Finite Group Databasebeta

← #29670554

Subgroups — #29670554

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#21458452C211yesmaximal, characteristic
2#21457752C211yesmaximal, characteristic, complemented
3D4×#11758202C211yesmaximal, characteristic, complemented
4#21445476C211yesmaximal, characteristic, complemented
5#21445387C211yesmaximal, characteristic, complemented
6#21445258C211yesmaximal, characteristic, complemented
7#21445246C211yesmaximal, characteristic, complemented
8#21445182C211yesmaximal, characteristic, complemented
9#21445089C211yesmaximal, characteristic, complemented
10#21445080C211yesmaximal, characteristic, complemented
11#21445011C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic, complemented
13#21444667C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3367C211yesmaximal, characteristic, complemented
16C22×#11758202—21no—
17#16840433C2211yescharacteristic
18D4×Dic122C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840141C2211yescharacteristic
21#16840110C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840107—21no—
25#16840086—21no—
26#16840044C2211yescharacteristic
27#16840042C2211yescharacteristic
28#16840041—21no—
29#16840001—21no—
30#16839993—21no—
31#16839972C2211yescharacteristic
32#16839969C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839955—21no—
36#16839922—21no—
37#16839918—21no—
38#16839917—21no—
39C8×Dic122C2211yescharacteristic
40C4×#11758202C2211yescharacteristic
41#16839842—21no—
42D4×#92418C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44#16839756—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#556C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#489—21no—
59C61×#487C2211yescharacteristic
60C61×#477—21no—
61C61×#474C2211yescharacteristic
62C61×#463—21no—
63C61×#461C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#440—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C22×Dic122—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758202—44no—
74C2×#11758202D411yescharacteristic
75C2×#11758201—21no—
76C2×#11758199—21no—
77C22×#92418—21no—
78C22×#92418—21no—
79#14519777C2311yescharacteristic
80#14519776—21no—
81#14519774C2311yescharacteristic
82#14519773—21no—
83#14519756—21no—
84#14519754—41no—
85#14519751—21no—
86#14519750—41no—
87C4×Dic122C2311yescharacteristic
88D4×Dic61—21no—
89#14519739—21no—
90#14519710—21no—
91#14519710D411yescharacteristic
92#14519708—21no—
93#14519708—41no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519706—21no—
98#14519701—21no—
99#14519701D411yescharacteristic
100#14519700—21no—
101#14519700—21no—
102#14519700D411yescharacteristic
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697—41no—
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C8×Dic61—21no—
116C22×Q8×C61—21no—
117C2×SD16×C61—21no—
118C2×D8×C61D411yescharacteristic
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158C22×Dic61—41no—
159#11758222—42no—
160#11758222—81no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C2×Dic122—42no—
167C2×Dic122—41no—
168C2×Dic122—81no—
169C2×Dic122C2×D411yescharacteristic
170SD16×C61—42no—
171D8×C61—22no—
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—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C22⋊C4×C61—41no—
190C42×C61—21no—
191C42×C61—41no—
192C42×C61C2411yescharacteristic
193C22⋊Dic61—41no—
194#11758202—22no—
195#11758202—42no—
196#11758202—42no—
197#11758202—42no—
198#11758202—81no—
199#11758201—81no—
200#11758199—42no—
201#11758198—41no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C2×D411yescharacteristic
205#11758196—21no—
206#11758196—41no—
207C4×Dic61—21no—
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—
218C22×D61—41no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—82no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61#13512yes—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—82no—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C22⋊D61—82no—
245C22⋊D61—161no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—82no—
250D244—41no—
251D244—41no—
252D244—41no—
253D244—41no—
254D244—81no—
255C4×D61—82no—
256Dic122—21no—
257Dic122—41no—
258Dic122—41no—
259Dic122—82no—
260Dic122AGL(1,ℤ/8)12yes—
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×C61D4211yescharacteristic
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61C2×#13511yescharacteristic
289C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
290C4×C61#57611yescharacteristic
291C4×C61#57811yescharacteristic
292Dic61—22no—
293Dic61—41no—
294Dic61—81no—
295#3367D6111yescharacteristic, complemented
296#3358—611no—
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#326411yescharacteristic
313C2×C61#328511yescharacteristic
314C2×C61#328911yescharacteristic
315D61—161no—
316C22×Q16—1221no—
317#556C2×D6111yescharacteristic
318#550—611no—
319D4×Q8—611no—
320D4×Q8C2×D6111yescharacteristic
321#498—611no—
322#494—611no—
323#489—21no—
324#487C2×D6111yescharacteristic
325#486—611no—
326#486—611no—
327#486—611no—
328#486—1221no—
329#482—1221no—
330#477—21no—
331#475—611no—
332#474—611no—
333#474—1221no—
334#474C2×D6111yescharacteristic
335#467—1221no—
336#465—1221no—
337#463—21no—
338#462—611no—
339#462—611no—
340#461C2×D6111yescharacteristic
341#460—611no—
342#459—611no—
343#459—1221no—
344#452—1221no—
345#451—1221no—
346#451—1221no—
347#446C2×D6111yescharacteristic
348C8×Q8—611no—
349#440—21no—
350C4×Q16—611no—
351C4×D8—1221no—
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—1221no—
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—21no—
378C2×SD16—1221no—
379C2×D8—1221no—
380C2×D8C22⋊D6111yescharacteristic
381C22×C8—1221no—
382C22×C8—1221no—
383C4⋊Q8—21no—
384C4⋊Q8—611no—
385C4⋊Q8—1221no—
386C4⋊Q8C22×D6111yescharacteristic
387C4⋊Q8C22×D6111yescharacteristic
388#127—611no—
389#127—1221no—
390#126—41no—
391#124—21no—
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394C22⋊Q8—41no—
395C22⋊Q8—1221no—
396C22⋊Q8—2441no—
397#121—1221no—
398#121—2441no—
399C4×Q8—21no—
400C4×Q8—611no—
401C4×Q8C22×D6111yescharacteristic
402C4×D4—21no—
403C4×D4—1221no—
404C4×D4—1221no—
405C4×D4C22×D6111yescharacteristic
406C4×D4C22×D6111yescharacteristic
407#117—21no—
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—21no—
421#103—21no—
422#103—21no—
423#103—611no—
424#103—611no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#102—21no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
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—21no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—1221no—
445C4×C8C22×D6111yescharacteristic
446#41—4881no—
447#41—2442no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—41no—
451C2×Q8—42no—
452C2×Q8—81no—
453C2×Q8—611no—
454C2×Q8—1221no—
455C2×Q8—4881no—
456C2×Q8—2441no—
457C2×Q8—2442no—
458C2×Q8D4×D6111yescharacteristic
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—2441no—
463C2×D4—2441no—
464C2×D4C2×C22⋊D6111yescharacteristic
465C2×D4C2×C22⋊D6111yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—41no—
469C22×C4—2441no—
470C22×C4—2441no—
471Q16—1222no—
472Q16—4881no—
473Q16—2442no—
474Q16—2442no—
475Q16—2442no—
476SD16—42no—
477SD16—4881no—
478D8—22no—
479D8—2442no—
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—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—611no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
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—41no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C42—21no—
514C42—41no—
515C42—1221no—
516C42C23×D6111yescharacteristic
517C23—21no—
518C23—21no—
519C23—2441no—
520Q8—21no—
521Q8—21no—
522Q8—41no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—82no—
527Q8—612no—
528Q8—1221no—
529Q8—4882no—
530Q8—2441no—
531Q8—2441no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—9761no—
536D4—4881no—
537D4—4882no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541D4—2441no—
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—41no—
553C2×C4—82no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—4882no—
557C2×C4—4882no—
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—4881no—
578C22—4881no—
579C22—4881no—
580C22D4×C22⋊D6111yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—1222no—
589C4—4881no—
590C4—2441no—
591C4C2×#1451979711yescharacteristic
592C4AGL(1,ℤ/8)×D6111yescharacteristic
593C4#1684056611yescharacteristic
594C4#1684057711yescharacteristic
595C2—42no—
596C2—81no—
597C2—9761no—
598C2#2145729311yescharacteristic
599C2#2145755611yescharacteristic
600C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3367C2
socleC22×C61D42
prime radical#21458452C2
hypercenter#3367D61
hypocenterC61#19835
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3367

Complements

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

NH ≅ G/Ntype
2. #21457752C2semidirect
3. D4×#11758202C2semidirect
4. #21445476C2semidirect
5. #21445387C2semidirect
6. #21445258C2semidirect
7. #21445246C2semidirect
8. #21445182C2semidirect
9. #21445089C2semidirect
10. #21445080C2semidirect
11. #21445011C2semidirect
12. #21444922C2semidirect
13. #21444667C2semidirect
14. #21444643C2semidirect
15. C61×#3367C2semidirect
295. #3367D61semidirect
367. C61#19835semidirect

Maximal subgroups

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

subgroupindex
#214584522
#214577522
D4×#117582022
#214454762
#214453872
#214452582
#214452462
#214451822
#214450892
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