Finite Group Databasebeta

← #29669555

Subgroups — #29669555

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458727C211yesmaximal, characteristic, complemented
2#21458486C211yesmaximal, characteristic, complemented
3#21458176C211yesmaximal, characteristic, complemented
4#21445438C211yesmaximal, characteristic, complemented
5#21445288C211yesmaximal, characteristic, complemented
6#21445226C211yesmaximal, characteristic, complemented
7#21445218C211yesmaximal, characteristic, complemented
8#21445199C211yesmaximal, characteristic, complemented
9#21445038C211yesmaximal, characteristic, complemented
10#21444970C211yesmaximal, characteristic, complemented
11#21444946C211yesmaximal, characteristic, complemented
12#21444889C211yesmaximal, characteristic, complemented
13#21444655C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3284C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840469C2211yescharacteristic
18#16840169—21no—
19#16840167C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840134C2211yescharacteristic
22#16840114C2211yescharacteristic
23#16840108—21no—
24#16840108C2211yescharacteristic
25#16840092—21no—
26#16840078—21no—
27#16840071—21no—
28#16840057C2211yescharacteristic
29#16840052—21no—
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16840052C2211yescharacteristic
33#16840044C2211yescharacteristic
34#16840040—21no—
35#16839989—21no—
36#16839959—21no—
37#16839959C2211yescharacteristic
38#16839892C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839863—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519708—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56C2×#135×C61—21no—
57C61×#550C2211yescharacteristic
58C61×#547C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#465—21no—
61C61×#462C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#452—21no—
64C61×#449—21no—
65C4×SD16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758200—41no—
69C2×#11758200D411yescharacteristic
70C22×#92418—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519776—21no—
75#14519776—41no—
76#14519768—21no—
77#14519756—21no—
78#14519754—41no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122C2311yescharacteristic
82D4×Dic61—21no—
83C2×C4⋊Dic61—21no—
84#14519711—21no—
85#14519711—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519700—21no—
101#14519700—41no—
102#14519700D411yescharacteristic
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519687—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C2×#41×C61—21no—
117#135×C61—44no—
118C2×Q16×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61D411yescharacteristic
122C2×D8×C61—21no—
123C22×C8×C61—21no—
124C4⋊Q8×C61C2311yescharacteristic
125#127×C61—21no—
126#127×C61C2311yescharacteristic
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—41no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—42no—
147#41×C61—81no—
148#41×C61—81no—
149C2×Q8×C61—21no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C22×C4×C61—41no—
163C22×Dic61—41no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—41no—
167C2×Dic122C2×D411yescharacteristic
168Q16×C61—42no—
169SD16×C61—42no—
170SD16×C61—81no—
171SD16×C61SD1614yescomplemented
172D8×C61—42no—
173C2×C8×C61—21no—
174C2×C8×C61—42no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
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—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61C2411yescharacteristic
190C22⋊Dic61—41no—
191#11758200—41no—
192#11758200—42no—
193#11758200—82no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61—41no—
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—21no—
202#11758196—41no—
203#11758196—41no—
204C4×Dic61—21no—
205C4×Dic61—41no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—42no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—41no—
216Q8×C61—21no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61C2×SD1612yes—
220D4×C61—21no—
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—82no—
230D4×C61—161no—
231D4×C61C2×SD1612yes—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—82no—
242C2×C4×C61—82no—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—161no—
251Dic122—21no—
252Dic122—21no—
253Dic122—81no—
254Dic122—81no—
255Dic122—161no—
256C8×C61—21no—
257C8×C61—41no—
258C8×C61C2×SD1612yes—
259#92418—21no—
260#92418—21no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#19768—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61D4211yescharacteristic
276C4×C61—22no—
277C4×C61—21no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—81no—
282C4×C61C22×SD1611yescharacteristic
283C4×C61#57511yescharacteristic
284C4×C61#57711yescharacteristic
285C4×C61#57811yescharacteristic
286Dic61—41no—
287Dic61—41no—
288Dic61—81no—
289Dic61—81no—
290#3385—611no—
291#3362—611no—
292#3335—611no—
293#3284D6111yescharacteristic
294#1714—611no—
295#1686—611no—
296#1677—611no—
297#1675—611no—
298#1672—611no—
299#1645—611no—
300#1635—611no—
301#1631—611no—
302#1622—611no—
303#1582—611no—
304SD16⋊C8—611no—
305C2×C61—42no—
306C2×C61—81no—
307C2×C61—161no—
308C2×C61#326911yescharacteristic
309C2×C61D4×SD1611yescharacteristic
310C2×C61#328811yescharacteristic
311C2×#135—21no—
312Q82—611no—
313#554—611no—
314#550C2×D6111yescharacteristic
315#547C2×D6111yescharacteristic
316#501—1221no—
317#500—611no—
318#498—611no—
319#492—611no—
320#492C2×D6111yescharacteristic
321#487—611no—
322#486—611no—
323#486—1221no—
324#483—1221no—
325#481—1221no—
326#479—1221no—
327#477—611no—
328#476—611no—
329#476—611no—
330#476—611no—
331#476—1221no—
332#475—611no—
333#474—1221no—
334#465—21no—
335#465—1221no—
336#462C2×D6111yescharacteristic
337#460—611no—
338#460—1221no—
339#459C2×D6111yescharacteristic
340#452—21no—
341#449—21no—
342#446—611no—
343C8×Q8—611no—
344#440—1221no—
345C4×SD16—611no—
346C4×SD16—1221no—
347C4×SD16C2×D6111yescharacteristic
348C8×D4—611no—
349C8×D4C2×D6111yescharacteristic
350C2×#106—1221no—
351#335—611no—
352#334—611no—
353#333—611no—
354#333—611no—
355#332—611no—
356#331—611no—
357Q8⋊C8—611no—
358Q8⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361#322—611no—
362C61#1976811yescharacteristic, complemented, Sylow
363C2×#41—21no—
364C2×#41—21no—
365#135—44no—
366C2×Q16—21no—
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—611no—
370C2×SD16—1221no—
371C2×SD16—2441no—
372C2×SD16C22⋊D6111yescharacteristic
373C2×D8—21no—
374C22×C8—21no—
375C22×C8—1221no—
376C4⋊Q8—611no—
377C4⋊Q8—611no—
378C4⋊Q8—1221no—
379C4⋊Q8—1221no—
380C4⋊Q8—2441no—
381C4⋊Q8C22×D6111yescharacteristic
382#127—21no—
383#127C22×D6111yescharacteristic
384#125—1221no—
385#123—41no—
386C22⋊Q8—21no—
387C22⋊Q8—1221no—
388C22⋊Q8—2441no—
389#121—21no—
390#121—21no—
391#121—41no—
392C4×Q8—611no—
393C4×Q8—1221no—
394C4×Q8—1221no—
395C4×Q8C22×D6111yescharacteristic
396C4×D4—21no—
397C4×D4—1221no—
398C4×D4C22×D6111yescharacteristic
399C4×D4C22×D6111yescharacteristic
400C2×C4⋊C4—21no—
401C2×C4⋊C4—1221no—
402#107—1221no—
403#107—1221no—
404#107—1221no—
405#106—611no—
406#106—1221no—
407#106—1221no—
408#106—1221no—
409#106—1221no—
410#106—1221no—
411#106—1221no—
412#106C22⋊D6111yescharacteristic
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8C22×D6111yescharacteristic
419#103—21no—
420#103—611no—
421#103—1221no—
422#103—1221no—
423#103—2441no—
424#103C22⋊D6111yescharacteristic
425#102—21no—
426#102—611no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102C22⋊D6111yescharacteristic
434C22⋊C8—21no—
435C22⋊C8—1221no—
436#97—1221no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—1221no—
440C4×C8C22×D6111yescharacteristic
441#41—42no—
442#41—42no—
443#41—81no—
444#41—81no—
445C2×Q8—21no—
446C2×Q8—611no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—2441no—
450C2×Q8C2×C22⋊D6111yescharacteristic
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4D4×D6111yescharacteristic
458C2×D4C2×C22⋊D6111yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—2441no—
464Q16—42no—
465SD16—42no—
466SD16—81no—
467SD16—4882no—
468SD16—2441no—
469SD16—2442no—
470SD16#1175820014yescomplemented
471D8—42no—
472C2×C8—21no—
473C2×C8—42no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—2442no—
481C2×C8C2×C22⋊D6111yescharacteristic
482C2×C8C2×C22⋊D6111yescharacteristic
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—41no—
487C4⋊C4—611no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—2441no—
494C4⋊C4—2441no—
495C4⋊C4—2441no—
496C4⋊C4—2441no—
497C4⋊C4D4×D6111yescharacteristic
498C4⋊C4C2×C22⋊D6111yescharacteristic
499C4⋊C4C2×C22⋊D6111yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—2441no—
505C42—21no—
506C42—1221no—
507C42—2441no—
508C42C23×D6111yescharacteristic
509C23—21no—
510C23—21no—
511C23—41no—
512Q8—21no—
513Q8—41no—
514Q8—41no—
515Q8—1221no—
516Q8—1221no—
517Q8—9761no—
518Q8—4881no—
519Q8—4881no—
520Q8C2×#1175820012yes—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—82no—
529D4—81no—
530D4—82no—
531D4—161no—
532D4C2×#1175820012yes—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—9761no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4#1451982211yescharacteristic
552C8—21no—
553C8—41no—
554C8—1221no—
555C8—1221no—
556C8—2441no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C8C2×#1175820012yes—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22D4×C22⋊D6111yescharacteristic
571C4—22no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—81no—
577C4—4881no—
578C4—4881no—
579C4—2441no—
580C4—2441no—
581C4C22×#1175820011yescharacteristic
582C4#1684055911yescharacteristic
583C4#1684057311yescharacteristic
584C4#1684057611yescharacteristic
585C2—42no—
586C2—81no—
587C2—161no—
588C2#2145737411yescharacteristic
589C2D4×#1175820011yescharacteristic
590C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3284C2
socleC22×C61D42
hypercenter#3284D61
hypocenterC61#19768
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3284

Complements

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

NH ≅ G/Ntype
1. #21458727C2semidirect
2. #21458486C2semidirect
3. #21458176C2semidirect
4. #21445438C2semidirect
5. #21445288C2semidirect
6. #21445226C2semidirect
7. #21445218C2semidirect
8. #21445199C2semidirect
9. #21445038C2semidirect
10. #21444970C2semidirect
11. #21444946C2semidirect
12. #21444889C2semidirect
13. #21444655C2semidirect
14. SD16⋊#92418C2semidirect
171. SD16×C61SD16semidirect
362. C61#19768semidirect
470. SD16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214587272
#214584862
#214581762
#214454382
#214452882
#214452262
#214452182
#214451992
#214450382
#214449702
#214449462
#214448892
#214446552
SD16⋊#924182
C61×#32842
#1976861