Finite Group Databasebeta

← #29670560

Subgroups — #29670560

Order 15616.

652 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#21458535C211yesmaximal, characteristic, complemented
2#21457758C211yesmaximal, characteristic, complemented
3D4×#11758202C211yesmaximal, characteristic, complemented
4#21445480C211yesmaximal, characteristic, complemented
5#21445384C211yesmaximal, characteristic, complemented
6#21445254C211yesmaximal, characteristic, complemented
7#21445249C211yesmaximal, characteristic, complemented
8#21445182C211yesmaximal, characteristic, complemented
9#21445088C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21445012C211yesmaximal, characteristic, complemented
12#21444921C211yesmaximal, characteristic, complemented
13#21444669C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3358C211yesmaximal, characteristic
16C22×#11758202—21no—
17#16840482C2211yescharacteristic
18D4×Dic122C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840141C2211yescharacteristic
21#16840120—21no—
22#16840113C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840083—21no—
26#16840056—21no—
27#16840047C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840001—21no—
30#16839989—21no—
31#16839975—21no—
32#16839972C2211yescharacteristic
33#16839969C2211yescharacteristic
34#16839967C2211yescharacteristic
35#16839959C2211yescharacteristic
36#16839924—21no—
37#16839918—21no—
38#16839917—21no—
39#16839892C2211yescharacteristic
40#16839859—21no—
41C4×#11758202C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44#16839758—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839509C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#550C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#486—21no—
59C61×#486C2211yescharacteristic
60C61×#474—21no—
61C61×#474C2211yescharacteristic
62C61×#459—21no—
63C61×#459C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—44no—
73C2×#11758202D411yescharacteristic
74C2×#11758200—21no—
75C2×#11758200—21no—
76C22×#92418—21no—
77C22×#92418—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519776—21no—
81#14519770—41no—
82#14519764—21no—
83#14519756—21no—
84#14519756—21no—
85#14519754—41no—
86C4×Dic122C2311yescharacteristic
87#14519742—21no—
88D4×Dic61—21no—
89#14519736—21no—
90#14519710—21no—
91#14519710D411yescharacteristic
92#14519708—21no—
93#14519708—41no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519706—21no—
98#14519702—21no—
99#14519702—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519700—21no—
105#14519700D411yescharacteristic
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698—21no—
110#14519698—41no—
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519687—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C22×Q8×C61—21no—
118C2×D8×C61—21no—
119C2×D8×C61D411yescharacteristic
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61—21no—
123#127×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126#121×C61—21no—
127#121×C61—41no—
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C4×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×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
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—42no—
167C2×Dic122—41no—
168C2×Dic122—81no—
169C2×Dic122C2×D411yescharacteristic
170D8×C61—22no—
171D8×C61—42no—
172C2×C8×C61—21no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
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—
189C22⋊Dic61—41no—
190C22⋊Dic61—41no—
191#11758202—22no—
192#11758202—42no—
193#11758202—42no—
194#11758202—42no—
195#11758202—81no—
196#11758200—42no—
197#11758200—81no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205#11758196—41no—
206C4×Dic61—21no—
207C4×Dic61—41no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—42no—
213C2×#92418—81no—
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—41no—
219Q8×C61—21no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—82no—
223Q8×C61AGL(1,ℤ/8)12yes—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—82no—
232D4×C61—81no—
233D4×C61—161no—
234D4×C61#13512yes—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—82no—
245C2×C4×C61—82no—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—82no—
254Dic122—21no—
255Dic122—21no—
256Dic122—41no—
257Dic122—41no—
258Dic122—41no—
259Dic122—41no—
260Dic122—82no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—41no—
264#92418—21no—
265#92418—21no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#19835—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61D4211yescharacteristic
281C4×C61—22no—
282C4×C61—21no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—81no—
287C4×C61C2×#13511yescharacteristic
288C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
289C4×C61#57611yescharacteristic
290C4×C61#57811yescharacteristic
291Dic61—41no—
292Dic61—41no—
293Dic61—81no—
294Dic61—81no—
295#3367—611no—
296#3358D6111yescharacteristic
297#3304—611no—
298D4×Q16—611no—
299#1720—611no—
300#1702—611no—
301#1681—611no—
302#1680—611no—
303#1669—611no—
304#1652—611no—
305#1651—611no—
306#1641—611no—
307#1627—611no—
308#1584—611no—
309#1580—611no—
310C2×C61—42no—
311C2×C61—81no—
312C2×C61—161no—
313C2×C61#326411yescharacteristic
314C2×C61#328511yescharacteristic
315C2×C61#328911yescharacteristic
316C22×Q16—1221no—
317#556—611no—
318#550C2×D6111yescharacteristic
319D4×Q8—611no—
320D4×Q8C2×D6111yescharacteristic
321#498—611no—
322#494—611no—
323#489—1221no—
324#487—611no—
325#486—21no—
326#486—611no—
327#486—611no—
328#486C2×D6111yescharacteristic
329#482—1221no—
330#477—1221no—
331#475—611no—
332#474—21no—
333#474—611no—
334#474C2×D6111yescharacteristic
335#467—1221no—
336#465—1221no—
337#463—1221no—
338#462—611no—
339#462—611no—
340#461—611no—
341#460—611no—
342#459—21no—
343#459C2×D6111yescharacteristic
344#452—1221no—
345#451—1221no—
346#451—1221no—
347#446—611no—
348C8×Q8C2×D6111yescharacteristic
349#440—1221no—
350C4×Q16—611no—
351C4×D8—21no—
352C4×D8C2×D6111yescharacteristic
353C8×D4—611no—
354C8×D4—611no—
355#416—1221no—
356#335—611no—
357#334—611no—
358#332—611no—
359#332—611no—
360#332—611no—
361#331—611no—
362Q8⋊C8—611no—
363Q8⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366#322—611no—
367C61#1983511yescharacteristic, complemented, Sylow
368C2×#41—21no—
369C22×Q8—21no—
370C22×Q8—1221no—
371C2×Q16—611no—
372C2×Q16—1221no—
373C2×Q16—1221no—
374C2×Q16—1221no—
375C2×Q16—1221no—
376C2×Q16—2444no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×D8—21no—
380C2×D8C22⋊D6111yescharacteristic
381C22×C8—1221no—
382C22×C8—1221no—
383C4⋊Q8—21no—
384C4⋊Q8—611no—
385C4⋊Q8—611no—
386C4⋊Q8—1221no—
387C4⋊Q8C22×D6111yescharacteristic
388#127—21no—
389#127C22×D6111yescharacteristic
390#126—2441no—
391#124—1221no—
392C22⋊Q8—21no—
393C22⋊Q8—41no—
394C22⋊Q8—1221no—
395C22⋊Q8—1221no—
396C22⋊Q8—2441no—
397#121—21no—
398#121—41no—
399C4×Q8—611no—
400C4×Q8—1221no—
401C4×Q8C22×D6111yescharacteristic
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—1221no—
405C4×D4C22×D6111yescharacteristic
406C4×D4C22×D6111yescharacteristic
407#117—1221no—
408#107—21no—
409#107—611no—
410#107—1221no—
411#107C22⋊D6111yescharacteristic
412#106—1221no—
413#106—2441no—
414C4⋊C8—21no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—1221no—
419C4⋊C8C22×D6111yescharacteristic
420#103—611no—
421#103—611no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#102—21no—
429#102—21no—
430#102—21no—
431#102—21no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—2441no—
437#102C22⋊D6111yescharacteristic
438#102C22⋊D6111yescharacteristic
439C22⋊C8—1221no—
440C22⋊C8—1221no—
441#97—1221no—
442C4×C8—21no—
443C4×C8—611no—
444C4×C8—611no—
445C4×C8C22×D6111yescharacteristic
446#41—42no—
447#41—81no—
448C2×Q8—21no—
449C2×Q8—41no—
450C2×Q8—42no—
451C2×Q8—81no—
452C2×Q8—611no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—4881no—
456C2×Q8—2441no—
457C2×Q8—2442no—
458C2×Q8D4×D6111yescharacteristic
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4C2×C22⋊D6111yescharacteristic
465C2×D4C2×C22⋊D6111yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—2441no—
471Q16—1222no—
472Q16—4881no—
473Q16—2442no—
474Q16—2442no—
475Q16—2442no—
476SD16—4881no—
477SD16—2442no—
478D8—22no—
479D8—42no—
480C2×C8—21no—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—4881no—
488C2×C8—2442no—
489C2×C8C2×C22⋊D6111yescharacteristic
490C2×C8C2×C22⋊D6111yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—611no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C4⋊C4D4×D6111yescharacteristic
504C4⋊C4C2×C22⋊D6111yescharacteristic
505C4⋊C4C2×C22⋊D6111yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C42—21no—
514C42—1221no—
515C42—2441no—
516C42C23×D6111yescharacteristic
517C23—21no—
518C23—21no—
519C23—41no—
520Q8—21no—
521Q8—41no—
522Q8—41no—
523Q8—82no—
524Q8—1221no—
525Q8—1221no—
526Q8—4882no—
527Q8—2441no—
528Q8—2441no—
529Q8—2441no—
530Q8—2441no—
531Q8#1451980112yes—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—82no—
540D4—81no—
541D4—161no—
542D4#1451979712yes—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—4882no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4C22×C22⋊D6111yescharacteristic
561C2×C4#1451982211yescharacteristic
562C8—21no—
563C8—21no—
564C8—41no—
565C8—1221no—
566C8—1221no—
567C8—2441no—
568C8—2441no—
569C8—2441no—
570C8—2441no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22D4×C22⋊D6111yescharacteristic
581C4—22no—
582C4—21no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—4881no—
588C4—4881no—
589C4—2441no—
590C4—2441no—
591C4C2×#1451979711yescharacteristic
592C4C2×#1451980111yescharacteristic
593C4#1684056311yescharacteristic
594C4#1684057711yescharacteristic
595C2—42no—
596C2—81no—
597C2—161no—
598C2#2145729911yescharacteristic
599C2#2145755611yescharacteristic
600C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3358C2
socleC22×C61D42
hypercenter#3358D61
hypocenterC61#19835
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3358

Complements

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

NH ≅ G/Ntype
1. #21458535C2semidirect
2. #21457758C2semidirect
3. D4×#11758202C2semidirect
4. #21445480C2semidirect
5. #21445384C2semidirect
6. #21445254C2semidirect
7. #21445249C2semidirect
8. #21445182C2semidirect
9. #21445088C2semidirect
10. #21445082C2semidirect
11. #21445012C2semidirect
12. #21444921C2semidirect
13. #21444669C2semidirect
14. #21444643C2semidirect
367. C61#19835semidirect

Maximal subgroups

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

subgroupindex
#214585352
#214577582
D4×#117582022
#214454802
#214453842
#214452542
#214452492
#214451822
#214450882
#214450822
#214450122
#214449212
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