Finite Group Databasebeta

← #29670171

Subgroups — #29670171

Order 15616.

632 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758202C211yesmaximal, characteristic, complemented
2#21458488C211yesmaximal, characteristic
3#21457481C211yesmaximal, characteristic
4#21445444C211yesmaximal, characteristic, complemented
5#21445340C211yesmaximal, characteristic, complemented
6#21445235C211yesmaximal, characteristic, complemented
7#21445198C211yesmaximal, characteristic, complemented
8#21445169C211yesmaximal, characteristic, complemented
9#21445039C211yesmaximal, characteristic, complemented
10#21444984C211yesmaximal, characteristic, complemented
11#21444945C211yesmaximal, characteristic
12#21444897C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444636C211yesmaximal, characteristic, complemented
15Q8×SD16×C61C211yesmaximal, characteristic, complemented
16C2×#14519808—21no—
17Q8×Dic122C2211yescharacteristic
18#16840433C2211yescharacteristic
19#16840165—21no—
20#16840165C2211yescharacteristic
21#16840136C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840109C2211yescharacteristic
25#16840107—21no—
26#16840057—21no—
27#16840057C2211yescharacteristic
28#16840054—21no—
29#16840054C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840051—21no—
32#16840010—21no—
33#16839969C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839962C2211yescharacteristic
36#16839960—21no—
37#16839922—21no—
38#16839901—21no—
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41C4×#11758202—21no—
42C4×#11758202C2211yescharacteristic
43#16839847—21no—
44#16839840C2211yescharacteristic
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#498—21no—
59C61×#498C2211yescharacteristic
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#475—21no—
63C61×#475C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758222—21no—
68#14519808—44no—
69C2×#11758202—21no—
70C2×#11758202D411yescharacteristic
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—41no—
76C2×#11758194—21no—
77#14519777—21no—
78#14519777C2311yescharacteristic
79#14519776—21no—
80#14519776—41no—
81#14519774C2311yescharacteristic
82#14519773—21no—
83#14519751—21no—
84#14519750—41no—
85Q8×Dic61—21no—
86C4×Dic122—21no—
87C4×Dic122C2311yescharacteristic
88#14519739—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710D411yescharacteristic
94#14519708—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519706—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519700—41no—
105#14519700D411yescharacteristic
106#14519699—21no—
107#14519699—21no—
108#14519698—21no—
109#14519697—21no—
110#14519697—21no—
111C22⋊#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C8×Dic61—21no—
116C22×Q8×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#106×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158#11758222—42no—
159#11758222—81no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×C4×D61—41no—
164C2×Dic122—21no—
165C2×Dic122—41no—
166C2×Dic122C2×D411yescharacteristic
167SD16×C61—22no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C4⋊C4×C61—21no—
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×C61—41no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—41no—
188C42×C61C2411yescharacteristic
189#11758202—42no—
190#11758202SD1614yescomplemented
191#11758201—42no—
192#11758201—42no—
193#11758201—42no—
194#11758201—81no—
195#11758201—82no—
196#11758198—41no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—41no—
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—21no—
202#11758196—41no—
203C4×Dic61—21no—
204C4×Dic61—41no—
205#11758194—42no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C23×C61—21no—
213C22×D61—41no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—82no—
219Q8×C61—81no—
220Q8×C61—161no—
221Q8×C61C2×SD1612yes—
222Q8×C61#13712yescomplemented
223D4×C61—21no—
224D4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—82no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C22⋊D61—82no—
239C22⋊D61—161no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—81no—
248C4×D61—82no—
249Dic122—21no—
250Dic122—81no—
251Dic122—161no—
252Dic122C2×SD1612yes—
253C8×C61—21no—
254C8×C61—21no—
255C8×C61—41no—
256#92418—21no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#92418—41no—
261#92418C2×SD1612yes—
262#19809—611nomaximal, Sylow
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61D4211yescharacteristic
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61C22×SD1611yescharacteristic
279C4×C61C2×#13711yescharacteristic
280C4×C61#57711yescharacteristic
281C4×C61#57811yescharacteristic
282Dic61—22no—
283Dic61—41no—
284Dic61—81no—
285Q8×Q16—611no—
286Q8×SD16D6111yescharacteristic, complemented
287#3362—611no—
288#3278—611no—
289#1714—611no—
290#1695—611no—
291#1678—611no—
292#1672—611no—
293#1666—611no—
294#1645—611no—
295#1637—611no—
296#1631—611no—
297#1623—611no—
298#1583—611no—
299#1579—611no—
300C2×C61—81no—
301C2×C61#327011yescharacteristic
302C2×C61D4×SD1611yescharacteristic
303C2×C61#328811yescharacteristic
304D61—161no—
305C2×#137—1221no—
306Q82—611no—
307Q82C2×D6111yescharacteristic
308#550—611no—
309D4×Q8C2×D6111yescharacteristic
310#500—611no—
311#500—1221no—
312#498—21no—
313#498C2×D6111yescharacteristic
314#492—611no—
315#492—611no—
316#487—611no—
317#486—611no—
318#486—1221no—
319#477—611no—
320#477—611no—
321#477—1221no—
322#477—1221no—
323#476—21no—
324#476—611no—
325#476—1221no—
326#476C2×D6111yescharacteristic
327#475—21no—
328#475C2×D6111yescharacteristic
329#468—1221no—
330#462—611no—
331#460—611no—
332#460—611no—
333#460—1221no—
334#452—1221no—
335#448—1221no—
336C8×Q8—611no—
337C8×Q8—611no—
338C8×Q8C2×D6111yescharacteristic
339C4×Q16—611no—
340C4×Q16—1221no—
341C4×SD16—21no—
342C4×SD16—1221no—
343C4×SD16C2×D6111yescharacteristic
344#436—611no—
345#334—611no—
346#334—611no—
347#333—611no—
348#333—611no—
349#332—611no—
350#331—611no—
351Q8⋊C8—611no—
352Q8⋊C8—611no—
353Q8⋊C8—611no—
354D4⋊C8—611no—
355#322—611no—
356C61#1980911yescharacteristic, complemented, Sylow
357C2×#41—1221no—
358C22×Q8—21no—
359#137—2444no—
360C2×Q16—611no—
361C2×Q16—1221no—
362C2×SD16—1221no—
363C2×SD16—1221no—
364C2×SD16—1221no—
365C2×SD16—1221no—
366C2×SD16—2441no—
367C2×SD16C22⋊D6111yescharacteristic
368C2×M16—1221no—
369C4⋊Q8—21no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—41no—
373C4⋊Q8—611no—
374C4⋊Q8—1221no—
375C4⋊Q8—1221no—
376C4⋊Q8—2441no—
377C4⋊Q8C22×D6111yescharacteristic
378C4⋊Q8C22×D6111yescharacteristic
379#127—611no—
380#127—1221no—
381C22⋊Q8—21no—
382C22⋊Q8—41no—
383#121—1221no—
384#121—2441no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8—611no—
388C4×Q8—1221no—
389C4×Q8—1221no—
390C4×Q8C22×D6111yescharacteristic
391C4×Q8C22×D6111yescharacteristic
392C4×D4—21no—
393C4×D4—1221no—
394C4×D4C22×D6111yescharacteristic
395#107—611no—
396#107—1221no—
397#107—1221no—
398#107—1221no—
399#107—1221no—
400#106—21no—
401#106—21no—
402#106—21no—
403#106—21no—
404#106—1221no—
405#106C22⋊D6111yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—1221no—
411C4⋊C8—1221no—
412C4⋊C8C22×D6111yescharacteristic
413#103—21no—
414#103—611no—
415#103—611no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103—2441no—
420#103C22⋊D6111yescharacteristic
421#102—21no—
422#102—1221no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102C22⋊D6111yescharacteristic
428C22⋊C8—1221no—
429C4×C8—21no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—1221no—
433C4×C8—1221no—
434C4×C8C22×D6111yescharacteristic
435#41—4881no—
436#41—2442no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—41no—
440C2×Q8—41no—
441C2×Q8—42no—
442C2×Q8—81no—
443C2×Q8—611no—
444C2×Q8—1221no—
445C2×Q8—2441no—
446C2×Q8D4×D6111yescharacteristic
447C2×Q8C2×C22⋊D6111yescharacteristic
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—2441no—
452C2×D4—2441no—
453C2×D4C2×C22⋊D6111yescharacteristic
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—2441no—
457Q16—614no—
458Q16—2442no—
459SD16—22no—
460SD16—4881no—
461SD16—4882no—
462SD16—2442no—
463SD16—2442no—
464SD16—2442no—
465M16—2442no—
466C2×C8—21no—
467C2×C8—611no—
468C2×C8—611no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8C2×C22⋊D6111yescharacteristic
474C2×C8C2×C22⋊D6111yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—611no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—2441no—
489C4⋊C4—2441no—
490C4⋊C4D4×D6111yescharacteristic
491C4⋊C4C2×C22⋊D6111yescharacteristic
492C4⋊C4C2×C22⋊D6111yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—41no—
495C22⋊C4—2441no—
496C42—21no—
497C42—21no—
498C42—41no—
499C42—1221no—
500C42—2441no—
501C42C23×D6111yescharacteristic
502C23—21no—
503C23—2441no—
504Q8—21no—
505Q8—21no—
506Q8—41no—
507Q8—41no—
508Q8—82no—
509Q8—81no—
510Q8—161no—
511Q8—612no—
512Q8—1221no—
513Q8—9761no—
514Q8—4881no—
515Q8SD16×D6112yes—
516Q8#1451980812yescomplemented
517D4—21no—
518D4—21no—
519D4—9761no—
520D4—4881no—
521D4—4882no—
522D4—2441no—
523D4—2441no—
524D4—2441no—
525D4—2441no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—4882no—
539C2×C4—2441no—
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4C22×C22⋊D6111yescharacteristic
542C2×C4#1451982211yescharacteristic
543C8—21no—
544C8—21no—
545C8—41no—
546C8—612no—
547C8—1221no—
548C8—2441no—
549C8—2441no—
550C8—2441no—
551C8—2441no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22D4×C22⋊D6111yescharacteristic
559C4—22no—
560C4—21no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—1222no—
568C4—4881no—
569C4—2441no—
570C4C2×SD16×D6111yescharacteristic
571C4C2×#1451980811yescharacteristic
572C4#1684056811yescharacteristic
573C4#1684057711yescharacteristic
574C2—81no—
575C2—9761no—
576C2#2145739011yescharacteristic
577C2SD16×C22⋊D6111yescharacteristic
578C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×SD16×C61C2
socleC22×C61D42
prime radical#16840433C22
hypercenterQ8×SD16D61
hypocenterC61#19809
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×#11758202C2semidirect
4. #21445444C2semidirect
5. #21445340C2semidirect
6. #21445235C2semidirect
7. #21445198C2semidirect
8. #21445169C2semidirect
9. #21445039C2semidirect
10. #21444984C2semidirect
12. #21444897C2semidirect
13. #21444663C2semidirect
14. #21444636C2semidirect
15. Q8×SD16×C61C2semidirect
190. #11758202SD16semidirect
222. Q8×C61#137semidirect
286. Q8×SD16D61semidirect
356. C61#19809semidirect
516. Q8#14519808semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582022
#214584882
#214574812
#214454442
#214453402
#214452352
#214451982
#214451692
#214450392
#214449842
#214449452
#214448972
#214446632
#214446362
Q8×SD16×C612
#1980961