes—
459C2×Q16—6124no—
460C2×Q16—6124no—
461C2×Q16—6148no—
462C2×Q16—6112no—
463C2×Q16—613no—
464C2×Q16—6112no—
465C2×Q16—616no—
466C2×Q16C22×D6116yes—
467C2×SD16—6124no—
468C2×SD16—6148no—
469C2×SD16—616no—
470C2×SD16C22×D61112yes—
471C2×D8—618no—
472C2×D8—611no—
473C2×D8C22×D6116yes—
474#131—618no—
475#131—6124no—
476#131—6112no—
477#131—614no—
478#131—6112no—
479#131—6112no—
480#131C22×D6114yes—
481#131C22×D6114yes—
482C2×M16—616no—
483C2×M16—6124no—
484C2×M16—6112no—
485C2×M16C22×D6113yes—
486C22×C8—6112no—
487C22×C8C22×D6113yes—
488#41—28no—
489#41—224no—
490#41—212no—
491#41—212no—
492#41—618no—
493#41—6124no—
494#41—6112no—
495#41—6112no—
496#41—6112no—
497#41—614no—
498#41—6116no—
499#41—6148no—
500#41—6124no—
501#41—12248no—
502#41—12248no—
503#41—12224no—
504#41—12248no—
505#41—1228no—
506#41—12248no—
507#41—12248no—
508#41—12248no—
509#41—12296no—
510#41—12224no—
511#41—12224no—
512#41—12248no—
513#41—12212no—
514#41D4×D6114yes—
515#41C23×D61124yes—
516#41C23×D6118yes—
517#41C23×D6114yes—
518C2×Q8—28no—
519C2×Q8—618no—
520C2×Q8—616no—
521C2×Q8—6148no—
522C2×Q8—6124no—
523C2×Q8—616no—
524C2×Q8—12224no—
525C2×Q8—12224no—
526C2×Q8—12248no—
527C2×Q8—12248no—
528C2×Q8—12224no—
529C2×Q8—12216no—
530C2×Q8—12212no—
531C2×Q8—1226no—
532C2×Q8D4×D6111yescharacteristic
533C2×Q8C23×D61112yes—
534C2×D4—212no—
535C2×D4—224no—
536C2×D4—6112no—
537C2×D4—6124no—
538C2×D4—6116no—
539C2×D4—616no—
540C2×D4—612no—
541C2×D4—1228no—
542C2×D4—12216no—
543C2×D4—12224no—
544C2×D4—12248no—
545C2×D4—12248no—
546C2×D4—12224no—
547C2×D4—1226no—
548C2×D4D4×D6113yes—
549C2×D4C23×D61112yes—
550C22×C4—212no—
551C22×C4—6112no—
552C22×C4—12248no—
553C22×C4—12224no—
554C22×C4—12224no—
555C22×C4—1226no—
556C22×C4—1226no—
557C22×C4C23×D6113yes—
558Q16—6148no—
559Q16—616no—
560Q16—6124no—
561Q16—616no—
562Q16C23×D6114yes—
563Q16C23×D61112yes—
564SD16—6116no—
565SD16—6148no—
566SD16—6124no—
567SD16C23×D61124yes—
568SD16C23×D6118yes—
569D8—6116no—
570D8—612no—
571D8—612no—
572D8C23×D6114yes—
573D8C23×D61112yes—
574M16—6112no—
575M16—614no—
576M16—6124no—
577M16—618no—
578M16C23×D61112yes—
579C2×C8—6112no—
580C2×C8—618no—
581C2×C8—6124no—
582C2×C8C23×D6116yes—
583C2×C8C23×D6111yescharacteristic
584C2×C8C23×D6116yes—
585C2×C8C23×D6113yes—
586C23—212no—
587C23—6112no—
588C23—12216no—
589C23—1222no—
590C23C2×D4×D6113yes—
591Q8—212no—
592Q8—614no—
593Q8—6112no—
594Q8—6112no—
595Q8—12248no—
596Q8—1226no—
597Q8—12224no—
598Q8C2×D4×D6114yes—
599Q8C24×D6118yes—
600D4—224no—
601D4—28no—
602D4—212no—
603D4—6124no—
604D4—618no—
605D4—614no—
606D4—6112no—
607D4—614no—
608D4—12216no—
609D4—12216no—
610D4—12248no—
611D4—12224no—
612D4—1222no—
613D4C2×D4×D61112yes—
614D4C24×D6118yes—
615C2×C4—212no—
616C2×C4—28no—
617C2×C4—224no—
618C2×C4—6112no—
619C2×C4—6124no—
620C2×C4—618no—
621C2×C4—12216no—
622C2×C4—12248no—
623C2×C4—12248no—
624C2×C4—12224no—
625C2×C4—1226no—
626C2×C4—12224no—
627C2×C4C2×D4×D6116yes—
628C2×C4C2×D4×D6113yes—
629C2×C4C24×D6116yes—
630C2×C4C24×D6111yescharacteristic
631C8—618no—
632C8C24×D6111yescharacteristic
633C8C24×D6116yes—
634C8C24×D6111yescharacteristic
635C22—28no—
636C22—43no—
637C22—812no—
638C22—618no—
639C22—1224no—
640C22—4882no—
641C22—48816no—
642C22—24412no—
643C22C22×D4×D6116yes—
644C22C22×D4×D6111yescharacteristic
645C4—28no—
646C4—618no—
647C4—12212no—
648C4C22×D4×D6111yescharacteristic
649C4C22×D4×D6116yes—
650C4C25×D6111yescharacteristic
651C2—21no—
652C2—26no—
653C2—48no—
654C2—1228no—
655C2—2444no—
656C2C23×D4×D6111yescharacteristic
subgroupG/H
centerC2C23×D4×D61
commutator, cosocleC4×C61C26
FrattiniC4C25×D61
FittingC61×#3584C2
socleC2×C61C24×D4
hypercenter#3584D61
hypocenterC61#63093
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×#41 ◁ #3584

Complements

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

NH ≅ G/Ntype
1. #21460632C2semidirect
2. #21460605C2semidirect
3. #21460603C2semidirect
4. #21460601C2semidirect
5. #21460588C2semidirect
6. #21460587C2semidirect
7. C2×#16840579C2semidirect
8. C2×#16840576C2semidirect
9. #578×D61C2semidirect
10. #21460530C2semidirect
11. C61×#3584C2semidirect
14. #16840598C22semidirect
16. #16840597C22semidirect
18. C2×#14519824C22semidirect
19. C2×#14519824C22semidirect
21. 2−1+4×D61C22semidirect
22. #16840579C22semidirect
23. #16840579C22semidirect
24. #16840578C22semidirect
25. #16840577C22semidirect
26. #16840576C22semidirect
27. #16840576C22semidirect
28. #16840576C22semidirect
29. #16840576C22semidirect
30. #16840575C22semidirect
31. #16840573C22semidirect
32. #16840572C22semidirect
33. #16840569C22semidirect
34. #16840568C22semidirect
35. #16840564C22semidirect
36. #16840563C22semidirect
37. #16840560C22semidirect
38. #16840559C22semidirect
39. #16840558C22semidirect
40. C2×#14519810C22semidirect
41. C2×#14519807C22semidirect
42. C2×#14519806C22semidirect
43. #137×D61C22semidirect
44. C2×#14519799C22semidirect
45. C2×#14519796C22semidirect
46. #135×D61C22semidirect
47. C22×Dic244C22semidirect
48. C2×Q16×D61C22semidirect
49. #16840529C22semidirect
50. #16840528C22semidirect
51. #16840527C22semidirect
52. C2×#14519783C22semidirect
53. #131×D61C22semidirect
54. C61×#585C22semidirect
55. C61×#578C22semidirect
56. C61×#577C22semidirect
57. C61×#576C22semidirect
58. C61×#575C22semidirect
59. C2×#135×C61C22semidirect
60. C2×#131×C61C22semidirect
329. #3584D61semidirect
345. #585C2×D61semidirect
360. #578C2×D61semidirect
365. #577C2×D61semidirect
368. #576C2×D61semidirect
372. #575C2×D61semidirect
380. C2×#135C2×D61semidirect
388. C2×#131C2×D61semidirect
389. C61#63093semidirect

Maximal subgroups

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

subgroupindex
#214606322
#214606052
#214606032
#214606012
#214605882
#214605872
C2×#168405792
C2×#168405762
#578×D612
#214605302
C61×#35842
#6309361