Finite Group Databasebeta

← #29670164

Subgroups — #29670164

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
1Q16×Dic122C211yesmaximal, characteristic, complemented
2Q8×#11758200C211yesmaximal, characteristic, complemented
3#21458486C211yesmaximal, characteristic
4#21445438C211yesmaximal, characteristic, complemented
5#21445342C211yesmaximal, characteristic, complemented
6#21445233C211yesmaximal, characteristic, complemented
7#21445199C211yesmaximal, characteristic, complemented
8#21445164C211yesmaximal, characteristic, complemented
9#21445038C211yesmaximal, characteristic, complemented
10#21444984C211yesmaximal, characteristic, complemented
11#21444946C211yesmaximal, characteristic
12#21444896C211yesmaximal, characteristic, complemented
13#21444660C211yesmaximal, characteristic, complemented
14#21444638C211yesmaximal, characteristic, complemented
15C61×#3278C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17Q8×Dic122C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840166—21no—
20#16840158—21no—
21#16840158C2211yescharacteristic
22#16840134C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840108—21no—
25#16840108C2211yescharacteristic
26#16840057C2211yescharacteristic
27#16840057C2211yescharacteristic
28#16840055—21no—
29#16840055—21no—
30#16840052—21no—
31#16840052—21no—
32#16840052C2211yescharacteristic
33#16840052C2211yescharacteristic
34#16840044—21no—
35#16840044C2211yescharacteristic
36#16839959—21no—
37#16839959C2211yescharacteristic
38Q8×#92418C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41Q16×Dic61—21no—
42C4×#11758200—21no—
43C4×#11758200—21no—
44C4×#11758200C2211yescharacteristic
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56C2×#137×C61—21no—
57C61×#550C2211yescharacteristic
58D4×Q8×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#468—21no—
61C61×#462C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#452—21no—
64C61×#448—21no—
65C4×Q16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758200—41no—
69C2×#11758200D411yescharacteristic
70#14519777—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519776—21no—
75#14519776—21no—
76#14519776—41no—
77#14519776—41no—
78Q8×Dic61—21no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519711—21no—
84#14519711—21no—
85#14519710—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708D411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519701—21no—
101#14519700—21no—
102#14519700—41no—
103#14519700D411yescharacteristic
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C4×#92418—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×#41×C61—21no—
116C22×Q8×C61—21no—
117#137×C61—44no—
118C2×Q16×C61—21no—
119C2×Q16×C61D411yescharacteristic
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—21no—
123C2×M16×C61—21no—
124C4⋊Q8×C61—21no—
125C4⋊Q8×C61C2311yescharacteristic
126#127×C61—21no—
127#127×C61C2311yescharacteristic
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—41no—
130#121×C61—21no—
131#121×C61—41no—
132C4×Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—41no—
157C2×D4×C61—41no—
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—41no—
165C2×Dic122—41no—
166C2×Dic122C2×D411yescharacteristic
167Q16×C61—42no—
168Q16×C61SD1614yescomplemented
169SD16×C61—42no—
170SD16×C61—42no—
171SD16×C61—81no—
172M16×C61—42no—
173C2×C8×C61—21no—
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61C2411yescharacteristic
190#11758200—22no—
191#11758200—42no—
192#11758200—82no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—41no—
198C4⋊Dic61—41no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—21no—
201#11758196—41no—
202#11758196—41no—
203C4×Dic61—21no—
204C4×Dic61—41no—
205C4×Dic61—41no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C23×C61—21no—
213C23×C61—41no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—82no—
219Q8×C61C2×SD1612yes—
220Q8×C61C2×SD1612yes—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—82no—
228D4×C61—81no—
229D4×C61—161no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—82no—
239C2×C4×C61—82no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247Dic122—21no—
248Dic122—81no—
249Dic122—81no—
250Dic122—161no—
251Dic122—161no—
252Dic122#13712yescomplemented
253C8×C61—21no—
254C8×C61—41no—
255C8×C61C2×SD1612yes—
256#92418—21no—
257#92418—21no—
258#92418—41no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#19809—611nomaximal, Sylow
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61D4211yescharacteristic
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—81no—
277C4×C61C22×SD1611yescharacteristic
278C4×C61C2×#13711yescharacteristic
279C4×C61#57711yescharacteristic
280C4×C61#57811yescharacteristic
281Dic61—22no—
282Dic61—41no—
283Dic61—81no—
284Dic61—81no—
285Q8×Q16—611no—
286Q8×SD16—611no—
287#3362—611no—
288#3278D6111yescharacteristic
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—161no—
302C2×C61#327011yescharacteristic
303C2×C61D4×SD1611yescharacteristic
304C2×C61#328811yescharacteristic
305C2×#137—21no—
306Q82—611no—
307Q82—611no—
308#550C2×D6111yescharacteristic
309D4×Q8C2×D6111yescharacteristic
310#500—611no—
311#500—1221no—
312#498—611no—
313#498—1221no—
314#492—611no—
315#492C2×D6111yescharacteristic
316#487—611no—
317#486—611no—
318#486—1221no—
319#477—611no—
320#477—611no—
321#477—1221no—
322#477—1221no—
323#476—611no—
324#476—611no—
325#476—1221no—
326#476—1221no—
327#475—611no—
328#475—1221no—
329#468—21no—
330#462C2×D6111yescharacteristic
331#460—611no—
332#460—1221no—
333#460C2×D6111yescharacteristic
334#452—21no—
335#448—21no—
336C8×Q8—611no—
337C8×Q8—611no—
338C8×Q8—611no—
339C4×Q16—1221no—
340C4×Q16C2×D6111yescharacteristic
341C4×SD16—611no—
342C4×SD16—1221no—
343C4×SD16—1221no—
344#436C2×D6111yescharacteristic
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—21no—
358C22×Q8—21no—
359#137—44no—
360C2×Q16—21no—
361C2×Q16C22⋊D6111yescharacteristic
362C2×SD16—21no—
363C2×SD16—21no—
364C2×SD16—21no—
365C2×SD16—611no—
366C2×SD16—1221no—
367C2×SD16—2441no—
368C2×M16—21no—
369C4⋊Q8—21no—
370C4⋊Q8—611no—
371C4⋊Q8—611no—
372C4⋊Q8—1221no—
373C4⋊Q8—1221no—
374C4⋊Q8—1221no—
375C4⋊Q8—1221no—
376C4⋊Q8—2441no—
377C4⋊Q8—2441no—
378C4⋊Q8C22×D6111yescharacteristic
379#127—21no—
380#127C22×D6111yescharacteristic
381C22⋊Q8—21no—
382C22⋊Q8—41no—
383#121—21no—
384#121—41no—
385C4×Q8—611no—
386C4×Q8—1221no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×Q8—1221no—
390C4×Q8C22×D6111yescharacteristic
391C4×Q8C22×D6111yescharacteristic
392C4×D4—21no—
393C4×D4—21no—
394C4×D4C22×D6111yescharacteristic
395#107—1221no—
396#107—1221no—
397#107—1221no—
398#107—1221no—
399#107C22⋊D6111yescharacteristic
400#106—611no—
401#106—1221no—
402#106—1221no—
403#106—1221no—
404#106—1221no—
405#106—1221no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—1221no—
410C4⋊C8—1221no—
411C4⋊C8—1221no—
412C4⋊C8C22×D6111yescharacteristic
413#103—21no—
414#103—611no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—2441no—
419#103C22⋊D6111yescharacteristic
420#103C22⋊D6111yescharacteristic
421#102—21no—
422#102—611no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428C22⋊C8—21no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—1221no—
432C4×C8—1221no—
433C4×C8—1221no—
434C4×C8C22×D6111yescharacteristic
435#41—42no—
436#41—81no—
437C2×Q8—21no—
438C2×Q8—41no—
439C2×Q8—42no—
440C2×Q8—81no—
441C2×Q8—611no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—2441no—
445C2×Q8—2441no—
446C2×Q8C2×C22⋊D6111yescharacteristic
447C2×Q8C2×C22⋊D6111yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4D4×D6111yescharacteristic
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457Q16—42no—
458Q16#1175820014yescomplemented
459SD16—42no—
460SD16—42no—
461SD16—81no—
462SD16—1222no—
463SD16—4882no—
464SD16—2442no—
465M16—42no—
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—41no—
480C4⋊C4—611no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—2441no—
487C4⋊C4—2441no—
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—41no—
496C42—21no—
497C42—21no—
498C42—1221no—
499C42—2441no—
500C42—2441no—
501C42C23×D6111yescharacteristic
502C23—21no—
503C23—41no—
504Q8—21no—
505Q8—21no—
506Q8—41no—
507Q8—41no—
508Q8—82no—
509Q8—612no—
510Q8—1221no—
511Q8—9761no—
512Q8—9761no—
513Q8—4881no—
514Q8—4881no—
515Q8C2×#1175820012yes—
516Q8C2×#1175820012yes—
517D4—21no—
518D4—21no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—82no—
524D4—81no—
525D4—161no—
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—82no—
535C2×C4—82no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4C22×C22⋊D6111yescharacteristic
542C2×C4#1451982211yescharacteristic
543C8—21no—
544C8—41no—
545C8—1221no—
546C8—1221no—
547C8—2441no—
548C8—2441no—
549C8—2441no—
550C8—2441no—
551C8C2×#1175820012yes—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22D4×C22⋊D6111yescharacteristic
559C4—22no—
560C4—22no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—1222no—
567C4—4881no—
568C4—4881no—
569C4—2441no—
570C4C22×#1175820011yescharacteristic
571C4#137×D6111yescharacteristic
572C4#1684057311yescharacteristic
573C4#1684057811yescharacteristic
574C2—81no—
575C2—161no—
576C2#2145738311yescharacteristic
577C2D4×#1175820011yescharacteristic
578C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3278C2
socleC22×C61D42
prime radicalC61×#550C22
hypercenter#3278D61
hypocenterC61#19809
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3278

Complements

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

NH ≅ G/Ntype
1. Q16×Dic122C2semidirect
2. Q8×#11758200C2semidirect
4. #21445438C2semidirect
5. #21445342C2semidirect
6. #21445233C2semidirect
7. #21445199C2semidirect
8. #21445164C2semidirect
9. #21445038C2semidirect
10. #21444984C2semidirect
12. #21444896C2semidirect
13. #21444660C2semidirect
14. #21444638C2semidirect
168. Q16×C61SD16semidirect
252. Dic122#137semidirect
356. C61#19809semidirect
458. Q16#11758200semidirect

Maximal subgroups

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

subgroupindex
Q16×Dic1222
Q8×#117582002
#214584862
#214454382
#214453422
#214452332
#214451992
#214451642
#214450382
#214449842
#214449462
#214448962
#214446602
#214446382
C61×#32782
#1980961