Finite Group Databasebeta

← #30626597

Subgroups — #30626597

Order 15616.

1746 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462771C213yesmaximal, complemented
2#21462765C213yesmaximal, complemented
3#21462749C213yesmaximal, complemented
4#21462737C211yesmaximal, characteristic, complemented
5#21462727C213yesmaximal, complemented
6#21462721C211yesmaximal, characteristic, complemented
7#21462713C211yesmaximal, characteristic, complemented
8#21462663C211yesmaximal, characteristic, complemented
9#21462655C213yesmaximal, complemented
10#21462636C213yesmaximal, complemented
11#21462627C213yesmaximal, complemented
12Q8×#11758213C211yesmaximal, characteristic, complemented
13#21462615C213yesmaximal, complemented
14Q8×D61⋊C8C211yesmaximal, characteristic, complemented
15#21459439C211yesmaximal, characteristic
16#16840813C2213yes—
17#16840813C2216yes—
18#16840813C2213yes—
19#16840812C2213yes—
20#16840812C2213yes—
21#16840812C2213yes—
22#16840812C2213yes—
23#16840811C2213yes—
24#16840811C2213yes—
25#16840811C2213yes—
26#16840811C2213yes—
27Q8×#92420C2211yescharacteristic
28Q8×#92420C2211yescharacteristic
29Q8×#92420C2211yescharacteristic
30Q8×#92420C2211yescharacteristic
31#16840796C2213yes—
32#16840796C2213yes—
33#16840796C2213yes—
34#16840796C2213yes—
35#16840792C2211yescharacteristic
36#16840792C2213yes—
37#16840792C2213yes—
38#16840792C2213yes—
39#16840792C2216yes—
40#16840790C2213yes—
41#16840790C2216yes—
42#16840790C2213yes—
43#16840787C2216yes—
44#16840787C2216yes—
45#16840784C2213yes—
46#16840784C2213yes—
47#16840781C2213yes—
48#16840781C2213yes—
49#16840780C2213yes—
50#16840780C2213yes—
51#16840769C2216yes—
52#16840766C2213yes—
53D61⋊(C4⋊C8)C2213yes—
54D61⋊(C4⋊C8)C2213yes—
55#16840727C2216yes—
56C4×#11758213C2213yes—
57C4×#11758213C2213yes—
58C4×D61⋊C8C2213yes—
59#16840485C413yes—
60#16840450C413yes—
61#16840450C413yes—
62Q8×D244C411yescharacteristic
63#16840424C411yescharacteristic
64#16840424C411yescharacteristic
65#16840415C413yes—
66#16840317C2213yes—
67#16840297C2213yes—
68#16840297C2213yes—
69C4×#11758226C2213yes—
70C4×#11758226C2211yescharacteristic
71C4×Q8×D61C2211yescharacteristic
72C4×Q8×D61C2211yescharacteristic
73C61×#557C411yescharacteristic
74C2×#11758213C2311yescharacteristic
75C2×#11758213C2313yes—
76C2×#11758213C2313yes—
77C2×#11758213C2313yes—
78C2×D61⋊C8C2311yescharacteristic
79C2×D61⋊C8C2313yes—
80C2×D61⋊C8C2316yes—
81#14519844C2313yes—
82#14519844C2316yes—
83#14519844C2313yes—
84#14519844C2311yescharacteristic
85#14519844C2313yes—
86C4⋊#92420C2313yes—
87C4⋊#92420C2313yes—
88C4⋊#92420C2313yes—
89C4⋊#92420C2316yes—
90C4⋊#92420C2313yes—
91C4⋊#92420C2313yes—
92C4⋊#92420C2313yes—
93#14519834C2311yescharacteristic
94#14519834C2313yes—
95#14519834C2313yes—
96#14519834C2313yes—
97#14519834C2311yescharacteristic
98#14519834C2313yes—
99#14519834C2316yes—
100#14519832C2311yescharacteristic
101#14519832C2313yes—
102#14519832C2313yes—
103#14519832C2313yes—
104#14519832C2316yes—
105C4×#92420C2313yes—
106C4×#92420C2313yes—
107C4×#92420C2313yes—
108C4×#92420C2313yes—
109C4×#92420C2313yes—
110C4×#92420C2313yes—
111C4×#92420C2316yes—
112C2×#11758226C2×C411yescharacteristic
113C2×#11758226C2×C413yes—
114C2×Q8×D61C2×C411yescharacteristic
115C2×Q8×D61C2×C411yescharacteristic
116#14519777C2×C413yes—
117#14519772C2×C416yes—
118#14519772C2×C416yes—
119#14519768C2×C416yes—
120#14519765C2×C413yes—
121#14519765C2×C413yes—
122#14519762C2×C416yes—
123#14519762C2×C416yes—
124#14519757C2×C413yes—
125#14519757C2×C413yes—
126#14519757C2×C413yes—
127#14519757C2×C413yes—
128#14519753C2×C416yes—
129#14519752C2×C413yes—
130#14519752C2×C413yes—
131Q8×Dic61C2311yescharacteristic
132Q8×Dic61C2313yes—
133C4×Dic122C2×C413yes—
134C4×Dic122C2×C413yes—
135#14519742C2313yes—
136#14519742C2313yes—
137C4×D244C2×C413yes—
138C4×D244C2×C413yes—
139C4×D244C2×C413yes—
140C4×D244C2×C413yes—
141#14519739C2311yescharacteristic
142#14519739C2313yes—
143#14519739C2313yes—
144#14519739C2313yes—
145#14519735C2×C413yes—
146#14519735C2×C413yes—
147#14519735C2×C413yes—
148#14519735C2×C413yes—
149#14519735C2316yes—
150C4⋊C4×D61C2×C413yes—
151C4⋊C4×D61C2×C413yes—
152C42×D61C2313yes—
153C42×D61C2313yes—
154C42×D61C2313yes—
155C4⋊Q8×C61C2×C413yes—
156#125×C61C2×C413yes—
157#125×C61C2×C413yes—
158C4×Q8×C61C2×C411yescharacteristic
159C4×Q8×C61C2×C411yescharacteristic
160C4×Q8×C61C2×C413yes—
161C4×Q8×C61C2×C411yescharacteristic
162C2×Q8×C61C22×C413yes—
163C2×Q8×C61C22×C411yescharacteristic
164#11758226—24no—
165#11758226—46no—
166Q8×D61—24no—
167Q8×D61—24no—
168C2×D244C22×C411yescharacteristic
169C2×D244C22×C413yes—
170C2×D244C22×C413yes—
171C2×D244C22×C413yes—
172C2×C4×D61C22×C413yes—
173C2×C4×D61C22×C413yes—
174C2×C4×D61C22×C413yes—
175C2×C4×D61C22×C413yes—
176C2×C4×D61C22×C413yes—
177C2×C4×D61C22×C413yes—
178C2×C4×D61C2411yescharacteristic
179C2×C4×D61C2413yes—
180C2×C4×D61C2411yescharacteristic
181C2×Dic122C22×C413yes—
182C2×Dic122C22×C413yes—
183C2×#92420C2411yescharacteristic
184C2×#92420C2413yes—
185C2×#92420C2411yescharacteristic
186C2×#92420C2413yes—
187C2×#92420C2411yescharacteristic
188C2×#92420C2413yes—
189C2×#92420C2413yes—
190C2×#92420C2411yescharacteristic
191#11758213—22no—
192#11758213—26no—
193#11758213—43no—
194#11758213—43no—
195D61⋊C8—22no—
196D61⋊C8—46no—
197D61⋊C8—43no—
198C4⋊C4×C61C22×C411yescharacteristic
199C4⋊C4×C61C22×C413yes—
200C4⋊C4×C61C22×C413yes—
201C4⋊C4×C61C22×C413yes—
202C4⋊C4×C61C22×C413yes—
203C4⋊C4×C61C22×C413yes—
204C4⋊C4×C61C22×C416yes—
205C42×C61C22×C413yes—
206C42×C61C22×C413yes—
207C42×C61C22×C413yes—
208#11758198C22×C411yescharacteristic
209#11758198C22×C411yescharacteristic
210#11758198C22×C413yes—
211#11758198C22×C413yes—
212#11758198C22×C416yes—
213#11758198C22×C413yes—
214#11758198C22×C413yes—
215#11758198C22×C416yes—
216#11758198C22×C416yes—
217C4⋊Dic61C22×C413yes—
218C4⋊Dic61C22×C413yes—
219C4⋊Dic61C22×C413yes—
220C4⋊Dic61C22×C413yes—
221C4⋊Dic61C22×C416yes—
222#11758196C22×C416yes—
223#11758196C22×C416yes—
224C4×Dic61C22×C413yes—
225C4×Dic61C22×C413yes—
226C4×Dic61C22×C413yes—
227C4×Dic61C2413yes—
228C4×Dic61C2413yes—
229C4×Dic61C2413yes—
230C4×Dic61C2411yescharacteristic
231C22×D61C23×C413yes—
232C22×D61C23×C411yescharacteristic
233C22×D61C23×C411yescharacteristic
234Q8×C61—43no—
235Q8×C61#13114yes—
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C23×C413yes—
239C2×C4×C61C23×C413yes—
240C2×C4×C61C23×C413yes—
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C23×C413yes—
243C2×Dic61C23×C413yes—
244C2×Dic61C23×C411yescharacteristic
245C2×Dic61C23×C413yes—
246C2×Dic61C23×C413yes—
247C2×Dic61C2511yescharacteristic
248D244—26no—
249D244—22no—
250D244—26no—
251D244—43no—
252C4×D61—26no—
253C4×D61—26no—
254C4×D61—26no—
255C4×D61—26no—
256C4×D61—21no—
257C4×D61—23no—
258C4×D61—21no—
259C4×D61—21no—
260C4×D61—23no—
261C4×D61—21no—
262C4×D61—43no—
263C4×D61—43no—
264Dic122—26no—
265Dic122—26no—
266#92420—21no—
267#92420—21no—
268#92420—23no—
269#92420—23no—
270#92420—23no—
271#92420—23no—
272#92420—21no—
273#924202−1+412yes—
274#60335—611nomaximal, Sylow
275C22×C61C24×C411yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—23no—
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—23no—
282C2×D61—21no—
283C2×D61—23no—
284C2×D61#56812yes—
285C4×C61—21no—
286C4×C61—23no—
287C4×C61—21no—
288C4×C61—23no—
289C4×C61—21no—
290C4×C61—23no—
291C4×C61C2×#13116yes—
292Dic61—23no—
293Dic61—23no—
294Dic61—21no—
295Dic61—23no—
296Dic61C2×2−1+411yescharacteristic
297Dic61#58511yescharacteristic
298#3482—611no—
299#2994—613no—
300#2993—613no—
301#2990—613no—
302#2988—611no—
303#2985—613no—
304#2984—611no—
305#2982—611no—
306#2971—611no—
307#2969—613no—
308#2966—613no—
309#2965—613no—
310Q8×M16—611no—
311#2962—613no—
312C2×C8×Q8—611no—
313C2×C61C4×2−1+411yescharacteristic
314C2×C61C22×#13111yescharacteristic
315C2×C61C2×#56811yescharacteristic
316D61—22no—
317D61—41no—
318D61—43no—
319#557#692311yescharacteristic
320#556—613no—
321#552—613no—
322#552—613no—
323D4×Q8—611no—
324#548—611no—
325#548—611no—
326#547—613no—
327#533—613no—
328#529—613no—
329#529—613no—
330C4×#41—613no—
331C4×#41—611no—
332C2×C4×Q8—611no—
333C2×C4×Q8—611no—
334#446—613no—
335#446—613no—
336#446—613no—
337#446—613no—
338#446—613no—
339#446—616no—
340#446—613no—
341C8×Q8—611no—
342C8×Q8—611no—
343C8×Q8—611no—
344C8×Q8—611no—
345C8×Q8—613no—
346C8×Q8—613no—
347C8×Q8—613no—
348C8×Q8—613no—
349#436—613no—
350#436—613no—
351#436—613no—
352#436—613no—
353#435—611no—
354#435—613no—
355#435—613no—
356#435—613no—
357#435—616no—
358C8×D4—613no—
359C8×D4—616no—
360C8×D4—613no—
361#433—616no—
362#433—616no—
363#432—613no—
364#432—613no—
365#431—613no—
366#431—613no—
367#431—613no—
368#431—613no—
369#424—616no—
370#423—613no—
371C2×C4⋊C8—613no—
372C2×C4⋊C8—613no—
373#405—616no—
374C4×M16—613no—
375C4×M16—613no—
376C2×C4×C8—613no—
377C61#6033511yescharacteristic, complemented, Sylow
378C2×#41—611no—
379C2×#41—613no—
380C22×Q8—611no—
381C22×Q8—611no—
382C2×M16—611no—
383C2×M16—613no—
384C2×M16—613no—
385C2×M16—613no—
386C22×C8—611no—
387C22×C8—613no—
388C22×C8—616no—
389C4⋊Q8—613no—
390C4⋊Q8C2×#692313yes—
391#126—616no—
392#126—616no—
393#125—616no—
394#125C2×#692313yes—
395#125C2×#692313yes—
396#124—613no—
397#124—613no—
398#123—616no—
399#123—616no—
400C22⋊Q8—616no—
401C22⋊Q8—613no—
402C22⋊Q8—613no—
403C22⋊Q8—613no—
404C22⋊Q8—613no—
405#121—613no—
406#121—613no—
407C4×Q8—611no—
408C4×Q8—613no—
409C4×Q8—613no—
410C4×Q8—613no—
411C4×Q8—613no—
412C4×Q8—613no—
413C4×Q8C2×#692311yescharacteristic
414C4×Q8C2×#692311yescharacteristic
415C4×Q8C2×#692311yescharacteristic
416C4×Q8C2×#692313yes—
417C4×D4—611no—
418C4×D4—613no—
419C4×D4—613no—
420C4×D4—613no—
421C4×D4—613no—
422C4×D4—613no—
423C4×D4—613no—
424C4×D4—613no—
425#117—613no—
426#117—613no—
427#117—613no—
428#117—613no—
429#117—616no—
430C2×C4⋊C4—613no—
431C2×C4⋊C4—613no—
432C2×C42—613no—
433C2×C42—613no—
434C2×C42—613no—
435C4⋊C8—613no—
436C4⋊C8—613no—
437C4⋊C8—613no—
438C4⋊C8—616no—
439C4⋊C8—613no—
440C4⋊C8—613no—
441C4⋊C8—613no—
442C4⋊C8—613no—
443C4⋊C8—616no—
444C4⋊C8—613no—
445C4⋊C8—611no—
446C4⋊C8—613no—
447C22⋊C8—611no—
448C22⋊C8—613no—
449C22⋊C8—611no—
450C22⋊C8—613no—
451C22⋊C8—613no—
452C22⋊C8—613no—
453C22⋊C8—616no—
454#97—611no—
455#97—613no—
456#97—613no—
457#97—613no—
458#97—616no—
459C4×C8—613no—
460C4×C8—613no—
461C4×C8—613no—
462C4×C8—613no—
463C4×C8—613no—
464C4×C8—613no—
465C4×C8—616no—
466#41—1224no—
467#41—2446no—
468C2×Q8—613no—
469C2×Q8—613no—
470C2×Q8—1224no—
471C2×Q8—1224no—
472C2×Q8C22×#692313yes—
473C2×Q8C22×#692311yescharacteristic
474C2×D4—611no—
475C2×D4—613no—
476C2×D4—613no—
477C2×D4—613no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—613no—
481C22×C4—613no—
482C22×C4—613no—
483C22×C4—613no—
484C22×C4—613no—
485C22×C4—613no—
486C22×C4—613no—
487M16—1222no—
488M16—1226no—
489M16—2443no—
490M16—2443no—
491C2×C8—611no—
492C2×C8—611no—
493C2×C8—613no—
494C2×C8—611no—
495C2×C8—613no—
496C2×C8—611no—
497C2×C8—613no—
498C2×C8—613no—
499C2×C8—1222no—
500C2×C8—2446no—
501C2×C8—2443no—
502C4⋊C4—613no—
503C4⋊C4—616no—
504C4⋊C4—616no—
505C4⋊C4—613no—
506C4⋊C4—613no—
507C4⋊C4—613no—
508C4⋊C4—616no—
509C4⋊C4C22×#692313yes—
510C4⋊C4C22×#692313yes—
511C4⋊C4C22×#692311yescharacteristic
512C4⋊C4C22×#692313yes—
513C4⋊C4C22×#692313yes—
514C4⋊C4C22×#692313yes—
515C4⋊C4C22×#692316yes—
516C22⋊C4—611no—
517C22⋊C4—611no—
518C22⋊C4—613no—
519C22⋊C4—613no—
520C22⋊C4—616no—
521C22⋊C4—613no—
522C22⋊C4—613no—
523C22⋊C4—616no—
524C22⋊C4—616no—
525C42—613no—
526C42—613no—
527C42—613no—
528C42—611no—
529C42—613no—
530C42—613no—
531C42—613no—
532C42C22×#692313yes—
533C42C22×#692313yes—
534C42C22×#692313yes—
535C23—611no—
536C23—613no—
537C23—611no—
538Q8—43no—
539Q8—1226no—
540Q8—1226no—
541Q8#1451986514yes—
542D4—1226no—
543D4—1222no—
544D4—1226no—
545D4—2443no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—613no—
549C2×C4—613no—
550C2×C4—613no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1223no—
554C2×C4—1221no—
555C2×C4—1223no—
556C2×C4—1221no—
557C2×C4—1226no—
558C2×C4—1226no—
559C2×C4—1226no—
560C2×C4—1226no—
561C2×C4—2443no—
562C2×C4—2443no—
563C2×C4C23×#692311yescharacteristic
564C2×C4C23×#692311yescharacteristic
565C2×C4C23×#692311yescharacteristic
566C2×C4C23×#692313yes—
567C2×C4C23×#692313yes—
568C2×C4C23×#692313yes—
569C2×C4C23×#692313yes—
570C8—612no—
571C8—1221no—
572C8—1223no—
573C8—1223no—
574C8—1223no—
575C8—1221no—
576C8—1223no—
577C8—1221no—
578C22—612no—
579C22—1221no—
580C22—1221no—
581C22—1223no—
582C22—1221no—
583C22—1221no—
584C22—1223no—
585C22—1221no—
586C22—1223no—
587C22C24×#692311yescharacteristic
588C4—21no—
589C4—23no—
590C4—21no—
591C4—23no—
592C4—21no—
593C4—23no—
594C4—611no—
595C4—611no—
596C4—1221no—
597C4—1223no—
598C4—1223no—
599C4—1223no—
600C4C2×#1451986516yes—
601C2—1222no—
602C2—2441no—
603C2—2443no—
604C22−1+4×#692311yescharacteristic
605C2C22×#1451986511yescharacteristic
606C2D61⋊#56811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#557C4
prime radical#21459439C2
cosocleC2×Dic61C25
hypercenter#557#6923
hypocenterC61#60335
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #21462771C2semidirect
2. #21462765C2semidirect
3. #21462749C2semidirect
4. #21462737C2semidirect
5. #21462727C2semidirect
6. #21462721C2semidirect
7. #21462713C2semidirect
8. #21462663C2semidirect
9. #21462655C2semidirect
10. #21462636C2semidirect
11. #21462627C2semidirect
12. Q8×#11758213C2semidirect
13. #21462615C2semidirect
14. Q8×D61⋊C8C2semidirect
377. C61#60335semidirect

Maximal subgroups

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

subgroupindex
#214627712
#214627652
#214627492
#214627372
#214627272
#214627212
#214627132
#214626632
#214626552
#214626362
#214626272
Q8×#117582132
#214626152
Q8×D61⋊C82
#214594392
#6033561