Finite Group Databasebeta

← #29669731

Subgroups — #29669731

Order 15616.

622 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458113C211yesmaximal, characteristic, complemented
2#21457749C211yesmaximal, characteristic
3#21457684C211yesmaximal, characteristic, complemented
4#21445464C211yesmaximal, characteristic, complemented
5#21445364C211yesmaximal, characteristic, complemented
6#21445277C211yesmaximal, characteristic, complemented
7#21445268C211yesmaximal, characteristic, complemented
8#21445205C211yesmaximal, characteristic, complemented
9#21445102C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21445046C211yesmaximal, characteristic, complemented
12#21444984C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15Q8×D8×C61C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17#16840416C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840114C2211yescharacteristic
20#16840113C2211yescharacteristic
21#16840112C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840105—21no—
24#16840100—21no—
25#16840090—21no—
26#16840057C2211yescharacteristic
27#16840052C2211yescharacteristic
28#16840047C2211yescharacteristic
29#16840010—21no—
30#16839993—21no—
31#16839990—21no—
32#16839967C2211yescharacteristic
33#16839964C2211yescharacteristic
34#16839964C2211yescharacteristic
35#16839904—21no—
36#16839901—21no—
37#16839901—21no—
38C4×#11758201C2211yescharacteristic
39#16839840C2211yescharacteristic
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42#16839771—21no—
43#16839755—21no—
44C2×#14519701—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56D4×Q8×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#500—21no—
59C61×#500C2211yescharacteristic
60C61×#474—21no—
61C61×#474—21no—
62C61×#474C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758224—21no—
68C2×#11758200—21no—
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×#11758201D411yescharacteristic
72C2×#11758194—21no—
73C22×#92418—21no—
74C2×C8×D61—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519763—41no—
78#14519761—41no—
79#14519757—21no—
80#14519751—21no—
81#14519751—21no—
82#14519748—21no—
83C4×D244C2311yescharacteristic
84#14519736—21no—
85#14519735—21no—
86C2×C4⋊Dic61—21no—
87#14519710—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—41no—
91#14519708D411yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519702—21no—
96#14519702—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519700—41no—
108#14519698—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C22×Q8×C61—21no—
117C2×D8×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#107×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#107×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—41no—
158C22×C4×C61—41no—
159C2×C22⋊D61—41no—
160C2×C22⋊D61—41no—
161C22×Dic61—41no—
162#11758224—42no—
163#11758224—81no—
164C2×D244C2×D411yescharacteristic
165C2×C4×D61—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168D8×C61—22no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189#11758201—22no—
190#11758201—81no—
191#11758201—81no—
192#11758200—81no—
193#11758198—21no—
194#11758198—41no—
195#11758198—41no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61C2×D411yescharacteristic
202C4×Dic61—41no—
203#11758194—81no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—81no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C8×D61—81no—
212C23×C61—21no—
213C23×C61—21no—
214C22×D61—21no—
215Q8×C61—21no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—82no—
221Q8×C61—82no—
222Q8×C61#13512yes—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61#13712yes—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—82no—
238C2×C4×C61—82no—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C22⋊D61—82no—
243C22⋊D61—161no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—82no—
248D244—21no—
249D244—21no—
250C4×D61—41no—
251C4×D61—41no—
252Dic122—41no—
253Dic122—41no—
254Dic122—81no—
255C8×C61—21no—
256C8×C61—21no—
257C8×C61—41no—
258#92418—21no—
259#92418—21no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#19780—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61D4211yescharacteristic
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—41no—
275C4×C61—22no—
276C4×C61—21no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61C2×#13511yescharacteristic
283C4×C61C2×#13711yescharacteristic
284C4×C61#57611yescharacteristic
285C4×C61#57711yescharacteristic
286Dic61—41no—
287Dic61—81no—
288Dic61—81no—
289Q8×D8D6111yescharacteristic, complemented
290#3329—611no—
291#3303—611no—
292#3298—611no—
293#1718—611no—
294#1699—611no—
295#1684—611no—
296#1683—611no—
297#1673—611no—
298#1654—611no—
299#1651—611no—
300#1646—611no—
301#1637—611no—
302#1581—611no—
303SD16⋊C8—611no—
304C2×C61—42no—
305C2×C61—81no—
306C2×C61#326611yescharacteristic
307C2×C61#328011yescharacteristic
308C2×C61#328911yescharacteristic
309D61—81no—
310#555—611no—
311D4×Q8C2×D6111yescharacteristic
312D4×Q8C2×D6111yescharacteristic
313#547—611no—
314#500—21no—
315#500C2×D6111yescharacteristic
316#498—611no—
317#487—611no—
318#487—611no—
319#487—611no—
320#487—611no—
321#485—1221no—
322#485—1221no—
323#483—1221no—
324#477—611no—
325#476—611no—
326#475—611no—
327#474—21no—
328#474—21no—
329#474C2×D6111yescharacteristic
330#474C2×D6111yescharacteristic
331#468—1221no—
332#465—1221no—
333#465—1221no—
334#461—611no—
335#461—611no—
336#461—611no—
337#449—1221no—
338#448—1221no—
339#448—1221no—
340C8×Q8C2×D6111yescharacteristic
341C4×SD16—611no—
342C4×D8—21no—
343C4×D8C2×D6111yescharacteristic
344#436—611no—
345C8×D4—611no—
346C8×D4—611no—
347#419—1221no—
348#416—1221no—
349C2×#103—1221no—
350#335—611no—
351#335—611no—
352#333—611no—
353#332—611no—
354#332—611no—
355#332—611no—
356Q8⋊C8—611no—
357D4⋊C8—611no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360#322—611no—
361C61#1978011yescharacteristic, complemented, Sylow
362C2×#41—1221no—
363C22×Q8—21no—
364C22×Q8—21no—
365C2×SD16—611no—
366C2×SD16—1221no—
367C2×SD16—1221no—
368C2×SD16—1221no—
369C2×D8C22⋊D6111yescharacteristic
370C2×M16—1221no—
371C22×C8—1221no—
372C22×C8—1221no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—611no—
376C4⋊Q8—611no—
377C4⋊Q8C22×D6111yescharacteristic
378C4⋊Q8C22×D6111yescharacteristic
379#124—2441no—
380#123—2441no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—41no—
384C22⋊Q8—41no—
385C22⋊Q8—1221no—
386#121—1221no—
387#121—1221no—
388#121—1221no—
389C4×Q8C22×D6111yescharacteristic
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—611no—
393C4×D4C22×D6111yescharacteristic
394C4×D4C22×D6111yescharacteristic
395#117—1221no—
396#117—1221no—
397C2×C4⋊C4—1221no—
398#107—21no—
399#107—21no—
400#107—21no—
401#107—21no—
402#107—1221no—
403#107C22⋊D6111yescharacteristic
404#106—611no—
405#106—1221no—
406#106—1221no—
407#106—2441no—
408C4⋊C8—21no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8C22×D6111yescharacteristic
413#103—611no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—2441no—
426#102—21no—
427#102—21no—
428#102—611no—
429#102—1221no—
430#102C22⋊D6111yescharacteristic
431#102C22⋊D6111yescharacteristic
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435C4×C8—21no—
436C4×C8—611no—
437C4×C8—611no—
438C4×C8C22×D6111yescharacteristic
439#41—4881no—
440#41—2442no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—81no—
448C2×Q8—81no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8D4×D6111yescharacteristic
452C2×D4—611no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4C2×C22⋊D6111yescharacteristic
456C2×D4C2×C22⋊D6111yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—1221no—
462C22×C4—2441no—
463SD16—1222no—
464SD16—4881no—
465SD16—4881no—
466SD16—4881no—
467D8—22no—
468M16—4881no—
469C2×C8—21no—
470C2×C8—611no—
471C2×C8—611no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—4881no—
477C2×C8—4881no—
478C2×C8C2×C22⋊D6111yescharacteristic
479C2×C8C2×C22⋊D6111yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—611no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—2441no—
493C4⋊C4D4×D6111yescharacteristic
494C4⋊C4C2×C22⋊D6111yescharacteristic
495C4⋊C4C2×C22⋊D6111yescharacteristic
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—1221no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C42—21no—
505C42—2441no—
506C42C23×D6111yescharacteristic
507C23—21no—
508C23—21no—
509C23—1221no—
510Q8—21no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—82no—
516Q8—82no—
517Q8—4881no—
518Q8—2441no—
519Q8—2441no—
520Q8#1451979612yes—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—1221no—
525D4—1221no—
526D4—9761no—
527D4—4882no—
528D4#1451980812yes—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—1221no—
542C2×C4—4882no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4C2×D4×D6111yescharacteristic
548C2×C4C22×C22⋊D6111yescharacteristic
549C2×C4#1451982211yescharacteristic
550C8—21no—
551C8—21no—
552C8—41no—
553C8—1221no—
554C8—1221no—
555C8—2441no—
556C8—2441no—
557C8—2441no—
558C8—2441no—
559C22—22no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—2441no—
566C22—2441no—
567C22—2441no—
568C22D4×C22⋊D6111yescharacteristic
569C4—22no—
570C4—21no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—4881no—
577C4—4881no—
578C4—2441no—
579C4C2×#1451979611yescharacteristic
580C4C2×#1451980811yescharacteristic
581C4#1684056311yescharacteristic
582C4#1684057011yescharacteristic
583C2—42no—
584C2—81no—
585C2—4881no—
586C2#2145733011yescharacteristic
587C2#2145750411yescharacteristic
588C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×D8×C61C2
socleC22×C61D42
prime radical#21457749C2
hypercenterQ8×D8D61
hypocenterC61#19780
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×D8

Complements

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

NH ≅ G/Ntype
1. #21458113C2semidirect
3. #21457684C2semidirect
4. #21445464C2semidirect
5. #21445364C2semidirect
6. #21445277C2semidirect
7. #21445268C2semidirect
8. #21445205C2semidirect
9. #21445102C2semidirect
10. #21445082C2semidirect
11. #21445046C2semidirect
12. #21444984C2semidirect
13. #21444649C2semidirect
14. #21444629C2semidirect
15. Q8×D8×C61C2semidirect
289. Q8×D8D61semidirect
361. C61#19780semidirect

Maximal subgroups

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

subgroupindex
#214581132
#214577492
#214576842
#214454642
#214453642
#214452772
#214452682
#214452052
#214451022
#214450822
#214450462
#214449842
#214446492
#214446292
Q8×D8×C612
#1978061