Finite Group Databasebeta

← #30613131

Subgroups — #30613131

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462614C211yesmaximal, characteristic, complemented
2#21462559C211yesmaximal, characteristic
3#21461454C211yesmaximal, characteristic, complemented
4#21461453C211yesmaximal, characteristic, complemented
5#21461427C211yesmaximal, characteristic, complemented
6#21461426C211yesmaximal, characteristic, complemented
7#21461425C211yesmaximal, characteristic, complemented
8#21461423C211yesmaximal, characteristic, complemented
9#21461404C211yesmaximal, characteristic, complemented
10#21461403C211yesmaximal, characteristic, complemented
11#21461386C211yesmaximal, characteristic, complemented
12#21461167C211yesmaximal, characteristic, complemented
13C2×#16840610C211yesmaximal, characteristic
14#21461020C211yesmaximal, characteristic, complemented
15#21455824C211yesmaximal, characteristic
16C2×D61⋊M16—21no—
17#16840814—21no—
18#16840813C2211yescharacteristic, complemented
19#16840812—21no—
20#16840812C2211yescharacteristic, complemented
21#16840811C2211yescharacteristic, complemented
22Q8×#92420C2211yescharacteristic, complemented
23#16840796C2211yescharacteristic
24#16840796C2211yescharacteristic
25#16840792—21no—
26#16840792—21no—
27#16840790C2211yescharacteristic
28#16840790C2211yescharacteristic
29#16840768—21no—
30#16840765C2211yescharacteristic
31C2×C4⋊#92420C2211yescharacteristic
32#16840734—21no—
33#16840726—21no—
34C4×#11758217C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840624C2211yescharacteristic, complemented
37#16840624C2211yescharacteristic, complemented
38#16840623C2211yescharacteristic, complemented
39#16840623C2211yescharacteristic, complemented
40#16840612C2211yescharacteristic, complemented
41#16840612C2211yescharacteristic, complemented
42Q8⋊#92420C2211yescharacteristic, complemented
43Q8⋊#92420C2211yescharacteristic, complemented
44#16840610C2211yescharacteristic
45#16840610C2211yescharacteristic
46#16840610C2211yescharacteristic
47#16840610C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic, complemented
49#92420⋊C8C2211yescharacteristic, complemented
50C8×#92420C2211yescharacteristic, complemented
51C8×#92420C2211yescharacteristic, complemented
52#16840314C2211yescharacteristic
53C2×#14519739C2211yescharacteristic
54#16840092C411yescharacteristic
55#16840051C411yescharacteristic
56#16840051C411yescharacteristic
57#16839966C411yescharacteristic
58#16839966C411yescharacteristic
59#16839915C411yescharacteristic
60#16839912C411yescharacteristic
61#16839849C2211yescharacteristic
62#16839849C2211yescharacteristic
63#16839847C2211yescharacteristic
64#16839847C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#483C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758213—21no—
69D61⋊M16—44no—
70C2×#11758217—21no—
71C2×#11758217D411yescharacteristic
72C2×#11758217D411yescharacteristic
73C2×D61⋊C8—21no—
74C2×D61⋊C8—21no—
75C22×#92420—21no—
76C22×#92420—21no—
77#14519844—21no—
78#14519844—21no—
79#14519844—21no—
80C4⋊#92420—21no—
81C4⋊#92420—21no—
82C4⋊#92420—21no—
83C4⋊#92420C2311yescharacteristic
84C4⋊#92420C2311yescharacteristic
85C4⋊#92420C2311yescharacteristic
86C4⋊#92420C2311yescharacteristic
87#14519834—21no—
88#14519834—21no—
89#14519834—21no—
90#14519834—21no—
91#14519832—21no—
92#14519832—21no—
93#14519832D411yescharacteristic
94#14519832D411yescharacteristic
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420—21no—
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C4×#92420C2311yescharacteristic
102C22×D244C2×C411yescharacteristic
103C22×C4×D61—21no—
104C2×#11758190C2×C411yescharacteristic
105C2×#11758190C2×C411yescharacteristic
106C2×#11758201C2×C411yescharacteristic
107C2×#11758201C2×C411yescharacteristic
108#14519776C2×C411yescharacteristic
109#14519775C2×C411yescharacteristic
110#14519767—21no—
111#14519756C2×C411yescharacteristic
112#14519754—21no—
113Q8×Dic61C2311yescharacteristic
114#14519742—21no—
115#14519742C2311yescharacteristic
116#14519739—21no—
117#14519739—21no—
118#14519739C2×C411yescharacteristic
119#14519739C2×C411yescharacteristic
120#14519739C2311yescharacteristic
121#14519739C2311yescharacteristic
122#14519733—21no—
123C2×#11758198—21no—
124C2×C4×Dic61C2311yescharacteristic
125#14519708C2×C411yescharacteristic
126#14519708C2×C411yescharacteristic
127#14519706C2×C411yescharacteristic
128#14519706C2×C411yescharacteristic
129#14519700C2×C411yescharacteristic
130#14519700C2×C411yescharacteristic
131#14519699C2×C411yescharacteristic
132#14519699C2×C411yescharacteristic
133#14519697C2×C411yescharacteristic
134#14519697C2×C411yescharacteristic
135C22⋊#92418C2×C411yescharacteristic
136C8×Dic61C2311yescharacteristic
137C8×Dic61C2311yescharacteristic
138C22⋊Q8×C61C2×C411yescharacteristic
139C2×C4⋊C4×C61C2×C411yescharacteristic
140#106×C61C2×C411yescharacteristic
141#106×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143#103×C61C2×C411yescharacteristic
144C22⋊C8×C61C2×C411yescharacteristic
145C23×D61—21no—
146C2×Q8×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61#4111yescharacteristic
151C2×D244—22no—
152C2×D244—21no—
153C2×D244—41no—
154C2×D244C22×C411yescharacteristic
155C2×D244C22×C411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×Dic122—21no—
163C2×Dic122C22×C411yescharacteristic
164#11758217—22no—
165#11758217—42no—
166#11758217SD1614yescomplemented
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—42no—
174C2×#92420—42no—
175C2×#92420C2×D411yescharacteristic
176C2×#92420C2×D411yescharacteristic
177C2×#92420C2×D411yescharacteristic
178C2×#92420C2×D411yescharacteristic
179#11758213—42no—
180#11758213—42no—
181D61⋊C8—42no—
182D61⋊C8—42no—
183C2×C8×C61C22×C411yescharacteristic
184C2×C8×C61C22×C411yescharacteristic
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61C22×C411yescharacteristic
188C4⋊C4×C61C22×C411yescharacteristic
189C4⋊C4×C61C22×C411yescharacteristic
190C22⋊C4×C61—21no—
191C22⋊Dic61—21no—
192#11758201—22no—
193#11758201—41no—
194#11758198—21no—
195#11758198—21no—
196#11758198—21no—
197#11758198—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C22×C411yescharacteristic
200#11758196—21no—
201#11758196—21no—
202#11758196—21no—
203#11758196—21no—
204#11758196—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61C22×C411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61#4111yescharacteristic
210C4×Dic61C2411yescharacteristic
211C2×#92418C22×C411yescharacteristic
212C2×#92418C22×C411yescharacteristic
213#11758190—22no—
214#11758190—41no—
215C23×C61C4×D411yescharacteristic
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—81no—
221Q8×C61—21no—
222Q8×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—42no—
230C2×C4×C61C4×D411yescharacteristic
231C2×C4×C61C23×C411yescharacteristic
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—41no—
235C2×Dic61C4×D411yescharacteristic
236C2×Dic61C4×D411yescharacteristic
237C2×Dic61C2×SD1612yes—
238C2×Dic61#13512yes—
239C2×Dic61C22×D411yescharacteristic
240C2×Dic61C2×#4111yescharacteristic
241D244—21no—
242D244—21no—
243D244—21no—
244D244—42no—
245D244#13112yes—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—42no—
251C4×D61—42no—
252Dic122—21no—
253Dic122—21no—
254Dic122—81no—
255#92420—22no—
256#92420—21no—
257#92420—21no—
258#92420—41no—
259#92420—41no—
260#92420—41no—
261#92420—41no—
262#92420—41no—
263#92420C2×SD1612yes—
264#92420C2×SD1612yes—
265C8×C61—21no—
266C8×C61—21no—
267#92418—21no—
268#92418—21no—
269#17995—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61C4×SD1612yes—
273C22×C61C2×C4×D411yescharacteristic
274C2×D61—22no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—84no—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61C2×#13111yescharacteristic
287C4×C61#56811yescharacteristic
288Dic61—21no—
289Dic61—21no—
290Dic61—41no—
291Dic61—41no—
292Dic61C4×SD1612yes—
293Dic61C22×SD1611yescharacteristic
294Dic61C2×#13511yescharacteristic
295#3101—611no—
296#2962—611no—
297D4×M16—611no—
298#1590—611no—
299#1590—611no—
300#1583—611no—
301#1583—611no—
302#1582—611no—
303#1582—611no—
304C8×SD16—611no—
305C8×SD16—611no—
306#1570—611no—
307#1493—611no—
308C2×D4⋊C8—611no—
309C8×M16—611no—
310C2×C61—22no—
311C2×C61#293311yescharacteristic
312C2×C61C2×C4×SD1611yescharacteristic
313C2×C61#295611yescharacteristic
314D61—42no—
315D61—81no—
316C22×M16—1221no—
317#533—611no—
318C2×C4×D4—611no—
319#483—611no—
320#483#692311yescharacteristic
321#476—611no—
322#476—611no—
323#461—611no—
324#461—611no—
325#450—611no—
326#450—611no—
327#446—611no—
328#446—611no—
329#446—1221no—
330#446—1221no—
331C8×Q8—611no—
332C8×Q8—611no—
333C4×SD16—611no—
334C4×SD16—611no—
335C4×SD16—611no—
336C4×SD16—611no—
337#436—611no—
338#436—611no—
339#435—1221no—
340#435—1221no—
341C8×D4—611no—
342C8×D4—611no—
343#431—611no—
344#424—1221no—
345#423—611no—
346C2×C4⋊C8—611no—
347#407—1221no—
348#405—1221no—
349C4×M16—611no—
350C2×C4×C8—611no—
351#334—611no—
352#334—611no—
353#334—611no—
354#334—611no—
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358Q8⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363#322—611no—
364#322—611no—
365C82—611no—
366C82—611no—
367C61#1799511yescharacteristic, complemented, Sylow
368C22×D4—611no—
369C23×C4—1221no—
370C2×SD16—611no—
371C2×SD16—611no—
372C2×SD16—611no—
373C2×SD16—611no—
374C2×M16—611no—
375C2×M16—611no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16—1221no—
379C2×M16—2444no—
380C22×C8—1221no—
381C22×C8—1221no—
382C22×C8—1221no—
383C22×C8—1221no—
384C4⋊Q8—611no—
385C4⋊Q8—611no—
386#125—1221no—
387C22⋊Q8—611no—
388C22⋊Q8—1221no—
389C22⋊Q8C2×#692311yescharacteristic
390C4×Q8—611no—
391C4×Q8—611no—
392C4×Q8—1221no—
393C4×D4—611no—
394C4×D4—611no—
395C4×D4—611no—
396C4×D4—611no—
397C4×D4—1221no—
398C4×D4—1221no—
399#117—1221no—
400C2×C4⋊C4C2×#692311yescharacteristic
401C2×C22⋊C4—1221no—
402C2×C42—611no—
403#106—611no—
404#106—611no—
405#106C2×#692311yescharacteristic
406#106C2×#692311yescharacteristic
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419#103—611no—
420#103—611no—
421#103C2×#692311yescharacteristic
422#103C2×#692311yescharacteristic
423#102—611no—
424#102—611no—
425#102—611no—
426#102—611no—
427C22⋊C8—611no—
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C22⋊C8C2×#692311yescharacteristic
433#97—611no—
434#97—611no—
435#97—1221no—
436#97—1221no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—1221no—
444C4×C8—1221no—
445C4×C8—1221no—
446C24—1221no—
447C2×Q8—611no—
448C2×Q8—1221no—
449C2×Q8C22×#692311yescharacteristic
450C2×D4—611no—
451C2×D4—611no—
452C2×D4—1222no—
453C2×D4—1221no—
454C2×D4—2441no—
455C22×C4—21no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4C22×#692311yescharacteristic
465SD16—1222no—
466SD16—1222no—
467SD16—2441no—
468SD16—2441no—
469M16—614no—
470M16—1222no—
471M16—2442no—
472M16—2442no—
473M16—2442no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—2442no—
487C2×C8—2442no—
488C2×C8—2442no—
489C2×C8—2442no—
490C2×C8C22×#692311yescharacteristic
491C2×C8C22×#692311yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—611no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4C22×#692311yescharacteristic
502C4⋊C4C22×#692311yescharacteristic
503C4⋊C4C22×#692311yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C22⋊C4—1221no—
509C22⋊C4—1221no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—1221no—
515C42—1221no—
516C23—1221no—
517C23—1221no—
518C23—4881no—
519C23—2442no—
520C23—2442no—
521C23D4×#692311yescharacteristic
522Q8—21no—
523Q8—21no—
524Q8—1221no—
525Q8—1221no—
526Q8—4881no—
527D4—612no—
528D4—1221no—
529D4—1221no—
530D4—1221no—
531D4—2442no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—42no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2441no—
554C2×C4D4×#692311yescharacteristic
555C2×C4C23×#692311yescharacteristic
556C8—21no—
557C8—21no—
558C8—612no—
559C8—612no—
560C8—1221no—
561C8—1221no—
562C8—1222no—
563C8—1221no—
564C8—1221no—
565C8—2441no—
566C8—2441no—
567C8—2441no—
568C8—2441no—
569C8—2441no—
570C22—21no—
571C22—21no—
572C22—1222no—
573C22—4884no—
574C22—2441no—
575C22—2441no—
576C22—2441no—
577C22—2441no—
578C22—2441no—
579C22SD16×#692312yes—
580C22C2×D4×#692311yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—611no—
587C4—611no—
588C4—612no—
589C4—1221no—
590C4—1221no—
591C4—2441no—
592C4—2441no—
593C4D61⋊#13111yescharacteristic
594C4#1684084111yescharacteristic
595C2—22no—
596C2—4881no—
597C2—2442no—
598C2#2146255311yescharacteristic
599C2C2×SD16×#692311yescharacteristic
600C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#483C4
socleC22×C61C2×C4×D4
prime radicalC22×D244C2×C4
cosocleC4×Dic61C24
hypercenter#483#6923
hypocenterC61#17995
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #21462614C2semidirect
3. #21461454C2semidirect
4. #21461453C2semidirect
5. #21461427C2semidirect
6. #21461426C2semidirect
7. #21461425C2semidirect
8. #21461423C2semidirect
9. #21461404C2semidirect
10. #21461403C2semidirect
11. #21461386C2semidirect
12. #21461167C2semidirect
14. #21461020C2semidirect
18. #16840813C22semidirect
20. #16840812C22semidirect
21. #16840811C22semidirect
22. Q8×#92420C22semidirect
36. #16840624C22semidirect
37. #16840624C22semidirect
38. #16840623C22semidirect
39. #16840623C22semidirect
40. #16840612C22semidirect
41. #16840612C22semidirect
42. Q8⋊#92420C22semidirect
43. Q8⋊#92420C22semidirect
48. #92420⋊C8C22semidirect
49. #92420⋊C8C22semidirect
50. C8×#92420C22semidirect
51. C8×#92420C22semidirect
166. #11758217SD16semidirect
367. C61#17995semidirect

Maximal subgroups

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

subgroupindex
#214626142
#214625592
#214614542
#214614532
#214614272
#214614262
#214614252
#214614232
#214614042
#214614032
#214613862
#214611672
C2×#168406102
#214610202
#214558242
#1799561