Finite Group Databasebeta

← D61×#1678 #29662385

Subgroups — D61×#1678 #29662385

Order 15616.

722 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#486×D61C211yesmaximal, characteristic
2#477×D61C211yesmaximal, characteristic, complemented
3#460×D61C211yesmaximal, characteristic
4C8×Q8×D61C211yesmaximal, characteristic, complemented
5#21445238C211yesmaximal, characteristic, complemented
6#21445237C211yesmaximal, characteristic, complemented
7#21445236C211yesmaximal, characteristic, complemented
8#21445235C211yesmaximal, characteristic, complemented
9#21445234C211yesmaximal, characteristic, complemented
10#21445233C211yesmaximal, characteristic, complemented
11#21445232C211yesmaximal, characteristic, complemented
12#334×D61C211yesmaximal, characteristic, complemented
13#333×D61C211yesmaximal, characteristic, complemented
14#331×D61C211yesmaximal, characteristic
15C61×#1678C211yesmaximal, characteristic, complemented
16C2×SD16×D61—21no—
17C4⋊Q8×D61C2211yescharacteristic
18#127×D61C2211yescharacteristic
19C4×Q8×D61C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840109C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840107—21no—
24#16840107—21no—
25#16840058C2211yescharacteristic, complemented
26#16840057C2211yescharacteristic, complemented
27#16840056—21no—
28#16840055—21no—
29#16840054C2211yescharacteristic, complemented
30#16839963C2211yescharacteristic
31#16839962C2211yescharacteristic
32#16839961—21no—
33#16839960—21no—
34#16839959C2211yescharacteristic
35Q8×#92418C2211yescharacteristic, complemented
36C8×Dic122—21no—
37C8×Dic122C2211yescharacteristic, complemented
38#16839886—21no—
39#16839886C2211yescharacteristic, complemented
40#107×D61—21no—
41C4⋊C8×D61—21no—
42C4⋊C8×D61C2211yescharacteristic
43C4⋊C8×D61C2211yescharacteristic
44C4⋊C8×D61C2211yescharacteristic
45#103×D61—21no—
46#102×D61—21no—
47#102×D61—21no—
48#102×D61—21no—
49C4×C8×D61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839510C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#477C2211yescharacteristic, complemented
62C61×#460C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Q8×D61D411yescharacteristic
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75SD16×D61—44no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78C2×C8×D61—21no—
79C2×C8×D61D411yescharacteristic
80C2×C8×D61D411yescharacteristic
81#14519777C2311yescharacteristic
82#14519776—21no—
83#14519776—21no—
84#14519775—21no—
85#14519774C2311yescharacteristic
86#14519773—21no—
87#14519773—21no—
88#14519773—21no—
89Q8×Dic61D411yescharacteristic
90C4×Dic122—21no—
91C4×Dic122C2311yescharacteristic
92#14519742—21no—
93#14519742D411yescharacteristic
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61D411yescharacteristic
98C42×D61—21no—
99C42×D61C2311yescharacteristic
100#14519711—21no—
101#14519710—21no—
102#14519710—21no—
103C4⋊#92418—21no—
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107#14519706—21no—
108#14519706—21no—
109#14519706—21no—
110#14519706—21no—
111#14519706—21no—
112#14519702—21no—
113#14519701—21no—
114#14519700—21no—
115#14519699—21no—
116#14519699—21no—
117#14519699—21no—
118#14519698—21no—
119#14519698—21no—
120#14519698—21no—
121#14519697—21no—
122#14519697—21no—
123#14519697—21no—
124C4×#92418—21no—
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×SD16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134#107×C61—21no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×C22⋊D61—41no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—41no—
154Q8×D61—22no—
155Q8×D61—82no—
156Q8×D61SD1614yes—
157D4×D61—42no—
158D4×D61—42no—
159D4×D61—42no—
160D4×D61—42no—
161D4×D61—82no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—41no—
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61#4111yescharacteristic
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—41no—
174C2×Dic122C2×D411yescharacteristic
175SD16×C61—42no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×D411yescharacteristic
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C2×D411yescharacteristic
185C42×C61—21no—
186C42×C61C2411yescharacteristic
187#11758201—42no—
188#11758200—42no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61C2×D411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196#11758196—41no—
197#11758196—41no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61#4111yescharacteristic
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209#11758190—42no—
210C8×D61—22no—
211C8×D61—22no—
212C8×D61—42no—
213C8×D61—42no—
214C8×D61—42no—
215C23×C61—41no—
216C22×D61—41no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61#12111yescharacteristic
221Q8×C61—21no—
222Q8×C61—81no—
223Q8×C61C2×SD1612yes—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—81no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—82no—
236C22⋊D61—82no—
237C22⋊D61—82no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—41no—
241C2×Dic61#12111yescharacteristic
242C2×Dic61#12111yescharacteristic
243C2×Dic61#12111yescharacteristic
244D244—41no—
245D244—41no—
246D244—41no—
247D244—41no—
248D244—81no—
249C4×D61—24no—
250C4×D61—22no—
251C4×D61—42no—
252C4×D61—82no—
253C4×D61C2×SD1612yes—
254C4×D61#13512yes—
255C4×D61#13512yes—
256C4×D61#13712yes—
257Dic122—21no—
258Dic122—24no—
259Dic122—82no—
260Dic122—81no—
261Dic122C2×SD1612yes—
262C8×C61—21no—
263C8×C61—21no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61—41no—
267#92418—21no—
268#92418—21no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272C2×#1678—611nomaximal, Sylow
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61C2×#12111yescharacteristic
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—164no—
281C2×D61#46412yes—
282C2×D61#46512yes—
283C2×D61#47212yes—
284C4×C61—22no—
285C4×C61—21no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61C22×SD1611yescharacteristic
289C4×C61C2×#13511yescharacteristic
290C4×C61C2×#13511yescharacteristic
291C4×C61C2×#13711yescharacteristic
292Dic61—22no—
293Dic61—21no—
294Dic61—41no—
295Dic61—81no—
296Dic61#46412yes—
297Dic61#46512yes—
298C2×#486—611no—
299C2×#477—611no—
300C2×#460—611no—
301C2×C8×Q8—611no—
302#1678—611no—
303#1678—611no—
304#1678—611no—
305#1678—611no—
306#1678—611no—
307#1678—611no—
308#1678—611no—
309#1678D6111yescharacteristic, direct product factor
310C2×#334—611no—
311C2×#333—611no—
312C2×#331—611no—
313C2×C61—161no—
314C2×C61C2×#46411yescharacteristic
315C2×C61C2×#46511yescharacteristic
316C2×C61C2×#47211yescharacteristic
317D61—161no—
318D61#167814yesdirect product factor
319C22×SD16—1221no—
320C2×C4⋊Q8—611no—
321C2×#127—611no—
322C2×C4×Q8—611no—
323#486—611no—
324#486—611no—
325#486—611no—
326#486—1221no—
327#486—1221no—
328#486C2×D6111yescharacteristic
329#477—611no—
330#477—611no—
331#477—611no—
332#477—1221no—
333#477—1221no—
334#477C2×D6111yescharacteristic, complemented
335#460—611no—
336#460—611no—
337#460—611no—
338#460—1221no—
339#460—1221no—
340#460C2×D6111yescharacteristic
341C8×Q8—611no—
342C8×Q8—611no—
343C8×Q8—611no—
344C8×Q8—1221no—
345C8×Q8—1221no—
346C8×Q8C2×D6111yescharacteristic, complemented
347C2×#107—1221no—
348C2×C4⋊C8—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8—1221no—
352C2×#103—1221no—
353C2×#102—1221no—
354C2×#102—1221no—
355C2×#102—1221no—
356C2×C4×C8—611no—
357C2×C4×C8—1221no—
358#334—611no—
359#334—611no—
360#334—611no—
361#334C2×D6111yescharacteristic, complemented
362#333—611no—
363#333—611no—
364#333—611no—
365#333C2×D6111yescharacteristic, complemented
366#331—611no—
367#331—611no—
368#331—611no—
369#331C2×D6111yescharacteristic
370C61C2×#167811yescharacteristic, complemented, Sylow
371C22×Q8—611no—
372C22×Q8—1221no—
373C22×D4—1221no—
374C22×D4—1221no—
375C22×D4—1221no—
376C2×SD16—21no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—2444no—
381C22×C8—611no—
382C22×C8—611no—
383C22×C8—1221no—
384C22×C8—1221no—
385C22×C8—1221no—
386C4⋊Q8—611no—
387C4⋊Q8—1221no—
388C4⋊Q8—1221no—
389C4⋊Q8—1221no—
390C4⋊Q8C22×D6111yescharacteristic
391#127—611no—
392#127—1221no—
393#127—1221no—
394#127—1221no—
395#127C22×D6111yescharacteristic
396C4×Q8—611no—
397C4×Q8—611no—
398C4×Q8—611no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8C22×D6111yescharacteristic
402C2×C4⋊C4—611no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C42—611no—
407C2×C42—1221no—
408#107—21no—
409#107—1221no—
410#107—1221no—
411#107—1221no—
412C4⋊C8—21no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419C4⋊C8—1221no—
420C4⋊C8—1221no—
421C4⋊C8—1221no—
422C4⋊C8C22×D6111yescharacteristic
423C4⋊C8C22×D6111yescharacteristic
424C4⋊C8C22×D6111yescharacteristic
425#103—21no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#102—21no—
430#102—21no—
431#102—21no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441C4×C8—21no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—611no—
445C4×C8—1221no—
446C4×C8—1221no—
447C4×C8—1221no—
448C4×C8C22×D6111yescharacteristic
449C24—2441no—
450C2×Q8—21no—
451C2×Q8—614no—
452C2×Q8—611no—
453C2×Q8—1221no—
454C2×Q8—1222no—
455C2×Q8—1221no—
456C2×Q8—4882no—
457C2×Q8—2441no—
458C2×Q8D4×D6111yescharacteristic
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—4882no—
466C2×D4—2442no—
467C2×D4—2442no—
468C2×D4—2442no—
469C2×D4—2442no—
470C2×D4—2441no—
471C2×D4—2441no—
472C2×D4—2441no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—2441no—
479SD16—42no—
480SD16—2442no—
481SD16—2442no—
482SD16—2442no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—2442no—
494C2×C8—2442no—
495C2×C8—2442no—
496C2×C8D4×D6111yescharacteristic
497C2×C8D4×D6111yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—611no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4D4×D6111yescharacteristic
511C42—21no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C42C23×D6111yescharacteristic
519C23—41no—
520C23—611no—
521C23—4882no—
522C23—4882no—
523C23—4882no—
524C23—2441no—
525Q8—21no—
526Q8—81no—
527Q8—612no—
528Q8—1224no—
529Q8—1221no—
530Q8—4881no—
531Q8—4882no—
532Q8SD16×D6112yes—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—81no—
538D4—4881no—
539D4—4882no—
540D4—4882no—
541D4—4882no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—611no—
556C2×C4—1222no—
557C2×C4—1224no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—4882no—
561C2×C4—2442no—
562C2×C4—2441no—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C8—21no—
567C8—21no—
568C8—41no—
569C8—41no—
570C8—41no—
571C8—1221no—
572C8—1221no—
573C8—2441no—
574C8—2441no—
575C8—2441no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—612no—
580C22—612no—
581C22—612no—
582C22—9764no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22#121×D6111yescharacteristic
587C4—22no—
588C4—21no—
589C4—41no—
590C4—81no—
591C4—612no—
592C4—612no—
593C4—1222no—
594C4—1221no—
595C4—4881no—
596C4—2441no—
597C4C2×SD16×D6111yescharacteristic
598C4#135×D6111yescharacteristic
599C4#135×D6111yescharacteristic
600C4#137×D6111yescharacteristic
601C2—161no—
602C2—614no—
603C2—9761no—
604C2#464×D6111yescharacteristic
605C2#465×D6111yescharacteristic
606C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1678C2
socleC22×C61C2×#121
prime radical#127×D61C22
hypercenter#1678D61
hypocenterC61C2×#1678
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1678

Complements

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

NH ≅ G/Ntype
2. #477×D61C2semidirect
4. C8×Q8×D61C2semidirect
5. #21445238C2semidirect
6. #21445237C2semidirect
7. #21445236C2semidirect
8. #21445235C2semidirect
9. #21445234C2semidirect
10. #21445233C2semidirect
11. #21445232C2semidirect
12. #334×D61C2semidirect
13. #333×D61C2semidirect
15. C61×#1678C2semidirect
25. #16840058C22semidirect
26. #16840057C22semidirect
29. #16840054C22semidirect
35. Q8×#92418C22semidirect
37. C8×Dic122C22semidirect
39. #16839886C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839510C22semidirect
56. #16839510C22semidirect
61. C61×#477C22semidirect
63. C8×Q8×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#333C22semidirect
309. #1678D61direct (N × H)
318. D61#1678direct (N × H)
334. #477C2×D61semidirect
346. C8×Q8C2×D61semidirect
361. #334C2×D61semidirect
365. #333C2×D61semidirect
370. C61C2×#1678semidirect

Maximal subgroups

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

subgroupindex
#486×D612
#477×D612
#460×D612
C8×Q8×D612
#214452382
#214452372
#214452362
#214452352
#214452342
#214452332
#214452322
#334×D612
#333×D612
#331×D612
C61×#16782
C2×#167861