Finite Group Databasebeta

← #29663767

Subgroups — #29663767

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456791C211yesmaximal, characteristic
2#21456538C211yesmaximal, characteristic, complemented
3#21454824C211yesmaximal, characteristic, complemented
4#21445389C211yesmaximal, characteristic, complemented
5#21445371C211yesmaximal, characteristic, complemented
6#21445330C211yesmaximal, characteristic, complemented
7#21445286C211yesmaximal, characteristic, complemented
8#21445258C211yesmaximal, characteristic, complemented
9#21445240C211yesmaximal, characteristic, complemented
10#21445211C211yesmaximal, characteristic, complemented
11#21445169C211yesmaximal, characteristic, complemented
12#21444569C211yesmaximal, characteristic, complemented
13#21444484C211yesmaximal, characteristic, complemented
14#21444469C211yesmaximal, characteristic, complemented
15C61×#3248C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840144—21no—
19#16840136C2211yescharacteristic
20#16840119—21no—
21#16840112C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840010—21no—
24#16840010—21no—
25#16839972C2211yescharacteristic
26#16839967C2211yescharacteristic
27#16839962C2211yescharacteristic
28#16839957C2211yescharacteristic
29#16839922—21no—
30#16839905—21no—
31#16839893C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33#16839860—21no—
34#16839860—21no—
35#16839840—21no—
36#16839840C2211yescharacteristic
37#16839834—21no—
38D4×#92418C2211yescharacteristic
39#16839823—21no—
40#16839815C2211yescharacteristic
41#16839815C2211yescharacteristic
42#16839718—21no—
43C4×#11758194C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#479—21no—
59C61×#475C2211yescharacteristic
60C61×#474C2211yescharacteristic
61C61×#467—21no—
62C61×#465—21no—
63C61×#464—21no—
64C61×#462C2211yescharacteristic
65C61×#461C2211yescharacteristic
66C2×C4⋊C8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758201—21no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758199—21no—
74C2×#11758194—21no—
75C2×#11758194D411yescharacteristic
76C2×#11758194D411yescharacteristic
77C22×#92418—21no—
78#14519777C2311yescharacteristic
79#14519774C2311yescharacteristic
80#14519765—21no—
81#14519751—21no—
82#14519751—21no—
83C4×D244—21no—
84C4×D244—21no—
85#14519710—21no—
86#14519710—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93#14519701—21no—
94#14519701—21no—
95#14519700—21no—
96#14519700—21no—
97#14519697—21no—
98#14519697—21no—
99#14519697—21no—
100#14519697—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519688—21no—
106#14519688D411yescharacteristic
107#14519688D411yescharacteristic
108C4×#92418—21no—
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×Q16×C61—21no—
114C2×SD16×C61—21no—
115C22×C8×C61—21no—
116C4⋊Q8×C61C2311yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118#126×C61—21no—
119#125×C61—21no—
120#125×C61D411yescharacteristic
121#124×C61D411yescharacteristic
122#123×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61D411yescharacteristic
126#121×C61D411yescharacteristic
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#107×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61—21no—
142#103×C61—21no—
143#102×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61#4111yescharacteristic
153C2×C22⋊D61—41no—
154#11758222—81no—
155#11758222—81no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×C4×D61—41no—
160C2×Dic122—21no—
161Q16×C61—81no—
162SD16×C61—81no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—81no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61#4111yescharacteristic
189C42×C61C2411yescharacteristic
190#11758202—81no—
191#11758201—42no—
192#11758201—42no—
193#11758201—81no—
194#11758200—81no—
195#11758199—81no—
196#11758198—41no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200#11758194—22no—
201#11758194—41no—
202#11758194—42no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—42no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C23×C61—21no—
213C23×C61#12111yescharacteristic
214C22×D61—41no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—81no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C22⋊D61—82no—
243C2×Dic61—41no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—41no—
248D244—81no—
249C4×D61—82no—
250Dic122—41no—
251Dic122—41no—
252C8×C61—41no—
253C8×C61—41no—
254#92418—21no—
255#92418—21no—
256#92418—21no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#92418—41no—
261#92418#13512yes—
262#19252—611nomaximal, Sylow
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61C2×#12111yescharacteristic
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61C2×#13511yescharacteristic
283C4×C61#57511yescharacteristic
284C4×C61#57611yescharacteristic
285C4×C61#57711yescharacteristic
286Dic61—81no—
287#3248D6111yescharacteristic, complemented
288#3212—611no—
289#3182—611no—
290#2969—611no—
291#1702—611no—
292#1700—611no—
293#1693—611no—
294#1685—611no—
295#1681—611no—
296#1679—611no—
297#1674—611no—
298#1666—611no—
299#1565—611no—
300#1544—611no—
301#1540—611no—
302C2×C61—41no—
303C2×C61—81no—
304C2×C61#303511yescharacteristic
305C2×C61#306211yescharacteristic
306C2×C61#306911yescharacteristic
307D61—161no—
308#533C2×D6111yescharacteristic
309#532—611no—
310#529C2×D6111yescharacteristic
311#500—611no—
312#495—1221no—
313#492—611no—
314#488—1221no—
315#487—611no—
316#486—611no—
317#479—21no—
318#475C2×D6111yescharacteristic
319#474C2×D6111yescharacteristic
320#468—1221no—
321#468—1221no—
322#467—21no—
323#465—21no—
324#464—21no—
325#462—611no—
326#462C2×D6111yescharacteristic
327#461—611no—
328#461C2×D6111yescharacteristic
329#460—611no—
330#459—611no—
331#452—1221no—
332#449—1221no—
333#446—611no—
334C8×Q8—611no—
335#440—1221no—
336#440—1221no—
337#436—611no—
338#436—1221no—
339#435—1221no—
340C8×D4—611no—
341#433—1221no—
342#431—611no—
343#431—611no—
344C2×C4⋊C8C2×D6111yescharacteristic
345#405—1221no—
346C4×M16—611no—
347#335—611no—
348#335—611no—
349#334—611no—
350#334—611no—
351#332—611no—
352#332—611no—
353#332—611no—
354#332—611no—
355#332—611no—
356#332—611no—
357#331—611no—
358#331—611no—
359C61#1925211yescharacteristic, complemented, Sylow
360C2×#41—1221no—
361C2×Q16—21no—
362C2×Q16—1221no—
363C2×SD16—21no—
364C2×SD16—1221no—
365C2×SD16—1221no—
366C2×SD16—1221no—
367C2×SD16—1221no—
368C2×D8—1221no—
369C2×M16—611no—
370C2×M16—611no—
371C2×M16—1221no—
372C22×C8—21no—
373C22×C8—1221no—
374C4⋊Q8—611no—
375C4⋊Q8C22×D6111yescharacteristic
376C4⋊Q8C22×D6111yescharacteristic
377#127—611no—
378#126—21no—
379#125—21no—
380#125C22⋊D6111yescharacteristic
381#124—1221no—
382#124C22⋊D6111yescharacteristic
383#123—21no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8C22⋊D6111yescharacteristic
387#121—1221no—
388#121—1221no—
389#121C22⋊D6111yescharacteristic
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8C22×D6111yescharacteristic
393C4×D4—21no—
394C4×D4—1221no—
395C4×D4—1221no—
396C4×D4C22×D6111yescharacteristic
397#117—21no—
398#117—21no—
399C2×C42C22×D6111yescharacteristic
400#107—21no—
401#107—1221no—
402#107—1221no—
403#106—21no—
404C4⋊C8—21no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—1221no—
410C4⋊C8—1221no—
411C4⋊C8C22×D6111yescharacteristic
412C4⋊C8C22×D6111yescharacteristic
413#103—21no—
414#103—21no—
415#103—21no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#102—21no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424#102—1221no—
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427C22⋊C8—1221no—
428C22⋊C8—1221no—
429#97—611no—
430#97—611no—
431#97—1221no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8—1221no—
435C4×C8—1221no—
436C4×C8—1221no—
437#41—4881no—
438#41—4881no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—1221no—
442C2×Q8C2×C22⋊D6111yescharacteristic
443C2×D4—21no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—2441no—
448C2×D4C2×C22⋊D6111yescharacteristic
449C22×C4—21no—
450C22×C4—2441no—
451C22×C4D4×D6111yescharacteristic
452C22×C4#1175822411yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454Q16—81no—
455Q16—4881no—
456SD16—81no—
457SD16—4881no—
458SD16—4881no—
459SD16—2442no—
460SD16—2442no—
461D8—4881no—
462M16—1222no—
463M16—2442no—
464M16—2441no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—81no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—611no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—2442no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4C2×C22⋊D6111yescharacteristic
490C4⋊C4C2×C22⋊D6111yescharacteristic
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—41no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C42—21no—
500C42—41no—
501C42D4×D6111yescharacteristic
502C42#1175822411yescharacteristic
503C42C2×C22⋊D6111yescharacteristic
504C42C23×D6111yescharacteristic
505C23—21no—
506C23—2441no—
507C23#1451974911yescharacteristic
508Q8—21no—
509Q8—21no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—2441no—
515Q8—2441no—
516D4—21no—
517D4—21no—
518D4—81no—
519D4—4881no—
520D4—4882no—
521D4—2441no—
522D4—2441no—
523D4—2441no—
524D4—2441no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—4882no—
538C2×C4—2441no—
539C2×C4#1451974911yescharacteristic
540C2×C4#1451974911yescharacteristic
541C2×C4#1451974911yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4C2×#1175822411yescharacteristic
545C8—41no—
546C8—41no—
547C8—612no—
548C8—1221no—
549C8—1221no—
550C8—1221no—
551C8—2441no—
552C8—2441no—
553C8—2441no—
554C8—2441no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—4881no—
562C22—4881no—
563C22—4881no—
564C22C2×#1451974911yescharacteristic
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—4881no—
575C4#135×D6111yescharacteristic
576C4#1684055911yescharacteristic
577C4#1684056511yescharacteristic
578C4#1684057011yescharacteristic
579C2—41no—
580C2—81no—
581C2—9761no—
582C2#2145527011yescharacteristic
583C2#2145550111yescharacteristic
584C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3248C2
socleC22×C61C2×#121
prime radical#21456791C2
hypercenter#3248D61
hypocenterC61#19252
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3248

Complements

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

NH ≅ G/Ntype
2. #21456538C2semidirect
3. #21454824C2semidirect
4. #21445389C2semidirect
5. #21445371C2semidirect
6. #21445330C2semidirect
7. #21445286C2semidirect
8. #21445258C2semidirect
9. #21445240C2semidirect
10. #21445211C2semidirect
11. #21445169C2semidirect
12. #21444569C2semidirect
13. #21444484C2semidirect
14. #21444469C2semidirect
15. C61×#3248C2semidirect
287. #3248D61semidirect
359. C61#19252semidirect

Maximal subgroups

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

subgroupindex
#214567912
#214565382
#214548242
#214453892
#214453712
#214453302
#214452862
#214452582
#214452402
#214452112
#214451692
#214445692
#214444842
#214444692
C61×#32482
#1925261