Finite Group Databasebeta

← #29669721

Subgroups — #29669721

Order 15616.

676 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458452C211yesmaximal, characteristic, complemented
2#21458140C211yesmaximal, characteristic, complemented
3#21457678C211yesmaximal, characteristic, complemented
4#21445482C211yesmaximal, characteristic, complemented
5#21445368C211yesmaximal, characteristic, complemented
6#21445258C211yesmaximal, characteristic, complemented
7#21445256C211yesmaximal, characteristic, complemented
8#21445186C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic, complemented
10#21445080C211yesmaximal, characteristic, complemented
11#21445017C211yesmaximal, characteristic, complemented
12#21444908C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15C61×#3283C211yesmaximal, characteristic, complemented
16#16840433C2211yescharacteristic
17D4×Dic122C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840114C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840110C2211yescharacteristic
22#16840109C2211yescharacteristic
23#16840107—21no—
24#16840092—21no—
25#16840080—21no—
26#16840044C2211yescharacteristic
27#16840044C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840041—21no—
30#16840013—21no—
31#16839989—21no—
32#16839967C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839955—21no—
36#16839905—21no—
37#16839900—21no—
38C8×Dic122C2211yescharacteristic
39C4×#11758200C2211yescharacteristic
40#16839842—21no—
41#16839840C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839761—21no—
44C2×#14519700—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839505C2211yescharacteristic
51#16839492C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C2×#135×C61—21no—
57C61×#547C2211yescharacteristic
58C61×#547C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#467—21no—
61C61×#461C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#452—21no—
64C61×#452—21no—
65C4×D8×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758222—21no—
68C22×Dic122—21no—
69C2×#11758200—21no—
70C2×#11758200D411yescharacteristic
71C2×#11758199—21no—
72C2×#11758199—21no—
73C2×#11758194—21no—
74C22×#92418—21no—
75#14519777C2311yescharacteristic
76#14519776—21no—
77#14519774C2311yescharacteristic
78#14519773—21no—
79#14519756—21no—
80#14519754—41no—
81#14519751—21no—
82#14519750—41no—
83C4×Dic122C2311yescharacteristic
84D4×Dic61—21no—
85#14519739—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—41no—
90#14519708D411yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519706—21no—
95#14519700—21no—
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519700D411yescharacteristic
101#14519699—21no—
102#14519699—21no—
103#14519698—21no—
104#14519698D411yescharacteristic
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×#41×C61—21no—
116C2×#41×C61—21no—
117#135×C61—44no—
118C2×Q16×C61—21no—
119C2×Q16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×D8×C61D411yescharacteristic
123C22×C8×C61—21no—
124C4⋊Q8×C61C2311yescharacteristic
125C4⋊Q8×C61C2311yescharacteristic
126#123×C61—41no—
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130#121×C61—21no—
131#121×C61—21no—
132C4×D4×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135C2×C4⋊C4×C61—21no—
136C2×C4⋊C4×C61—21no—
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—42no—
147#41×C61—81no—
148#41×C61—81no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61C2×D411yescharacteristic
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—41no—
160C22×C4×C61—41no—
161C2×C22⋊D61—41no—
162C2×C22⋊D61—41no—
163C22×Dic61—41no—
164#11758222—42no—
165#11758222—81no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×C4×D61—41no—
170C2×Dic122—21no—
171C2×Dic122—42no—
172C2×Dic122—41no—
173C2×Dic122—81no—
174C2×Dic122C2×D411yescharacteristic
175Q16×C61—42no—
176Q16×C61—81no—
177SD16×C61—42no—
178SD16×C61—42no—
179D8×C61—22no—
180C2×C8×C61—21no—
181C2×C8×C61—42no—
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61C2×D411yescharacteristic
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—41no—
189C4⋊C4×C61—41no—
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—41no—
197C22⋊C4×C61—41no—
198C42×C61C2411yescharacteristic
199C22⋊Dic61—41no—
200#11758200—22no—
201#11758200—81no—
202#11758199—42no—
203#11758199—81no—
204#11758198—41no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C2×D411yescharacteristic
208#11758196—21no—
209#11758196—41no—
210C4×Dic61—21no—
211#11758194—81no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—81no—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C22×D61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—82no—
233D4×C61—82no—
234D4×C61AGL(1,ℤ/8)12yes—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—82no—
247C2×C4×C61—82no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C22⋊D61—82no—
252C22⋊D61—161no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256C2×Dic61—82no—
257D244—41no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—81no—
262C4×D61—82no—
263Dic122—21no—
264Dic122—41no—
265Dic122—41no—
266Dic122—82no—
267Dic122#13512yescomplemented
268C8×C61—21no—
269C8×C61—21no—
270C8×C61—41no—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#19779—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61D4211yescharacteristic
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—81no—
296C4×C61—81no—
297C4×C61C2×#13511yescharacteristic
298C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C4×C61#57611yescharacteristic
300C4×C61#57711yescharacteristic
301Dic61—22no—
302Dic61—41no—
303Dic61—81no—
304#3358—611no—
305#3332—611no—
306#3298—611no—
307#3283D6111yescharacteristic, complemented
308#1720—611no—
309#1699—611no—
310#1681—611no—
311#1681—611no—
312#1670—611no—
313#1654—611no—
314#1651—611no—
315#1642—611no—
316#1625—611no—
317#1581—611no—
318SD16⋊C8—611no—
319C2×C61—42no—
320C2×C61—81no—
321C2×C61—81no—
322C2×C61#326511yescharacteristic
323C2×C61#328011yescharacteristic
324C2×C61#328911yescharacteristic
325D61—161no—
326C2×#135—21no—
327#550—611no—
328D4×Q8—611no—
329#547C2×D6111yescharacteristic
330#547C2×D6111yescharacteristic
331#498—611no—
332#494C2×D6111yescharacteristic
333#487—611no—
334#486—611no—
335#486—611no—
336#486—611no—
337#486—1221no—
338#483—1221no—
339#481—1221no—
340#475—611no—
341#475—611no—
342#474—611no—
343#474—1221no—
344#469—1221no—
345#467—21no—
346#465—1221no—
347#461—611no—
348#461C2×D6111yescharacteristic
349#461C2×D6111yescharacteristic
350#459—611no—
351#459—611no—
352#459—1221no—
353#452—21no—
354#452—21no—
355#449—1221no—
356#448—1221no—
357C8×Q8—611no—
358C4×SD16—611no—
359C4×D8—1221no—
360C4×D8C2×D6111yescharacteristic
361#436—611no—
362C8×D4—611no—
363C8×D4C2×D6111yescharacteristic
364#417—1221no—
365C2×#103—1221no—
366#335—611no—
367#335—611no—
368#332—611no—
369#332—611no—
370#332—611no—
371#331—611no—
372Q8⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375D4⋊C8—611no—
376#322—611no—
377C61#1977911yescharacteristic, complemented, Sylow
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—1221no—
381C22×Q8—1221no—
382#135—44no—
383C2×Q16—21no—
384C2×Q16—21no—
385C2×SD16—21no—
386C2×SD16—21no—
387C2×SD16—611no—
388C2×SD16—1221no—
389C2×D8—1221no—
390C2×D8—1221no—
391C2×D8C22⋊D6111yescharacteristic
392C2×M16—1221no—
393C22×C8—21no—
394C22×C8—1221no—
395C4⋊Q8—611no—
396C4⋊Q8—1221no—
397C4⋊Q8C22×D6111yescharacteristic
398C4⋊Q8C22×D6111yescharacteristic
399#127—611no—
400#127—1221no—
401#123—41no—
402#123—41no—
403C22⋊Q8—21no—
404C22⋊Q8—21no—
405C22⋊Q8—1221no—
406C22⋊Q8—2441no—
407#121—21no—
408#121—21no—
409#121—1221no—
410#121—2441no—
411C4×Q8—611no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4C22×D6111yescharacteristic
415C4×D4C22×D6111yescharacteristic
416C4×D4C22×D6111yescharacteristic
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419#107—1221no—
420#107C22⋊D6111yescharacteristic
421#106—611no—
422#106—1221no—
423#106—1221no—
424#106—2441no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—1221no—
429C4⋊C8C22×D6111yescharacteristic
430#103—21no—
431#103—21no—
432#103—611no—
433#103—1221no—
434#103—1221no—
435#103—1221no—
436#103—1221no—
437#103—1221no—
438#102—611no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—2441no—
447#102C22⋊D6111yescharacteristic
448#102C22⋊D6111yescharacteristic
449C22⋊C8—21no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8—1221no—
455C4×C8C22×D6111yescharacteristic
456#41—42no—
457#41—42no—
458#41—81no—
459#41—81no—
460#41—4881no—
461#41—2442no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—611no—
465C2×Q8—1221no—
466C2×Q8—4881no—
467C2×Q8—2441no—
468C2×Q8—2442no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—2441no—
475C2×D4—2441no—
476C2×D4D4×D6111yescharacteristic
477C2×D4C2×C22⋊D6111yescharacteristic
478C2×D4C2×C22⋊D6111yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—2441no—
485C22×C4—2441no—
486Q16—42no—
487Q16—81no—
488SD16—42no—
489SD16—42no—
490SD16—1222no—
491SD16—4881no—
492D8—22no—
493D8—4881no—
494D8—2442no—
495M16—4881no—
496C2×C8—21no—
497C2×C8—42no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—4881no—
505C2×C8C2×C22⋊D6111yescharacteristic
506C2×C8C2×C22⋊D6111yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—611no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—2441no—
518C4⋊C4D4×D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C4⋊C4C2×C22⋊D6111yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—2441no—
527C22⋊C4—2441no—
528C42—1221no—
529C42C23×D6111yescharacteristic
530C23—21no—
531C23—21no—
532C23—21no—
533C23—2441no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—612no—
539Q8—1221no—
540Q8—4882no—
541Q8—2441no—
542Q8—2441no—
543D4—21no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—82no—
549D4—82no—
550D4—9761no—
551D4—4881no—
552D4—4882no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4#1451980312yes—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—4882no—
574C2×C4—4882no—
575C2×C4—2441no—
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4C22×C22⋊D6111yescharacteristic
578C2×C4#1451982211yescharacteristic
579C8—21no—
580C8—21no—
581C8—41no—
582C8—1221no—
583C8—1221no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22D4×C22⋊D6111yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—1222no—
608C4—4881no—
609C4—2441no—
610C4#135×D6111yescharacteristic
611C4C2×#1451980311yescharacteristic
612C4#1684056411yescharacteristic
613C4#1684057311yescharacteristic
614C2—42no—
615C2—81no—
616C2—81no—
617C2—9761no—
618C2#2145731011yescharacteristic
619C2#2145749811yescharacteristic
620C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3283C2
socleC22×C61D42
hypercenter#3283D61
hypocenterC61#19779
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. #21458452C2semidirect
2. #21458140C2semidirect
3. #21457678C2semidirect
4. #21445482C2semidirect
5. #21445368C2semidirect
6. #21445258C2semidirect
7. #21445256C2semidirect
8. #21445186C2semidirect
9. #21445103C2semidirect
10. #21445080C2semidirect
11. #21445017C2semidirect
12. #21444908C2semidirect
13. #21444649C2semidirect
14. #21444631C2semidirect
15. C61×#3283C2semidirect
267. Dic122#135semidirect
307. #3283D61semidirect
377. C61#19779semidirect

Maximal subgroups

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

subgroupindex
#214584522
#214581402
#214576782
#214454822
#214453682
#214452582
#214452562
#214451862
#214451032
#214450802
#214450172
#214449082
#214446492
#214446312
C61×#32832
#1977961