Finite Group Databasebeta

← #29669746

Subgroups — #29669746

Order 15616.

616 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#21458684C211yesmaximal, characteristic, complemented
2#21458157C211yesmaximal, characteristic, complemented
3#21457695C211yesmaximal, characteristic, complemented
4#21445471C211yesmaximal, characteristic, complemented
5#21445392C211yesmaximal, characteristic, complemented
6#21445260C211yesmaximal, characteristic, complemented
7#21445226C211yesmaximal, characteristic, complemented
8#21445223C211yesmaximal, characteristic, complemented
9#21445032C211yesmaximal, characteristic, complemented
10#21444984C211yesmaximal, characteristic, complemented
11#21444946C211yesmaximal, characteristic, complemented
12#21444889C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444636C211yesmaximal, characteristic, complemented
15C61×#3358C211yesmaximal, characteristic
16#16840469C2211yescharacteristic
17#16840469C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840163—21no—
20#16840134C2211yescharacteristic
21#16840114C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840095—21no—
25#16840073—21no—
26#16840060—21no—
27#16840060—21no—
28#16840057C2211yescharacteristic
29#16840057C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16840006—21no—
33#16840001—21no—
34#16839969C2211yescharacteristic
35#16839959C2211yescharacteristic
36#16839924—21no—
37#16839918—21no—
38C8×Dic122C2211yescharacteristic
39C4×#11758202C2211yescharacteristic
40D8×Dic61—21no—
41#16839840C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839763—21no—
44C2×#14519701—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
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
67C2×#11758202—21no—
68C2×#11758202D411yescharacteristic
69C2×#11758200—21no—
70C2×#11758200—41no—
71C2×#11758194—21no—
72C22×#92418—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519768—21no—
76#14519768—21no—
77#14519756—21no—
78#14519756—21no—
79#14519754—41no—
80#14519754—41no—
81C4×Dic122C2311yescharacteristic
82D4×Dic61—21no—
83D4×Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86#14519711—21no—
87#14519711—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701D411yescharacteristic
103#14519700—41no—
104#14519700D411yescharacteristic
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698—21no—
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×#41×C61—21no—
116C22×Q8×C61—21no—
117C2×D8×C61—21no—
118C2×D8×C61D411yescharacteristic
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61—21no—
122#127×C61C2311yescharacteristic
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61D411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#102×C61—21no—
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144#41×C61—42no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C22×Dic61—41no—
163C22×Dic61—41no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122C2×D411yescharacteristic
167D8×C61—22no—
168D8×C61—42no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61C2411yescharacteristic
185C22⋊Dic61—41no—
186C22⋊Dic61—41no—
187#11758202—22no—
188#11758202—81no—
189#11758200—81no—
190#11758200—82no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—41no—
196C4⋊Dic61—41no—
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—21no—
199#11758196—41no—
200#11758196—41no—
201C4×Dic61—21no—
202#11758194—81no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—81no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—41no—
213Q8×C61—21no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—82no—
217Q8×C61#13512yes—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—82no—
226D4×C61—81no—
227D4×C61—161no—
228D4×C61#13712yes—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—82no—
239C2×C4×C61—82no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—161no—
248C2×Dic61—161no—
249Dic122—21no—
250Dic122—21no—
251Dic122—81no—
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#19781—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61D4211yescharacteristic
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61C2×#13511yescharacteristic
280C4×C61C2×#13711yescharacteristic
281C4×C61#57611yescharacteristic
282C4×C61#57811yescharacteristic
283Dic61—41no—
284Dic61—41no—
285Dic61—81no—
286Dic61—81no—
287#3380—611no—
288#3358D6111yescharacteristic
289#3333—611no—
290#3299—611no—
291#1719—611no—
292#1703—611no—
293#1682—611no—
294#1677—611no—
295#1676—611no—
296#1644—611no—
297#1637—611no—
298#1631—611no—
299#1622—611no—
300#1581—611no—
301#1579—611no—
302C2×C61—42no—
303C2×C61—81no—
304C2×C61—161no—
305C2×C61#327011yescharacteristic
306C2×C61#328011yescharacteristic
307C2×C61#328811yescharacteristic
308#554—611no—
309#554—611no—
310#550C2×D6111yescharacteristic
311D4×Q8C2×D6111yescharacteristic
312#500—611no—
313#499—1221no—
314#492—611no—
315#487—611no—
316#487—611no—
317#486—21no—
318#486—611no—
319#486C2×D6111yescharacteristic
320#484—1221no—
321#480—1221no—
322#478—1221no—
323#478—1221no—
324#477—611no—
325#477—611no—
326#476—611no—
327#476—611no—
328#474—21no—
329#474C2×D6111yescharacteristic
330#468—1221no—
331#467—1221no—
332#462—611no—
333#460—611no—
334#459—21no—
335#459C2×D6111yescharacteristic
336#452—1221no—
337#451—1221no—
338C8×Q8—611no—
339C8×Q8C2×D6111yescharacteristic
340C4×Q16—611no—
341C4×D8—21no—
342C4×D8—1221no—
343C4×D8C2×D6111yescharacteristic
344#436—611no—
345C8×D4—611no—
346#417—1221no—
347C2×#103—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#1978111yescharacteristic, complemented, Sylow
360C2×#41—21no—
361C22×Q8—21no—
362C2×Q16—611no—
363C2×Q16—1221no—
364C2×SD16—1221no—
365C2×SD16—2441no—
366C2×D8—21no—
367C2×D8C22⋊D6111yescharacteristic
368C2×M16—1221no—
369C22×C8—1221no—
370C4⋊Q8—21no—
371C4⋊Q8—611no—
372C4⋊Q8—611no—
373C4⋊Q8C22×D6111yescharacteristic
374#127—21no—
375#127C22×D6111yescharacteristic
376#125—1221no—
377#125—1221no—
378C22⋊Q8—21no—
379C22⋊Q8—41no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—2441no—
383C22⋊Q8—2441no—
384#121—21no—
385#121—41no—
386C4×Q8—611no—
387C4×Q8C22×D6111yescharacteristic
388C4×D4—21no—
389C4×D4—21no—
390C4×D4—1221no—
391C4×D4—1221no—
392C4×D4C22×D6111yescharacteristic
393C4×D4C22×D6111yescharacteristic
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396#107—21no—
397#107—611no—
398#107—1221no—
399#107—1221no—
400#107—1221no—
401#107—1221no—
402#107C22⋊D6111yescharacteristic
403#106—1221no—
404#106—1221no—
405C4⋊C8—21no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—1221no—
410C4⋊C8C22×D6111yescharacteristic
411#103—611no—
412#103—611no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—2441no—
419#102—21no—
420#102—21no—
421#102—21no—
422#102—21no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102C22⋊D6111yescharacteristic
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—42no—
438#41—81no—
439C2×Q8—21no—
440C2×Q8—41no—
441C2×Q8—42no—
442C2×Q8—81no—
443C2×Q8—611no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8D4×D6111yescharacteristic
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4C2×C22⋊D6111yescharacteristic
453C2×D4C2×C22⋊D6111yescharacteristic
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—2441no—
459C22×C4—2441no—
460Q16—1222no—
461Q16—4881no—
462SD16—4881no—
463SD16—4882no—
464D8—22no—
465D8—42no—
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—41no—
481C4⋊C4—611no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—2441no—
488C4⋊C4—2441no—
489C4⋊C4—2441no—
490C4⋊C4—2441no—
491C4⋊C4D4×D6111yescharacteristic
492C4⋊C4C2×C22⋊D6111yescharacteristic
493C4⋊C4C2×C22⋊D6111yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C42—21no—
501C42—1221no—
502C42C23×D6111yescharacteristic
503C23—21no—
504C23—21no—
505C23—41no—
506Q8—21no—
507Q8—41no—
508Q8—41no—
509Q8—82no—
510Q8—1221no—
511Q8—1221no—
512Q8—4881no—
513Q8—4881no—
514Q8#1451979612yes—
515D4—21no—
516D4—21no—
517D4—21no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—82no—
523D4—81no—
524D4—161no—
525D4#1451980712yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—9761no—
540C2×C4—9761no—
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—22no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22D4×C22⋊D6111yescharacteristic
565C4—22no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—4881no—
572C4—4881no—
573C4—2441no—
574C4—2441no—
575C4C2×#1451979611yescharacteristic
576C4C2×#1451980711yescharacteristic
577C4#1684056311yescharacteristic
578C4#1684057711yescharacteristic
579C2—42no—
580C2—81no—
581C2—161no—
582C2#2145738511yescharacteristic
583C2#2145750411yescharacteristic
584C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3358C2
socleC22×C61D42
hypercenter#3358D61
hypocenterC61#19781
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. #21458684C2semidirect
2. #21458157C2semidirect
3. #21457695C2semidirect
4. #21445471C2semidirect
5. #21445392C2semidirect
6. #21445260C2semidirect
7. #21445226C2semidirect
8. #21445223C2semidirect
9. #21445032C2semidirect
10. #21444984C2semidirect
11. #21444946C2semidirect
12. #21444889C2semidirect
13. #21444649C2semidirect
14. #21444636C2semidirect
359. C61#19781semidirect

Maximal subgroups

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

subgroupindex
#214586842
#214581572
#214576952
#214454712
#214453922
#214452602
#214452262
#214452232
#214450322
#214449842
#214449462
#214448892
#214446492
#214446362
C61×#33582
#1978161