Finite Group Databasebeta

← #29670556

Subgroups — #29670556

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#21458537C211yesmaximal, characteristic, complemented
2#21458454C211yesmaximal, characteristic
3#21457760C211yesmaximal, characteristic, complemented
4#21445482C211yesmaximal, characteristic, complemented
5#21445389C211yesmaximal, characteristic, complemented
6#21445256C211yesmaximal, characteristic, complemented
7#21445251C211yesmaximal, characteristic, complemented
8#21445181C211yesmaximal, characteristic, complemented
9#21445087C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21445010C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic, complemented
13#21444670C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15D4×Q16×C61C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17#16840434C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840124—21no—
20#16840114C2211yescharacteristic
21#16840110—21no—
22#16840110C2211yescharacteristic
23#16840110C2211yescharacteristic
24#16840109C2211yescharacteristic
25#16840089—21no—
26#16840054—21no—
27#16840047C2211yescharacteristic
28#16840039—21no—
29#16840039C2211yescharacteristic
30#16840039C2211yescharacteristic
31#16839994—21no—
32#16839977—21no—
33#16839967C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839957—21no—
36#16839957C2211yescharacteristic
37#16839923—21no—
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839862—21no—
41C4×#11758199—21no—
42C4×#11758199C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#16839757—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56C22×Q16×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58D4×Q8×C61C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#467—21no—
61C61×#462C2211yescharacteristic
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C61×#451—21no—
65C4×Q16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758226—21no—
68C2×#11758200—21no—
69C2×#11758201—21no—
70C2×#11758199—21no—
71C2×#11758199D411yescharacteristic
72C2×C8×D61—21no—
73#14519777C2311yescharacteristic
74#14519774—21no—
75#14519774C2311yescharacteristic
76#14519772—41no—
77#14519765—21no—
78#14519753—21no—
79#14519752—21no—
80#14519752—41no—
81C4×Dic122—21no—
82C4×D244—21no—
83C4×D244C2311yescharacteristic
84#14519735—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710D411yescharacteristic
88#14519708—21no—
89#14519708—41no—
90C4⋊#92418—21no—
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519699—21no—
100#14519699—21no—
101#14519698D411yescharacteristic
102#14519697—21no—
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—41no—
109#14519697D411yescharacteristic
110#14519689—21no—
111#14519688—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C22×Q8×C61—21no—
117C2×Q16×C61—21no—
118C2×Q16×C61—21no—
119C2×Q16×C61—21no—
120C2×Q16×C61—21no—
121C2×Q16×C61—44no—
122C2×Q16×C61D411yescharacteristic
123C22×C8×C61—21no—
124C4⋊Q8×C61—21no—
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127C4⋊Q8×C61C2311yescharacteristic
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131C22⋊Q8×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×C61—21no—
141#103×C61D411yescharacteristic
142#103×C61D411yescharacteristic
143C22⋊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—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61—81no—
153C2×Q8×C61C2×D411yescharacteristic
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—41no—
158C22×C4×C61—41no—
159#11758226—42no—
160#11758226—81no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—41no—
165C2×D244—41no—
166C2×D244C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170Q16×C61—22no—
171Q16×C61—42no—
172Q16×C61—42no—
173Q16×C61—42no—
174Q16×C61—81no—
175C2×C8×C61—21no—
176C2×C8×C61—42no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D411yescharacteristic
190C22⋊C4×C61—21no—
191C22⋊C4×C61—41no—
192C22⋊C4×C61—41no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61C2411yescharacteristic
196#11758201—81no—
197#11758200—42no—
198#11758199—22no—
199#11758199—42no—
200#11758198—21no—
201#11758198—41no—
202#11758198—41no—
203#11758198—41no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C2×D411yescharacteristic
207#11758196—41no—
208C4×Dic61—41no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C8×D61—81no—
216C23×C61—21no—
217C22×D61—21no—
218C22×D61—41no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—21no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—82no—
227Q8×C61—82no—
228Q8×C61AGL(1,ℤ/8)12yes—
229D4×C61—21no—
230D4×C61#13512yes—
231C2×C4×C61—21no—
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—82no—
241C2×C4×C61—82no—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—41no—
247D244—21no—
248D244—21no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—41no—
253D244—82no—
254D244—81no—
255D244—161no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—82no—
259Dic122—41no—
260Dic122—41no—
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×C61D4211yescharacteristic
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C4×C61—22no—
282C4×C61—21no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61C2×#13511yescharacteristic
290C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
291C4×C61#57611yescharacteristic
292C4×C61#57811yescharacteristic
293Dic61—41no—
294Dic61—81no—
295#3367—611no—
296#3358—611no—
297#3304—611no—
298D4×Q16D6111yescharacteristic, complemented
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#326411yescharacteristic
312C2×C61#328511yescharacteristic
313C2×C61#328911yescharacteristic
314D61—81no—
315D61—161no—
316C22×Q16—21no—
317#556—611no—
318#550—611no—
319D4×Q8C2×D6111yescharacteristic
320D4×Q8C2×D6111yescharacteristic
321#498—611no—
322#494C2×D6111yescharacteristic
323#489—1221no—
324#487—611no—
325#486—611no—
326#486—611no—
327#486—611no—
328#486—1221no—
329#482—1221no—
330#477—1221no—
331#475—611no—
332#474—611no—
333#474—611no—
334#474—1221no—
335#467—21no—
336#465—1221no—
337#463—1221no—
338#462C2×D6111yescharacteristic
339#462C2×D6111yescharacteristic
340#461—611no—
341#460—611no—
342#459—611no—
343#459—1221no—
344#452—1221no—
345#451—21no—
346#451—21no—
347#446—611no—
348C8×Q8—611no—
349#440—1221no—
350C4×Q16C2×D6111yescharacteristic
351C4×D8—611no—
352C4×D8—1221no—
353C8×D4—611no—
354C8×D4C2×D6111yescharacteristic
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—21no—
371C2×Q16—21no—
372C2×Q16—21no—
373C2×Q16—21no—
374C2×Q16—21no—
375C2×Q16—44no—
376C2×Q16C22⋊D6111yescharacteristic
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×D8—611no—
380C2×D8—1221no—
381C22×C8—21no—
382C22×C8—1221no—
383C4⋊Q8—21no—
384C4⋊Q8—21no—
385C4⋊Q8—611no—
386C4⋊Q8C22×D6111yescharacteristic
387C4⋊Q8C22×D6111yescharacteristic
388#127—611no—
389#127—1221no—
390#126—2441no—
391#124—1221no—
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394C22⋊Q8—41no—
395C22⋊Q8—41no—
396C22⋊Q8—1221no—
397#121—1221no—
398#121—2441no—
399C4×Q8—1221no—
400C4×Q8C22×D6111yescharacteristic
401C4×Q8C22×D6111yescharacteristic
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—611no—
405C4×D4—1221no—
406C4×D4C22×D6111yescharacteristic
407#117—1221no—
408#107—611no—
409#107—1221no—
410#107—1221no—
411#107C22⋊D6111yescharacteristic
412#106—1221no—
413#106—2441no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419C4⋊C8C22×D6111yescharacteristic
420#103—21no—
421#103—21no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103C22⋊D6111yescharacteristic
427#103C22⋊D6111yescharacteristic
428#102—611no—
429#102—611no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—2441no—
439C22⋊C8—21no—
440C22⋊C8—1221no—
441#97—1221no—
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—41no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×Q8—81no—
456C2×Q8—1221no—
457C2×Q8C2×C22⋊D6111yescharacteristic
458C2×Q8C2×C22⋊D6111yescharacteristic
459C2×D4—611no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—2441no—
464C2×D4—2441no—
465C2×D4D4×D6111yescharacteristic
466C22×C4—21no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—1221no—
470C22×C4—2441no—
471Q16—22no—
472Q16—42no—
473Q16—42no—
474Q16—42no—
475Q16—81no—
476SD16—4881no—
477SD16—2442no—
478D8—1222no—
479D8—2442no—
480C2×C8—21no—
481C2×C8—42no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—4881no—
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—21no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—611no—
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—41no—
508C22⋊C4—41no—
509C22⋊C4—1221no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C42—21no—
514C42—21no—
515C42—2441no—
516C42C23×D6111yescharacteristic
517C23—21no—
518C23—1221no—
519C23—2441no—
520Q8—21no—
521Q8—21no—
522Q8—21no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—82no—
528Q8—82no—
529Q8—2441no—
530Q8—2441no—
531Q8#1451980312yes—
532D4—21no—
533D4—1221no—
534D4—1221no—
535D4—9761no—
536D4—4881no—
537D4—4882no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541D4—2441no—
542D4#1451980012yes—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—1221no—
555C2×C4—4882no—
556C2×C4—2441no—
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—4881no—
575C22—4881no—
576C22—4881no—
577C22—2441no—
578C22—2441no—
579C22—2441no—
580C22D4×C22⋊D6111yescharacteristic
581C4—22no—
582C4—21no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—4881no—
590C4—2441no—
591C4C2×#1451980011yescharacteristic
592C4C2×#1451980311yescharacteristic
593C4#1684056511yescharacteristic
594C4#1684057511yescharacteristic
595C2—42no—
596C2—9761no—
597C2—4881no—
598C2#2145729511yescharacteristic
599C2#2145756011yescharacteristic
600C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×Q16×C61C2
socleC22×C61D42
prime radical#21458454C2
hypercenterD4×Q16D61
hypocenterC61#19835
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. #21458537C2semidirect
3. #21457760C2semidirect
4. #21445482C2semidirect
5. #21445389C2semidirect
6. #21445256C2semidirect
7. #21445251C2semidirect
8. #21445181C2semidirect
9. #21445087C2semidirect
10. #21445082C2semidirect
11. #21445010C2semidirect
12. #21444922C2semidirect
13. #21444670C2semidirect
14. #21444641C2semidirect
15. D4×Q16×C61C2semidirect
298. D4×Q16D61semidirect
367. C61#19835semidirect

Maximal subgroups

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

subgroupindex
#214585372
#214584542
#214577602
#214454822
#214453892
#214452562
#214452512
#214451812
#214450872
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