Finite Group Databasebeta

← #30570414

Subgroups — #30570414

Order 15616.

1704 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460217C211yesmaximal, characteristic
2#21459629C211yesmaximal, characteristic, complemented
3#21458769C214yesmaximal, complemented
4#21458728C212yesmaximal, complemented
5#21458693C212yesmaximal, complemented
6#21458552C212yesmaximal, complemented
7#21458549C214yesmaximal, complemented
8#21458485C212yesmaximal, complemented
9#21457068C214yesmaximal, complemented
10#21457019C211yesmaximal, characteristic, complemented
11#21456951C211yesmaximal, characteristic, complemented
12#21456104C214yesmaximal, complemented
13#21455850C211yesmaximal, characteristic, complemented
14#21454916C211yesmaximal, characteristic, complemented
15C61×#3112C211yesmaximal, characteristic, complemented
16#16840506—22no—
17Q8×Dic122—21no—
18Q8×Dic122C2212yes—
19#16840491—21no—
20#16840491C2212yes—
21#16840488—22no—
22#16840488C2214yes—
23#16840473C2212yes—
24#16840469C2212yes—
25#16840433C2212yes—
26#16840432C2212yes—
27#16840417—21no—
28#16840416—21no—
29#16840393—22no—
30#16840393C2212yes—
31#16840365C2214yes—
32#16840341—22no—
33#16840340—22no—
34#16840319—22no—
35#16840319C2211yescharacteristic
36#16840319C2211yescharacteristic
37#16840312C2211yescharacteristic
38#16840240C2212yes—
39#16840171C2212yes—
40#16840170C2214yes—
41#16840165C2211yescharacteristic
42#16840163C2214yes—
43#16840158C2211yescharacteristic
44#16840127C2212yes—
45#16840125C2214yes—
46#16840108C2211yescharacteristic
47#16840108C2211yescharacteristic
48#16840107C2214yes—
49#16840087C2214yes—
50#16840083C2214yes—
51#16840080C2212yes—
52#16840071C2214yes—
53#16840067C2214yes—
54#16840065C2214yes—
55#16840062C2214yes—
56#16840060C2214yes—
57#16840053C2212yes—
58#16840050C2214yes—
59#16840047C2212yes—
60#16840043C2212yes—
61#16840041C2214yes—
62#16839962C2212yes—
63#16839960C2214yes—
64#16839960C2214yes—
65#16839901C2212yes—
66#16839890C2214yes—
67#16839889C2212yes—
68#16839886C2212yes—
69#16839865C2214yes—
70#16839863C2214yes—
71SD16×Dic61C2214yes—
72C4×#11758201C2212yes—
73C4×#11758201C2212yes—
74#16839847C2214yes—
75#16839820C2214yes—
76#16839815C2211yescharacteristic
77C61×#532C2211yescharacteristic
78C2×C4×Q8×C61C2211yescharacteristic
79C61×#480C2212yes—
80C61×#477C2212yes—
81C61×#460C2212yes—
82C61×#448C2212yes—
83C61×#440C2214yes—
84C61×#432C2211yescharacteristic
85C2×#11758222—21no—
86C2×#11758222—22no—
87C2×#11758224—22no—
88C2×#11758201C2312yes—
89C2×#11758201C2312yes—
90#14519777—22no—
91#14519777—22no—
92#14519777D411yescharacteristic
93#14519777D411yescharacteristic
94#14519777C2311yescharacteristic
95#14519777C2311yescharacteristic
96#14519777C2311yescharacteristic
97#14519776—24no—
98#14519776—22no—
99#14519776—24no—
100#14519776C2314yes—
101#14519774C2311yescharacteristic
102#14519773C2314yes—
103#14519769—24no—
104#14519769D412yes—
105#14519769C2312yes—
106#14519768—22no—
107#14519768—24no—
108#14519768—24no—
109#14519768C2314yes—
110#14519768C2314yes—
111#14519765—21no—
112#14519761—24no—
113#14519759—24no—
114#14519757—22no—
115#14519756—21no—
116#14519756—21no—
117#14519756—22no—
118#14519756—24no—
119#14519756D412yes—
120#14519756D412yes—
121#14519754—24no—
122#14519753—24no—
123#14519751—21no—
124#14519751—22no—
125#14519751C2312yes—
126#14519750—24no—
127#14519749—24no—
128#14519749—24no—
129#14519748—24no—
130#14519748—22no—
131Q8×Dic61—22no—
132Q8×Dic61—22no—
133Q8×Dic61C2314yes—
134Q8×Dic61C2314yes—
135C4×Dic122—22no—
136C4×Dic122—21no—
137C4×Dic122—21no—
138C4×Dic122—22no—
139C4×Dic122—24no—
140C4×Dic122C2312yes—
141C4×Dic122C2312yes—
142C4×D244—21no—
143C4×D244C2312yes—
144D4×Dic61C2314yes—
145#14519739C2314yes—
146#14519736—24no—
147#14519736C2312yes—
148#14519736C2312yes—
149C2×C4⋊Dic61—22no—
150C4⋊C4×D61—22no—
151#14519711C2314yes—
152#14519710C2314yes—
153#14519709C2314yes—
154#14519708C2312yes—
155#14519708C2312yes—
156#14519708C2314yes—
157C4⋊#92418C2312yes—
158#14519706C2314yes—
159#14519706C2314yes—
160#14519701C2312yes—
161#14519701C2312yes—
162#14519699C2314yes—
163#14519699C2314yes—
164#14519698C2314yes—
165#14519698C2314yes—
166#14519697C2312yes—
167#14519697C2312yes—
168C22⋊#92418C2312yes—
169#14519687C2314yes—
170C4×#92418C2311yescharacteristic
171C4×#92418C2311yescharacteristic
172C8×Dic61C2314yes—
173C2×#41×C61—21no—
174C22×Q8×C61C2311yescharacteristic
175C2×SD16×C61C2314yes—
176C4⋊Q8×C61C2311yescharacteristic
177#127×C61C2311yescharacteristic
178#124×C61—21no—
179#121×C61—21no—
180#121×C61C2312yes—
181C4×Q8×C61—21no—
182C4×Q8×C61—21no—
183C4×Q8×C61C2312yes—
184C4×Q8×C61C2312yes—
185C4×D4×C61—21no—
186C4×D4×C61C2312yes—
187C2×C4⋊C4×C61—21no—
188C2×C4⋊C4×C61C2311yescharacteristic
189C2×C42×C61—21no—
190C2×C42×C61C2311yescharacteristic
191#107×C61C2314yes—
192C4⋊C8×C61C2312yes—
193#103×C61C2314yes—
194#102×C61C2314yes—
195C22⋊C8×C61C2312yes—
196#97×C61C2312yes—
197#41×C61—82no—
198C2×Q8×C61—24no—
199C2×Q8×C61—21no—
200C2×Q8×C61—21no—
201C2×Q8×C61C2412yes—
202C2×D4×C61—22no—
203C2×D4×C61—21no—
204C2×D4×C61—21no—
205C2×D4×C61C2412yes—
206C22×C4×C61—21no—
207C22×C4×C61—21no—
208C22×C4×C61—22no—
209C22×C4×C61C2×D411yescharacteristic
210C22×C4×C61C2×D411yescharacteristic
211C22×C4×C61C2411yescharacteristic
212C2×C22⋊D61—22no—
213C2×C22⋊D61—24no—
214C2×C22⋊D61—24no—
215C22×Dic61—24no—
216#11758224—84no—
217#11758222—84no—
218#11758222—82no—
219C2×D244—21no—
220C2×D244—21no—
221C2×D244C2412yes—
222C2×C4×D61—22no—
223C2×C4×D61—24no—
224C2×Dic122—21no—
225C2×Dic122—21no—
226C2×Dic122—21no—
227C2×Dic122—24no—
228C2×Dic122C2×D412yes—
229C2×Dic122C2×D412yes—
230SD16×C61—28no—
231C2×C8×C61C2414yes—
232C4⋊C4×C61—22no—
233C4⋊C4×C61—22no—
234C4⋊C4×C61—21no—
235C4⋊C4×C61—21no—
236C4⋊C4×C61C2412yes—
237C4⋊C4×C61C2412yes—
238C22⋊C4×C61—22no—
239C22⋊C4×C61—22no—
240C42×C61—22no—
241C42×C61—22no—
242C42×C61C2×D411yescharacteristic
243C42×C61C2×D411yescharacteristic
244C42×C61C2411yescharacteristic
245C42×C61C2411yescharacteristic
246C22⋊Dic61—22no—
247C22⋊Dic61—24no—
248C22⋊Dic61C2×D414yes—
249#11758201—24no—
250#11758201—24no—
251#11758198—22no—
252#11758198—22no—
253#11758198—24no—
254C4⋊Dic61—21no—
255C4⋊Dic61—21no—
256C4⋊Dic61—21no—
257C4⋊Dic61—24no—
258C4⋊Dic61—24no—
259C4⋊Dic61—24no—
260C4⋊Dic61—24no—
261C4⋊Dic61C2×D412yes—
262C4⋊Dic61C2×D412yes—
263C4⋊Dic61C2×D412yes—
264C4⋊Dic61C2×D412yes—
265C4⋊Dic61C2412yes—
266C4⋊Dic61C2412yes—
267C4⋊Dic61C2412yes—
268#11758196—22no—
269#11758196—22no—
270#11758196—24no—
271#11758196—24no—
272#11758196—24no—
273#11758196C2×D414yes—
274#11758196C2×D414yes—
275C4×Dic61—22no—
276C4×Dic61—24no—
277C4×Dic61—24no—
278C4×Dic61C2414yes—
279C2×#92418C2412yes—
280C2×#92418C2412yes—
281C23×C61—22no—
282C23×C61C22×D411yescharacteristic
283C22×D61—22no—
284Q8×C61—24no—
285Q8×C61—22no—
286Q8×C61—81no—
287Q8×C612−1+414yes—
288D4×C61—24no—
289D4×C61—42no—
290D4×C61—81no—
291D4×C61—82no—
292C2×C4×C61—24no—
293C2×C4×C61—22no—
294C2×C4×C61—22no—
295C2×C4×C61—22no—
296C2×C4×C61—22no—
297C2×C4×C61—22no—
298C2×C4×C61—41no—
299C2×C4×C61—44no—
300C2×C4×C61—81no—
301C2×C4×C61C22×D411yescharacteristic
302C2×C4×C61C22×D411yescharacteristic
303C2×C4×C61C22×D411yescharacteristic
304C2×C4×C61C22×D411yescharacteristic
305C2×C4×C61C22×D411yescharacteristic
306C2×C4×C61C2511yescharacteristic
307C22⋊D61—84no—
308C22⋊D61—84no—
309C22⋊D61—82no—
310C2×Dic61—22no—
311C2×Dic61—22no—
312C2×Dic61—22no—
313C2×Dic61—84no—
314C2×Dic61C22×D414yes—
315C2×Dic61C22×D414yes—
316D244—22no—
317D244—22no—
318D244—42no—
319D244—81no—
320C4×D61—44no—
321C4×D61—84no—
322Dic122—24no—
323Dic122—42no—
324Dic122—42no—
325Dic122—84no—
326Dic122—81no—
327Dic122—81no—
328C8×C61—24no—
329#92418—22no—
330#92418—22no—
331#61851—611nomaximal, Sylow
332C22×C61—21no—
333C22×C61—21no—
334C22×C61—21no—
335C22×C61—42no—
336C22×C61—42no—
337C22×C61—42no—
338C22×C61C23×D411yescharacteristic
339C2×D61—42no—
340C2×D61—42no—
341C2×D61—42no—
342C4×C61—21no—
343C4×C61—21no—
344C4×C61—24no—
345C4×C61—21no—
346C4×C61—21no—
347C4×C61—42no—
348C4×C61—42no—
349C4×C61#57712yes—
350C4×C61C2×2−1+412yes—
351Dic61—24no—
352Dic61—24no—
353Dic61—42no—
354Dic61—42no—
355Dic61—42no—
356#3548—611no—
357#3502—611no—
358#3388—614no—
359#3385—612no—
360#3381—612no—
361#3368—612no—
362#3368—614no—
363#3362—612no—
364#3241—614no—
365#3235—611no—
366#3228—611no—
367#3126—614no—
368#3112D6111yescharacteristic, complemented
369#3103—611no—
370#2985—611no—
371C2×C61—41no—
372C2×C61—82no—
373C2×C61#348611yescharacteristic
374C2×C61C2×#57711yescharacteristic
375C2×C61#358511yescharacteristic
376D61—82no—
377#562—1222no—
378Q82—612no—
379Q82—1221no—
380#557—614no—
381#557—612no—
382#557—1222no—
383#557—1221no—
384#554—612no—
385#554—612no—
386#550—612no—
387#550—612no—
388#547—1221no—
389#547—1221no—
390#544—612no—
391#544—1222no—
392#540—614no—
393#537—1222no—
394#537—1222no—
395#533—611no—
396#533—611no—
397#533—1222no—
398#532—611no—
399#532C2×D6111yescharacteristic
400#519—612no—
401C2×C4×Q8C2×D6111yescharacteristic
402#501—614no—
403#501—612no—
404#500—611no—
405#499—614no—
406#498—611no—
407#490—614no—
408#490—612no—
409#486—611no—
410#486—611no—
411#486—614no—
412#482—614no—
413#482—614no—
414#481—612no—
415#480C2×D6112yes—
416#479—614no—
417#479—614no—
418#478—614no—
419#478—614no—
420#478—614no—
421#477C2×D6112yes—
422#476—612no—
423#476—614no—
424#475—612no—
425#474—614no—
426#474—612no—
427#460—614no—
428#460—614no—
429#460—612no—
430#460C2×D6112yes—
431#448—612no—
432#448C2×D6112yes—
433#446—612no—
434#446—614no—
435C8×Q8—612no—
436#440—614no—
437#440—614no—
438#440C2×D6114yes—
439C4×SD16—614no—
440C4×SD16—612no—
441C4×SD16—612no—
442C4×SD16—614no—
443#433—614no—
444#432C2×D6111yescharacteristic
445#431—611no—
446C61#6185111yescharacteristic, complemented, Sylow
447C2×#41—21no—
448C2×#41—1222no—
449C2×#41—1222no—
450C2×#41—1221no—
451C22×Q8C22×D6111yescharacteristic
452C2×SD16—612no—
453C2×SD16—612no—
454C2×SD16C22×D6114yes—
455C4⋊Q8—614no—
456C4⋊Q8—611no—
457C4⋊Q8—611no—
458C4⋊Q8—611no—
459C4⋊Q8—611no—
460C4⋊Q8—611no—
461C4⋊Q8—1222no—
462C4⋊Q8—1222no—
463C4⋊Q8—1222no—
464C4⋊Q8—1224no—
465C4⋊Q8—1224no—
466C4⋊Q8C22×D6111yescharacteristic
467#127—614no—
468#127—611no—
469#127C22×D6111yescharacteristic
470#125—614no—
471#125—614no—
472#125—612no—
473#125—612no—
474#125—1224no—
475#125—1222no—
476#125—1224no—
477#125—1224no—
478#124—21no—
479#124—1221no—
480#123—1224no—
481#123—1224no—
482C22⋊Q8—612no—
483C22⋊Q8—612no—
484C22⋊Q8—1224no—
485C22⋊Q8—1224no—
486C22⋊Q8—1222no—
487C22⋊Q8—1224no—
488C22⋊Q8—1222no—
489C22⋊Q8—1221no—
490C22⋊Q8—1221no—
491#121—21no—
492#121—612no—
493#121—1224no—
494#121—1224no—
495#121—1224no—
496#121—1222no—
497#121—1222no—
498#121—1224no—
499#121—1221no—
500#121C22×D6112yes—
501C4×Q8—21no—
502C4×Q8—21no—
503C4×Q8—612no—
504C4×Q8—612no—
505C4×Q8—614no—
506C4×Q8—614no—
507C4×Q8—1224no—
508C4×Q8—1222no—
509C4×Q8—1222no—
510C4×Q8—1222no—
511C4×Q8—1222no—
512C4×Q8—1221no—
513C4×Q8—1221no—
514C4×Q8C22×D6112yes—
515C4×Q8C22×D6112yes—
516C4×D4—21no—
517C4×D4—614no—
518C4×D4—614no—
519C4×D4—612no—
520C4×D4—1221no—
521C4×D4C22×D6112yes—
522#117—612no—
523#117—612no—
524#117—1224no—
525C2×C4⋊C4—21no—
526C2×C4⋊C4—1222no—
527C2×C4⋊C4—1222no—
528C2×C4⋊C4C22×D6111yescharacteristic
529C2×C42—21no—
530C2×C42C22×D6111yescharacteristic
531#107—614no—
532#107—614no—
533#107C22×D6114yes—
534#106—614no—
535#106—612no—
536#106—612no—
537#106—614no—
538C4⋊C8—614no—
539C4⋊C8—614no—
540C4⋊C8—612no—
541C4⋊C8C22×D6112yes—
542#103—612no—
543#103—612no—
544#103C22×D6114yes—
545#102—614no—
546#102—614no—
547#102—614no—
548#102—614no—
549#102—612no—
550#102—612no—
551#102C22×D6114yes—
552C22⋊C8—612no—
553C22⋊C8C22×D6112yes—
554#97—614no—
555#97C22×D6112yes—
556C4×C8—614no—
557C4×C8—611no—
558C4×C8—611no—
559#41—82no—
560#41—4882no—
561#41—4884no—
562#41—4884no—
563C2×Q8—24no—
564C2×Q8—21no—
565C2×Q8—21no—
566C2×Q8—612no—
567C2×Q8—612no—
568C2×Q8—1224no—
569C2×Q8—1221no—
570C2×Q8—1221no—
571C2×Q8—1221no—
572C2×Q8C23×D6112yes—
573C2×D4—22no—
574C2×D4—21no—
575C2×D4—21no—
576C2×D4—612no—
577C2×D4—1224no—
578C2×D4—1224no—
579C2×D4—1222no—
580C2×D4—1221no—
581C2×D4—1221no—
582C2×D4C23×D6112yes—
583C22×C4—21no—
584C22×C4—21no—
585C22×C4—22no—
586C22×C4—1224no—
587C22×C4—1224no—
588C22×C4—1222no—
589C22×C4D4×D6111yescharacteristic
590C22×C4D4×D6111yescharacteristic
591C22×C4C23×D6111yescharacteristic
592SD16—28no—
593SD16—1224no—
594SD16—1224no—
595C2×C8—612no—
596C2×C8—612no—
597C2×C8C23×D6114yes—
598C4⋊C4—22no—
599C4⋊C4—22no—
600C4⋊C4—21no—
601C4⋊C4—21no—
602C4⋊C4—612no—
603C4⋊C4—612no—
604C4⋊C4—614no—
605C4⋊C4—614no—
606C4⋊C4—612no—
607C4⋊C4—612no—
608C4⋊C4—612no—
609C4⋊C4—612no—
610C4⋊C4—612no—
611C4⋊C4—1224no—
612C4⋊C4—1224no—
613C4⋊C4—1224no—
614C4⋊C4—1224no—
615C4⋊C4—1224no—
616C4⋊C4—1224no—
617C4⋊C4—1224no—
618C4⋊C4—1221no—
619C4⋊C4—1222no—
620C4⋊C4—1222no—
621C4⋊C4—1221no—
622C4⋊C4—1221no—
623C4⋊C4C23×D6112yes—
624C4⋊C4C23×D6112yes—
625C22⋊C4—22no—
626C22⋊C4—22no—
627C22⋊C4—614no—
628C22⋊C4—1224no—
629C22⋊C4—1224no—
630C22⋊C4—1222no—
631C22⋊C4—1222no—
632C22⋊C4—1222no—
633C42—22no—
634C42—22no—
635C42—614no—
636C42—1224no—
637C42—1224no—
638C42—1222no—
639C42D4×D6111yescharacteristic
640C42D4×D6111yescharacteristic
641C42C23×D6111yescharacteristic
642C42C23×D6111yescharacteristic
643C23—22no—
644C23—1222no—
645C23C2×D4×D6111yescharacteristic
646Q8—24no—
647Q8—22no—
648Q8—81no—
649Q8—1224no—
650Q8—4881no—
651Q8—4881no—
652Q8—4884no—
653Q8—2442no—
654Q8—2442no—
655Q8#1451982414yes—
656D4—24no—
657D4—42no—
658D4—81no—
659D4—82no—
660D4—1222no—
661D4—1222no—
662D4—4881no—
663D4—4884no—
664D4—4884no—
665D4—4882no—
666D4—2442no—
667C2×C4—24no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—41no—
674C2×C4—44no—
675C2×C4—81no—
676C2×C4—614no—
677C2×C4—614no—
678C2×C4—1222no—
679C2×C4—1222no—
680C2×C4—1222no—
681C2×C4—4884no—
682C2×C4—4884no—
683C2×C4—2444no—
684C2×C4C2×D4×D6111yescharacteristic
685C2×C4C2×D4×D6111yescharacteristic
686C2×C4C2×D4×D6111yescharacteristic
687C2×C4C2×D4×D6111yescharacteristic
688C2×C4C2×D4×D6111yescharacteristic
689C2×C4C24×D6111yescharacteristic
690C8—24no—
691C8—1222no—
692C8—1222no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—2442no—
700C22—2442no—
701C22—2442no—
702C22C22×D4×D6111yescharacteristic
703C4—24no—
704C4—21no—
705C4—21no—
706C4—21no—
707C4—21no—
708C4—42no—
709C4—42no—
710C4—1224no—
711C4—1224no—
712C4—2442no—
713C4—2442no—
714C4—2442no—
715C4#1684057112yes—
716C4C2×#1451982412yes—
717C2—41no—
718C2—82no—
719C2—4882no—
720C2#2145946611yescharacteristic
721C2C2×#1684057111yescharacteristic
722C2#2146059511yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3112C2
socleC22×C61C23×D4
prime radical#21460217C2
hypercenter#3112D61
hypocenterC61#61851
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
2. #21459629C2semidirect
3. #21458769C2semidirect
4. #21458728C2semidirect
5. #21458693C2semidirect
6. #21458552C2semidirect
7. #21458549C2semidirect
8. #21458485C2semidirect
9. #21457068C2semidirect
10. #21457019C2semidirect
11. #21456951C2semidirect
12. #21456104C2semidirect
13. #21455850C2semidirect
14. #21454916C2semidirect
15. C61×#3112C2semidirect
368. #3112D61semidirect
446. C61#61851semidirect

Maximal subgroups

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

subgroupindex
#214602172
#214596292
#214587692
#214587282
#214586932
#214585522
#214585492
#214584852
#214570682
#214570192
#214569512
#214561042
#214558502
#214549162
C61×#31122
#6185161