Finite Group Databasebeta

← #30625701

Subgroups — #30625701

Order 15616.

1950 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462665C211yesmaximal, characteristic, complemented
2#21462663C211yesmaximal, characteristic, complemented
3#21462652C213yesmaximal, complemented
4#21462651C213yesmaximal, complemented
5#21462635C213yesmaximal, complemented
6#21462633C213yesmaximal, complemented
7#21462625C211yesmaximal, characteristic, complemented
8#41×#92420C211yesmaximal, characteristic, complemented
9Q8×#11758217C211yesmaximal, characteristic, complemented
10C2×Q8×#92420C211yesmaximal, characteristic, complemented
11#21462561C213yesmaximal, complemented
12#21462542C213yesmaximal, complemented
13#21462514C213yesmaximal, complemented
14C4×#14519864C213yesmaximal, complemented
15#21459409C211yesmaximal, characteristic
16C2×#14519864C2213yes—
17C2×#14519864C2213yes—
18#16840813C2213yes—
19#16840813C2213yes—
20Q8×#92420C2211yescharacteristic
21Q8×#92420C2213yes—
22Q8×#92420C2213yes—
23Q8×#92420C2211yescharacteristic
24Q8×#92420C2211yescharacteristic
25Q8×#92420C2211yescharacteristic
26#16840797C2213yes—
27#16840797C2213yes—
28#16840796C2213yes—
29#16840796C2213yes—
30#16840795C2216yes—
31#16840794C2216yes—
32#16840791C2216yes—
33#16840790C2213yes—
34#16840790C2213yes—
35#16840789C2216yes—
36D4×#92420C2213yes—
37D4×#92420C2213yes—
38#16840787C2216yes—
39#16840786C2216yes—
40#16840784C2213yes—
41#16840783C2213yes—
42#16840781C2213yes—
43#16840778C2213yes—
44#16840765C2213yes—
45C2×C4⋊#92420C2213yes—
46#16840735C2216yes—
47#16840727C2213yes—
48#16840727C2213yes—
49#16840726C2213yes—
50#16840726C2211yescharacteristic
51#16840726C2213yes—
52#16840726C2213yes—
53C4×#11758213C2213yes—
54C4×#11758217C2213yes—
55C4×#11758217C2213yes—
56C4×D61⋊C8C2213yes—
57C2×C4×#92420C2213yes—
58C2×C4×#92420C2213yes—
59#16840494C413yes—
60#16840488C411yescharacteristic
61#16840483C413yes—
62#16840464C413yes—
63#16840458C413yes—
64#16840431C411yescharacteristic
65#16840423C411yescharacteristic
66#16840307C2213yes—
67#16840298C2213yes—
68#16840291C2213yes—
69#41×Dic61C2211yescharacteristic
70C4×#11758226C2211yescharacteristic
71C4×#11758224C2213yes—
72C2×Q8×Dic61C2211yescharacteristic
73C61×#548C411yescharacteristic
74#14519864—212no—
75#14519864—212no—
76C2×#11758213C2313yes—
77C2×#11758213C2313yes—
78C2×#11758217C2313yes—
79C2×#11758217C2313yes—
80C2×#11758217C2313yes—
81C2×#11758217C2311yescharacteristic
82C2×D61⋊C8C2313yes—
83C2×D61⋊C8C2313yes—
84C22×#92420C2313yes—
85C22×#92420C2313yes—
86C22×#92420C2313yes—
87C22×#92420C2311yescharacteristic
88#14519844C2316yes—
89C4⋊#92420C2313yes—
90C4⋊#92420C2313yes—
91C4⋊#92420C2316yes—
92C4⋊#92420C2313yes—
93C4⋊#92420C2313yes—
94#14519834C2316yes—
95#14519834C2316yes—
96C22⋊#92420C2316yes—
97C22⋊#92420C2316yes—
98#14519832C2311yescharacteristic
99#14519832C2313yes—
100#14519832C2313yes—
101#14519832C2313yes—
102#14519832C2311yescharacteristic
103#14519832C2313yes—
104C4×#92420C2311yescharacteristic
105C4×#92420C2313yes—
106C4×#92420C2313yes—
107C4×#92420C2313yes—
108C4×#92420C2313yes—
109C4×#92420C2313yes—
110C4×#92420C2313yes—
111C4×#92420C2311yescharacteristic
112C4×#92420C2313yes—
113C4×#92420C2313yes—
114C2×#11758226C2×C411yescharacteristic
115C2×#11758224C2313yes—
116#14519776C2×C413yes—
117#14519773C2×C413yes—
118#14519772C2×C416yes—
119#14519770C2×C416yes—
120#14519769C2×C413yes—
121#14519768C2×C413yes—
122#14519766C2×C413yes—
123#14519763C2×C416yes—
124#14519761C2×C416yes—
125#14519759C2×C416yes—
126#14519753C2×C413yes—
127#14519751C2×C413yes—
128#14519751C2×C413yes—
129#14519749C2×C416yes—
130#14519748C2×C413yes—
131Q8×Dic61C2×C413yes—
132Q8×Dic61C2×C411yescharacteristic
133Q8×Dic61C2×C411yescharacteristic
134Q8×Dic61C2311yescharacteristic
135Q8×Dic61C2313yes—
136C4×Dic122C2×C413yes—
137#14519742C2×C413yes—
138C4×D244C2×C413yes—
139D4×Dic61C2×C413yes—
140D4×Dic61C2×C413yes—
141D4×Dic61C2313yes—
142#14519739C2313yes—
143C4×C22⋊D61C2×C413yes—
144C4×C22⋊D61C2×C413yes—
145#14519737C2×C413yes—
146#14519737C2×C413yes—
147#14519737C2316yes—
148#14519736C2×C413yes—
149#14519735C2×C413yes—
150#14519735C2313yes—
151#14519733C2311yescharacteristic
152#14519733C2313yes—
153C2×C4⋊Dic61C2×C413yes—
154C2×C4×Dic61C2313yes—
155C2×C4×Dic61C2313yes—
156C42×D61C2313yes—
157C2×#41×C61C2×C411yescharacteristic
158C22×Q8×C61C2×C411yescharacteristic
159#124×C61C2×C413yes—
160C22⋊Q8×C61C2×C413yes—
161#121×C61C2×C413yes—
162C4×Q8×C61C2×C411yescharacteristic
163C4×D4×C61C2×C413yes—
164#41×C61—24no—
165C2×Q8×C61—24no—
166C2×Q8×C61C22×C413yes—
167C2×Q8×C61C22×C411yescharacteristic
168C2×D4×C61C22×C413yes—
169C2×D4×C61C22×C413yes—
170C22×C4×C61C22×C413yes—
171C22×C4×C61C22×C413yes—
172C2×C22⋊D61C22×C413yes—
173C2×C22⋊D61C22×C413yes—
174C2×C22⋊D61C22×C416yes—
175C22×Dic61C22×C413yes—
176C22×Dic61C22×C413yes—
177C22×Dic61C2413yes—
178C22×Dic61C2411yescharacteristic
179#11758226—24no—
180#11758224—26no—
181#11758224—26no—
182C2×D244C22×C413yes—
183C2×C4×D61C22×C413yes—
184C2×C4×D61C22×C413yes—
185C2×C4×D61C2413yes—
186C2×Dic122C22×C413yes—
187#11758217—26no—
188#11758217—26no—
189#11758217—26no—
190#11758217#4114yescomplemented
191C2×#92420—26no—
192C2×#92420—26no—
193C2×#92420—26no—
194C2×#92420#4114yescomplemented
195C2×#92420C2413yes—
196C2×#92420C2413yes—
197C2×#92420C2413yes—
198C2×#92420C2413yes—
199C2×#92420C2411yescharacteristic
200C2×#92420C2411yescharacteristic
201C2×#92420C2411yescharacteristic
202C2×#92420C2411yescharacteristic
203#11758213—26no—
204#11758213—26no—
205D61⋊C8—26no—
206D61⋊C8—26no—
207C4⋊C4×C61C22×C413yes—
208C4⋊C4×C61C22×C413yes—
209C22⋊C4×C61C22×C411yescharacteristic
210C22⋊C4×C61C22×C413yes—
211C22⋊C4×C61C22×C416yes—
212C42×C61C22×C413yes—
213C22⋊Dic61C22×C411yescharacteristic
214C22⋊Dic61C22×C413yes—
215C22⋊Dic61C22×C416yes—
216#11758198C22×C416yes—
217#11758198C22×C413yes—
218#11758198C22×C413yes—
219#11758198C22×C416yes—
220C4⋊Dic61C22×C413yes—
221C4⋊Dic61C22×C413yes—
222C4⋊Dic61C22×C416yes—
223C4⋊Dic61C22×C413yes—
224C4⋊Dic61C22×C413yes—
225#11758196C22×C411yescharacteristic
226#11758196C22×C413yes—
227#11758196C22×C411yescharacteristic
228#11758196C22×C413yes—
229#11758196C22×C416yes—
230C4×Dic61C22×C413yes—
231C4×Dic61C22×C413yes—
232C4×Dic61C22×C413yes—
233C4×Dic61C22×C413yes—
234C4×Dic61C22×C413yes—
235C4×Dic61C2411yescharacteristic
236C4×Dic61C2411yescharacteristic
237C4×Dic61C2413yes—
238C4×Dic61C2413yes—
239C23×C61C23×C413yes—
240C23×C61C23×C411yescharacteristic
241C22×D61C23×C413yes—
242Q8×C61—26no—
243Q8×C61#13114yescomplemented
244D4×C61—26no—
245D4×C61—26no—
246C2×C4×C61—26no—
247C2×C4×C61—26no—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C23×C413yes—
250C2×C4×C61C23×C413yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C23×C413yes—
253C22⋊D61—26no—
254C22⋊D61—26no—
255C22⋊D61—46no—
256C2×Dic61—23no—
257C2×Dic61—23no—
258C2×Dic61—43no—
259C2×Dic61—43no—
260C2×Dic61C23×C411yescharacteristic
261C2×Dic61C23×C411yescharacteristic
262C2×Dic61C23×C413yes—
263C2×Dic61C23×C413yes—
264C2×Dic61C23×C413yes—
265C2×Dic61C23×C411yescharacteristic
266C2×Dic61C2×#4112yes—
267C2×Dic61C2×#4112yes—
268C2×Dic61C2511yescharacteristic
269D244—26no—
270C4×D61—26no—
271C4×D61—23no—
272C4×D61—23no—
273C4×D61—43no—
274Dic122—26no—
275#92420—23no—
276#92420—23no—
277#92420—23no—
278#92420—23no—
279#92420C2×#4112yes—
280#92420C2×#4112yes—
281#92420C2×#4112yes—
282#92420C2×#4112yes—
283#60218—611nomaximal, Sylow
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—23no—
287C22×C61—23no—
288C22×C61—23no—
289C22×C61C4×#4112yes—
290C22×C61C24×C411yescharacteristic
291C2×D61—23no—
292C2×D61—23no—
293C2×D61—23no—
294C4×C61—23no—
295C4×C61—21no—
296C4×C61—23no—
297C4×C61—21no—
298C4×C61C2×#13116yes—
299Dic61—23no—
300Dic61—21no—
301Dic61—23no—
302Dic61—21no—
303Dic61—23no—
304Dic61C4×#4112yes—
305Dic61C22×#4111yescharacteristic
306Dic61C22×#4111yescharacteristic
307#3480—611no—
308#2971—611no—
309#2971—611no—
310#2969—613no—
311#2969—613no—
312#2966—613no—
313#2966—613no—
314C8×#41—611no—
315C8×#41—611no—
316Q8×M16—611no—
317C2×C8×Q8—611no—
318#2935—613no—
319#2931—613no—
320#2925—613no—
321C4×#131—613no—
322C2×C61—22no—
323C2×C61—43no—
324C2×C61C2×C4×#4111yescharacteristic
325C2×C61C22×#13111yescharacteristic
326C2×C61#357311yescharacteristic
327D61—43no—
328C2×#131—613no—
329C2×#131—613no—
330#559—613no—
331#557—611no—
332#556—613no—
333#553—613no—
334#553—613no—
335#550—611no—
336#548—611no—
337#548#692311yescharacteristic
338#532—613no—
339#529—613no—
340#529—613no—
341C4×#41—611no—
342C4×#41—611no—
343C4×#41—613no—
344C2×C4×Q8—611no—
345#446—613no—
346#446—613no—
347C8×Q8—611no—
348C8×Q8—613no—
349C8×Q8—613no—
350C8×Q8—611no—
351C8×Q8—611no—
352C8×Q8—611no—
353#436—616no—
354#436—613no—
355#436—613no—
356#436—613no—
357#436—613no—
358#435—616no—
359#435—616no—
360C8×D4—613no—
361C8×D4—613no—
362C8×D4—616no—
363C8×D4—613no—
364C8×D4—613no—
365#433—616no—
366#433—616no—
367#432—613no—
368#432—613no—
369#431—613no—
370#431—613no—
371#423—613no—
372C2×C4⋊C8—613no—
373#408—616no—
374#405—611no—
375#405—613no—
376#405—613no—
377#405—613no—
378#405—613no—
379#405—613no—
380C4×M16—613no—
381C4×M16—613no—
382C4×M16—613no—
383C2×C4×C8—613no—
384C2×C4×C8—613no—
385C2×C4×C8—613no—
386C61#6021811yescharacteristic, complemented, Sylow
387C2×#41—611no—
388C2×#41—613no—
389C2×#41C2×#692311yescharacteristic
390C22×Q8C2×#692311yescharacteristic
391#131—12212no—
392#131—12212no—
393C2×M16—611no—
394C2×M16—613no—
395C2×M16—613no—
396C2×M16—613no—
397C2×M16—613no—
398C2×M16—613no—
399C22×C8—611no—
400C22×C8—613no—
401C22×C8—613no—
402C22×C8—613no—
403C22×C8—613no—
404C22×C8—613no—
405C4⋊Q8—613no—
406#127—613no—
407#126—616no—
408#126—616no—
409#125—613no—
410#125—613no—
411#124—613no—
412#124—616no—
413#124C2×#692313yes—
414#123—616no—
415#123—616no—
416C22⋊Q8—613no—
417C22⋊Q8C2×#692313yes—
418#121—613no—
419#121—616no—
420#121—613no—
421#121—613no—
422#121C2×#692313yes—
423C4×Q8—611no—
424C4×Q8—613no—
425C4×Q8—611no—
426C4×Q8—613no—
427C4×Q8—611no—
428C4×Q8—613no—
429C4×Q8—613no—
430C4×Q8C2×#692311yescharacteristic
431C4×D4—613no—
432C4×D4—613no—
433C4×D4—613no—
434C4×D4—613no—
435C4×D4—613no—
436C4×D4—613no—
437C4×D4—613no—
438C4×D4—613no—
439C4×D4—616no—
440C4×D4—613no—
441C4×D4C2×#692313yes—
442#117—613no—
443#117—613no—
444#117—613no—
445#117—611no—
446#117—613no—
447C2×C4⋊C4—613no—
448C2×C42—613no—
449C2×C42—613no—
450C2×C42—613no—
451C4⋊C8—613no—
452C4⋊C8—613no—
453C4⋊C8—616no—
454C4⋊C8—613no—
455C4⋊C8—613no—
456C4⋊C8—616no—
457C22⋊C8—616no—
458C22⋊C8—616no—
459C22⋊C8—616no—
460C22⋊C8—616no—
461#97—611no—
462#97—611no—
463#97—613no—
464#97—613no—
465#97—613no—
466#97—613no—
467C4×C8—611no—
468C4×C8—611no—
469C4×C8—613no—
470C4×C8—613no—
471C4×C8—613no—
472C4×C8—613no—
473C4×C8—613no—
474C4×C8—613no—
475C4×C8—613no—
476C4×C8—613no—
477#41—24no—
478#41—1226no—
479#41—1226no—
480#41—1224no—
481C2×Q8—24no—
482C2×Q8—613no—
483C2×Q8C22×#692313yes—
484C2×Q8C22×#692311yescharacteristic
485C2×D4—613no—
486C2×D4—613no—
487C2×D4—616no—
488C2×D4—613no—
489C2×D4C22×#692313yes—
490C2×D4C22×#692313yes—
491C22×C4—611no—
492C22×C4—613no—
493C22×C4—613no—
494C22×C4—613no—
495C22×C4—613no—
496C22×C4—613no—
497C22×C4—613no—
498C22×C4C22×#692313yes—
499C22×C4C22×#692313yes—
500M16—614no—
501M16—1226no—
502M16—1226no—
503M16—1226no—
504M16—1226no—
505M16—1226no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—614no—
511C2×C8—613no—
512C2×C8—613no—
513C2×C8—613no—
514C2×C8—613no—
515C2×C8—1226no—
516C2×C8—1226no—
517C2×C8—1226no—
518C2×C8—1226no—
519C2×C8—1226no—
520C4⋊C4—613no—
521C4⋊C4—613no—
522C4⋊C4—616no—
523C4⋊C4—611no—
524C4⋊C4—611no—
525C4⋊C4—613no—
526C4⋊C4—613no—
527C4⋊C4—616no—
528C4⋊C4—613no—
529C4⋊C4—613no—
530C4⋊C4C22×#692313yes—
531C4⋊C4C22×#692313yes—
532C22⋊C4—611no—
533C22⋊C4—613no—
534C22⋊C4—616no—
535C22⋊C4—613no—
536C22⋊C4—616no—
537C22⋊C4—613no—
538C22⋊C4—616no—
539C22⋊C4C22×#692311yescharacteristic
540C22⋊C4C22×#692313yes—
541C22⋊C4C22×#692316yes—
542C42—611no—
543C42—611no—
544C42—613no—
545C42—613no—
546C42—613no—
547C42—613no—
548C42—613no—
549C42—613no—
550C42—613no—
551C42C22×#692313yes—
552C23—613no—
553C23C23×#692313yes—
554C23C23×#692311yescharacteristic
555Q8—26no—
556Q8—1226no—
557Q8#1451986414yescomplemented
558D4—26no—
559D4—26no—
560D4—1226no—
561D4—1226no—
562D4—1226no—
563D4—2446no—
564C2×C4—26no—
565C2×C4—26no—
566C2×C4—612no—
567C2×C4—612no—
568C2×C4—611no—
569C2×C4—611no—
570C2×C4—611no—
571C2×C4—611no—
572C2×C4—613no—
573C2×C4—613no—
574C2×C4—613no—
575C2×C4—1223no—
576C2×C4—1223no—
577C2×C4—1223no—
578C2×C4—1223no—
579C2×C4—1226no—
580C2×C4—2443no—
581C2×C4—2443no—
582C2×C4—2443no—
583C2×C4C23×#692311yescharacteristic
584C2×C4C23×#692313yes—
585C2×C4C23×#692311yescharacteristic
586C2×C4C23×#692313yes—
587C2×C4C23×#692313yes—
588C8—612no—
589C8—612no—
590C8—612no—
591C8—612no—
592C8—1223no—
593C8—1223no—
594C8—1223no—
595C8—1223no—
596C22—21no—
597C22—21no—
598C22—23no—
599C22—23no—
600C22—23no—
601C22—1223no—
602C22—1223no—
603C22—1223no—
604C22#41×#692312yes—
605C22C24×#692311yescharacteristic
606C4—23no—
607C4—21no—
608C4—23no—
609C4—21no—
610C4—612no—
611C4—611no—
612C4—611no—
613C4—1223no—
614C4—1221no—
615C4—1223no—
616C4—1221no—
617C4—1223no—
618C4C2×#1451986416yes—
619C2—22no—
620C2—43no—
621C2—2443no—
622C2C2×#41×#692311yescharacteristic
623C2C22×#1451986411yescharacteristic
624C2#2146281511yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#548C4
prime radical#21459409C2
cosocleC2×Dic61C25
hypercenter#548#6923
hypocenterC61#60218
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #21462665C2semidirect
2. #21462663C2semidirect
3. #21462652C2semidirect
4. #21462651C2semidirect
5. #21462635C2semidirect
6. #21462633C2semidirect
7. #21462625C2semidirect
8. #41×#92420C2semidirect
9. Q8×#11758217C2semidirect
10. C2×Q8×#92420C2semidirect
11. #21462561C2semidirect
12. #21462542C2semidirect
13. #21462514C2semidirect
14. C4×#14519864C2semidirect
190. #11758217#41semidirect
194. C2×#92420#41semidirect
243. Q8×C61#131semidirect
386. C61#60218semidirect
557. Q8#14519864semidirect

Maximal subgroups

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

subgroupindex
#214626652
#214626632
#214626522
#214626512
#214626352
#214626332
#214626252
#41×#924202
Q8×#117582172
C2×Q8×#924202
#214625612
#214625422
#214625142
C4×#145198642
#214594092
#6021861