Finite Group Databasebeta

← #29670451

Subgroups — #29670451

Order 15616.

614 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#21458490C211yesmaximal, characteristic, complemented
2#21458113C211yesmaximal, characteristic, complemented
3#21457771C211yesmaximal, characteristic
4#21445477C211yesmaximal, characteristic, complemented
5#21445354C211yesmaximal, characteristic, complemented
6#21445275C211yesmaximal, characteristic, complemented
7#21445235C211yesmaximal, characteristic, complemented
8#21445191C211yesmaximal, characteristic, complemented
9#21445018C211yesmaximal, characteristic, complemented
10#21444963C211yesmaximal, characteristic, complemented
11#21444959C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15Q8×SD16×C61C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17#16840434C2211yescharacteristic
18#16840141C2211yescharacteristic
19#16840112C2211yescharacteristic
20#16840110—21no—
21#16840110C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840100—21no—
25#16840088—21no—
26#16840057C2211yescharacteristic
27#16840057C2211yescharacteristic
28#16840049—21no—
29#16840049C2211yescharacteristic
30#16840010—21no—
31#16839997—21no—
32#16839964C2211yescharacteristic
33#16839962—21no—
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839906—21no—
37#16839901—21no—
38Q8×#92418C2211yescharacteristic
39C4×#11758201—21no—
40C4×#11758201C2211yescharacteristic
41#16839840C2211yescharacteristic
42C8×D244C2211yescharacteristic
43#16839771—21no—
44#16839757—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#498—21no—
59C61×#498C2211yescharacteristic
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#475—21no—
63C61×#475C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758226—21no—
68C2×#11758202—41no—
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×#11758201D411yescharacteristic
72C2×#11758199—21no—
73C2×#11758194—21no—
74C2×C8×D61—21no—
75#14519777C2311yescharacteristic
76#14519774—21no—
77#14519774C2311yescharacteristic
78#14519763—41no—
79#14519752—21no—
80#14519752—41no—
81#14519751—21no—
82#14519748—21no—
83C4×D244—21no—
84C4×D244C2311yescharacteristic
85#14519736—21no—
86#14519735—21no—
87#14519710—21no—
88#14519710—21no—
89#14519708—21no—
90#14519708D411yescharacteristic
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519701—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701D411yescharacteristic
101#14519699—21no—
102#14519699—21no—
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—41no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C2×SD16×C61D411yescharacteristic
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—41no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131#106×C61—21no—
132#106×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C2×C22⋊D61—41no—
156#11758226—42no—
157#11758226—81no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—41no—
162C2×D244—41no—
163C2×D244C2×D411yescharacteristic
164C2×C4×D61—21no—
165C2×C4×D61—41no—
166C2×Dic122—21no—
167SD16×C61—22no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—41no—
188C42×C61C2411yescharacteristic
189C22⋊Dic61—41no—
190#11758202—82no—
191#11758201—22no—
192#11758201—42no—
193#11758201—81no—
194#11758199—81no—
195#11758198—21no—
196#11758198—41no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C2×D411yescharacteristic
201C4×Dic61—41no—
202#11758194—81no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C8×D61—81no—
210C23×C61—21no—
211C22×D61—21no—
212C22×D61—41no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—82no—
218Q8×C61—81no—
219Q8×C61—161no—
220Q8×C61#13512yes—
221Q8×C61#13712yes—
222D4×C61—21no—
223D4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—82no—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C22⋊D61—161no—
238C2×Dic61—21no—
239C2×Dic61—41no—
240D244—21no—
241D244—21no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—82no—
247D244—81no—
248D244—161no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—82no—
252Dic122—81no—
253C8×C61—21no—
254C8×C61—21no—
255C8×C61—41no—
256#92418—21no—
257#92418—21no—
258#92418—41no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#19828—611nomaximal, Sylow
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61D4211yescharacteristic
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C4×C61—22no—
274C4×C61—22no—
275C4×C61—21no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61C2×#13511yescharacteristic
282C4×C61C2×#13711yescharacteristic
283C4×C61#57711yescharacteristic
284C4×C61#57711yescharacteristic
285Dic61—41no—
286Dic61—81no—
287Q8×SD16D6111yescharacteristic, complemented
288#3362—611no—
289#3329—611no—
290#3305—611no—
291#1720—611no—
292#1697—611no—
293#1684—611no—
294#1678—611no—
295#1671—611no—
296#1642—611no—
297#1634—611no—
298#1633—611no—
299#1627—611no—
300#1583—611no—
301SD16⋊C8—611no—
302C2×C61—81no—
303C2×C61#326611yescharacteristic
304C2×C61#328211yescharacteristic
305C2×C61#328811yescharacteristic
306D61—81no—
307D61—161no—
308Q82C2×D6111yescharacteristic
309#555—611no—
310#550—611no—
311D4×Q8C2×D6111yescharacteristic
312#498—21no—
313#498C2×D6111yescharacteristic
314#494—611no—
315#487—611no—
316#486—611no—
317#486—611no—
318#486—611no—
319#486—1221no—
320#485—1221no—
321#482—1221no—
322#477—611no—
323#477—611no—
324#476—21no—
325#476—611no—
326#476—1221no—
327#476C2×D6111yescharacteristic
328#475—21no—
329#475C2×D6111yescharacteristic
330#468—1221no—
331#466—1221no—
332#461—611no—
333#460—611no—
334#460—1221no—
335#459—611no—
336#449—1221no—
337#448—1221no—
338C8×Q8—611no—
339C8×Q8C2×D6111yescharacteristic
340C4×SD16—21no—
341C4×SD16—611no—
342C4×SD16—1221no—
343C4×SD16C2×D6111yescharacteristic
344#436—611no—
345C8×D4—611no—
346#419—1221no—
347#416—1221no—
348#335—611no—
349#334—611no—
350#333—611no—
351#333—611no—
352#332—611no—
353#331—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—611no—
358#322—611no—
359C61#1982811yescharacteristic, complemented, Sylow
360C2×#41—1221no—
361C22×Q8—21no—
362C2×Q16—2441no—
363C2×SD16—611no—
364C2×SD16—1221no—
365C2×SD16—1221no—
366C2×SD16C22⋊D6111yescharacteristic
367C2×D8—1221no—
368C2×M16—1221no—
369C22×C8—1221no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—41no—
374C4⋊Q8—611no—
375C4⋊Q8C22×D6111yescharacteristic
376C4⋊Q8C22×D6111yescharacteristic
377#127—611no—
378#127—1221no—
379#124—2441no—
380C22⋊Q8—21no—
381C22⋊Q8—41no—
382#121—1221no—
383#121—1221no—
384#121—1221no—
385#121—2441no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8C22×D6111yescharacteristic
389C4×Q8C22×D6111yescharacteristic
390C4×D4—21no—
391C4×D4—611no—
392C4×D4—1221no—
393C4×D4C22×D6111yescharacteristic
394#117—1221no—
395#117—1221no—
396#107—1221no—
397#107—1221no—
398#106—21no—
399#106—21no—
400#106—21no—
401#106—21no—
402#106—611no—
403#106—1221no—
404#106C22⋊D6111yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—1221no—
410C4⋊C8C22×D6111yescharacteristic
411#103—21no—
412#103—611no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103C22⋊D6111yescharacteristic
419#102—21no—
420#102—611no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—2441no—
429#102C22⋊D6111yescharacteristic
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C4×C8—21no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8—1221no—
436C4×C8C22×D6111yescharacteristic
437#41—4881no—
438#41—2442no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—41no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×Q8—81no—
445C2×Q8—1221no—
446C2×Q8D4×D6111yescharacteristic
447C2×Q8C2×C22⋊D6111yescharacteristic
448C2×D4—611no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—2441no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4C2×C22⋊D6111yescharacteristic
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—1221no—
459C22×C4—2441no—
460Q16—4882no—
461SD16—22no—
462SD16—1222no—
463SD16—4881no—
464SD16—2442no—
465D8—4881no—
466M16—4881no—
467C2×C8—21no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—4881no—
475C2×C8C2×C22⋊D6111yescharacteristic
476C2×C8C2×C22⋊D6111yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—611no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4D4×D6111yescharacteristic
490C4⋊C4C2×C22⋊D6111yescharacteristic
491C4⋊C4C2×C22⋊D6111yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—41no—
494C22⋊C4—1221no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C42—21no—
499C42—21no—
500C42—41no—
501C42—2441no—
502C42C23×D6111yescharacteristic
503C23—21no—
504C23—1221no—
505C23—2441no—
506Q8—21no—
507Q8—21no—
508Q8—41no—
509Q8—41no—
510Q8—82no—
511Q8—81no—
512Q8—161no—
513Q8—4881no—
514Q8#1451979812yes—
515Q8#1451980812yes—
516D4—21no—
517D4—21no—
518D4—1221no—
519D4—1221no—
520D4—9761no—
521D4—9761no—
522D4—4881no—
523D4—4882no—
524D4—2441no—
525D4—2441no—
526D4—2441no—
527D4—2441no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—82no—
538C2×C4—1221no—
539C2×C4—4882no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C22×C22⋊D6111yescharacteristic
545C2×C4#1451982211yescharacteristic
546C8—21no—
547C8—21no—
548C8—41no—
549C8—1221no—
550C8—1221no—
551C8—2441no—
552C8—2441no—
553C8—2441no—
554C8—2441no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—2441no—
562C22—2441no—
563C22—2441no—
564C22D4×C22⋊D6111yescharacteristic
565C4—22no—
566C4—21no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—4881no—
574C4—2441no—
575C4C2×#1451979811yescharacteristic
576C4C2×#1451980811yescharacteristic
577C4#1684056811yescharacteristic
578C4#1684057011yescharacteristic
579C2—81no—
580C2—9761no—
581C2—4881no—
582C2#2145733011yescharacteristic
583C2#2145752611yescharacteristic
584C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×SD16×C61C2
socleC22×C61D42
prime radical#21457771C2
hypercenterQ8×SD16D61
hypocenterC61#19828
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. #21458490C2semidirect
2. #21458113C2semidirect
4. #21445477C2semidirect
5. #21445354C2semidirect
6. #21445275C2semidirect
7. #21445235C2semidirect
8. #21445191C2semidirect
9. #21445018C2semidirect
10. #21444963C2semidirect
11. #21444959C2semidirect
12. #21444922C2semidirect
13. #21444663C2semidirect
14. #21444629C2semidirect
15. Q8×SD16×C61C2semidirect
287. Q8×SD16D61semidirect
359. C61#19828semidirect

Maximal subgroups

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

subgroupindex
#214584902
#214581132
#214577712
#214454772
#214453542
#214452752
#214452352
#214451912
#214450182
#214449632
#214449592
#214449222
#214446632
#214446292
Q8×SD16×C612
#1982861