Finite Group Databasebeta

← #29670080

Subgroups — #29670080

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457684C211yesmaximal, characteristic, complemented
2D4×#11758201C211yesmaximal, characteristic, complemented
3#21457420C211yesmaximal, characteristic
4#21445442C211yesmaximal, characteristic, complemented
5#21445329C211yesmaximal, characteristic, complemented
6#21445240C211yesmaximal, characteristic, complemented
7#21445167C211yesmaximal, characteristic, complemented
8#21445162C211yesmaximal, characteristic, complemented
9#21445102C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21444997C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic, complemented
13#21444662C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15Q8×D8×C61C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17C22×#11758201—21no—
18#16840416C2211yescharacteristic
19D4×D244C2211yescharacteristic
20#16840158C2211yescharacteristic, complemented
21#16840136C2211yescharacteristic
22#16840135C2211yescharacteristic
23#16840112C2211yescharacteristic, complemented
24#16840110C2211yescharacteristic
25#16840090—21no—
26#16840047C2211yescharacteristic, complemented
27#16840009—21no—
28#16839993—21no—
29#16839993—21no—
30#16839967C2211yescharacteristic
31#16839964C2211yescharacteristic
32#16839964C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839914—21no—
36#16839912—21no—
37#16839908—21no—
38#16839901—21no—
39#16839901—21no—
40C4×#11758201C2211yescharacteristic
41#16839841C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43D4×#92418C2211yescharacteristic, complemented
44C2×#14519701—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56D4×Q8×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic, complemented
58C61×#500—21no—
59C61×#500C2211yescharacteristic, complemented
60C61×#474—21no—
61C61×#474—21no—
62C61×#474C2211yescharacteristic
63C61×#474C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758224—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519801—44no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78C2×#11758201—21no—
79C2×#11758201—44no—
80C2×#11758201D411yescharacteristic
81C2×#11758199—21no—
82C2×#11758189—21no—
83C22×#92418—21no—
84C22×#92418—21no—
85#14519777C2311yescharacteristic
86#14519774C2311yescharacteristic
87#14519761—41no—
88#14519757—21no—
89#14519752—21no—
90#14519751—21no—
91#14519751—21no—
92#14519747—41no—
93C4×D244C2311yescharacteristic
94C2×C4⋊Dic61—21no—
95#14519708—21no—
96#14519708—41no—
97#14519708D411yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701—21no—
106#14519701D411yescharacteristic
107#14519700—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697—41no—
111#14519697D411yescharacteristic
112C22⋊#92418—21no—
113C22⋊#92418—21no—
114#14519689—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C22×Q8×C61—21no—
118C22×Q8×C61—21no—
119C2×D8×C61D411yescharacteristic
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—21no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127C22⋊Q8×C61—41no—
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#107×C61—21no—
134#107×C61—21no—
135#107×C61—21no—
136#107×C61—21no—
137#107×C61D411yescharacteristic
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143#102×C61D411yescharacteristic
144C4×C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—41no—
151C2×Q8×C61—42no—
152C2×Q8×C61—42no—
153C2×Q8×C61—81no—
154C2×Q8×C61—81no—
155C2×Q8×C61C2×D411yescharacteristic
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C2×C22⋊D61—41no—
163C2×C22⋊D61—41no—
164C22×Dic61—41no—
165#11758224—42no—
166#11758224—81no—
167D4×D61—42no—
168D4×D61—81no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—42no—
173C2×D244—41no—
174C2×D244—81no—
175C2×D244C2×D411yescharacteristic
176C2×C4×D61—21no—
177C2×Dic122—21no—
178D8×C61SD1614yescomplemented
179C2×C8×C61—21no—
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—41no—
188C4⋊C4×C61—41no—
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—41no—
195C22⋊C4×C61—41no—
196C42×C61—21no—
197C42×C61C2411yescharacteristic
198#11758201—22no—
199#11758201—42no—
200#11758201—42no—
201#11758201—42no—
202#11758201—81no—
203#11758201—81no—
204#11758200—42no—
205#11758199—42no—
206#11758198—21no—
207#11758198—41no—
208#11758198—41no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—41no—
212C4⋊Dic61C2×D411yescharacteristic
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—42no—
218C2×#92418—81no—
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221#11758190—42no—
222#11758190—81no—
223#11758189—42no—
224C23×C61—21no—
225C23×C61—21no—
226C22×D61—21no—
227C22×D61—41no—
228C22×D61—82no—
229C22×D61—82no—
230C22×D61—161no—
231Q8×C61—21no—
232Q8×C61—41no—
233Q8×C61—41no—
234Q8×C61—41no—
235Q8×C61—41no—
236Q8×C61—82no—
237Q8×C61—82no—
238Q8×C61AGL(1,ℤ/8)12yescomplemented
239D4×C61—21no—
240D4×C61—21no—
241D4×C61C2×SD1612yes—
242D4×C61C2×SD1612yescomplemented
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—82no—
254C2×C4×C61—82no—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C22⋊D61—82no—
259C22⋊D61—82no—
260C2×Dic61—21no—
261C2×Dic61—41no—
262C2×Dic61—82no—
263D244—21no—
264D244—21no—
265D244—41no—
266D244—41no—
267D244—41no—
268D244—41no—
269D244—81no—
270D244—82no—
271C4×D61—41no—
272C4×D61—41no—
273Dic122—41no—
274Dic122—41no—
275C8×C61—21no—
276C8×C61—41no—
277C8×C61C2×SD1612yescomplemented
278#92418—21no—
279#92418—21no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#19803—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61D4211yescharacteristic
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—164no—
299C4×C61—22no—
300C4×C61—21no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—81no—
305C4×C61—81no—
306C4×C61C22×SD1611yescharacteristic
307C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C61#57611yescharacteristic
309C4×C61#57711yescharacteristic
310Dic61—41no—
311Dic61—81no—
312Q8×D8D6111yescharacteristic, complemented
313#3298—611no—
314D4×SD16—611no—
315#3272—611no—
316#1714—611no—
317#1693—611no—
318#1679—611no—
319#1666—611no—
320#1665—611no—
321#1654—611no—
322#1651—611no—
323#1639—611no—
324#1627—611no—
325#1583—611no—
326#1580—611no—
327C2×C61—42no—
328C2×C61—42no—
329C2×C61#326411yescharacteristic
330C2×C61D4×SD1611yescharacteristic
331C2×C61#328911yescharacteristic
332D61—81no—
333D61—161no—
334C2×AGL(1,ℤ/8)—1221no—
335C22×SD16—1221no—
336D4×Q8C2×D6111yescharacteristic
337D4×Q8C2×D6111yescharacteristic, complemented
338#547—611no—
339D42—611no—
340#500—21no—
341#500C2×D6111yescharacteristic, complemented
342#498—611no—
343#492—611no—
344#492—611no—
345#487—611no—
346#486—611no—
347#483—1221no—
348#475—611no—
349#474—21no—
350#474—21no—
351#474C2×D6111yescharacteristic
352#474C2×D6111yescharacteristic, complemented
353#468—1221no—
354#465—1221no—
355#465—1221no—
356#461—611no—
357#461—611no—
358#461—611no—
359#460—611no—
360#459—611no—
361#450—1221no—
362#450—1221no—
363#449—1221no—
364#448—1221no—
365#448—1221no—
366C8×Q8C2×D6111yescharacteristic, complemented
367C4×SD16—611no—
368C4×D8—21no—
369C4×D8C2×D6111yescharacteristic
370#436—611no—
371C8×D4—611no—
372C8×D4—611no—
373C2×#103—1221no—
374#334—611no—
375#334—611no—
376#332—611no—
377#332—611no—
378#332—611no—
379#331—611no—
380Q8⋊C8—611no—
381D4⋊C8—611no—
382D4⋊C8—611no—
383D4⋊C8—611no—
384#322—611no—
385C61#1980311yescharacteristic, complemented, Sylow
386C2×#41—1221no—
387C22×Q8—21no—
388C22×Q8—21no—
389C22×D4—1221no—
390C22×D4—1221no—
391AGL(1,ℤ/8)—2444no—
392C2×SD16—611no—
393C2×SD16—1221no—
394C2×SD16—1221no—
395C2×SD16—1221no—
396C2×SD16—1221no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—1221no—
401C2×SD16—2444no—
402C2×D8—1221no—
403C2×D8C22⋊D6111yescharacteristic
404C2×M16—1221no—
405C22×C8—1221no—
406C22×C8—1221no—
407C4⋊Q8—21no—
408C4⋊Q8—21no—
409C4⋊Q8—611no—
410C4⋊Q8C22×D6111yescharacteristic
411C4⋊Q8C22×D6111yescharacteristic
412#127—611no—
413#123—2441no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—41no—
417C22⋊Q8—41no—
418C22⋊Q8—1221no—
419#121—1221no—
420#121—1221no—
421#121—1221no—
422#120—2441no—
423C4×Q8C22×D6111yescharacteristic
424C4×D4—21no—
425C4×D4—21no—
426C4×D4—611no—
427C4×D4C22×D6111yescharacteristic
428C4×D4C22×D6111yescharacteristic
429C2×C4⋊C4—1221no—
430#107—21no—
431#107—21no—
432#107—21no—
433#107—21no—
434#107C22⋊D6111yescharacteristic
435#106—611no—
436#106—1221no—
437#106—2441no—
438C4⋊C8—21no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—611no—
442C4⋊C8C22×D6111yescharacteristic
443#103—611no—
444#103—1221no—
445#103—1221no—
446#103—1221no—
447#103—1221no—
448#103—1221no—
449#103—1221no—
450#102—21no—
451#102—21no—
452#102—611no—
453#102—1221no—
454#102—1221no—
455#102—2441no—
456#102C22⋊D6111yescharacteristic
457#102C22⋊D6111yescharacteristic
458C22⋊C8—1221no—
459C22⋊C8—1221no—
460C22⋊C8—1221no—
461C4×C8—21no—
462C4×C8—611no—
463C4×C8—611no—
464C4×C8C22×D6111yescharacteristic
465C24—2441no—
466#41—4881no—
467#41—2442no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—41no—
471C2×Q8—41no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8—81no—
475C2×Q8—81no—
476C2×Q8—1221no—
477C2×Q8D4×D6111yescharacteristic
478C2×D4—611no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—4881no—
483C2×D4—4881no—
484C2×D4—2441no—
485C2×D4—2442no—
486C2×D4—2441no—
487C2×D4—2441no—
488C2×D4—2442no—
489C2×D4C2×C22⋊D6111yescharacteristic
490C2×D4C2×C22⋊D6111yescharacteristic
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—1221no—
496C22×C4—2441no—
497SD16—1222no—
498SD16—4881no—
499SD16—4881no—
500SD16—4881no—
501SD16—2442no—
502SD16—2442no—
503SD16—2442no—
504SD16—2442no—
505SD16—2442no—
506D8—2442no—
507D8#1175820114yescomplemented
508M16—2442no—
509C2×C8—21no—
510C2×C8—611no—
511C2×C8—611no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—4881no—
517C2×C8—2442no—
518C2×C8C2×C22⋊D6111yescharacteristic
519C2×C8C2×C22⋊D6111yescharacteristic
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—611no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—2441no—
531C4⋊C4D4×D6111yescharacteristic
532C4⋊C4C2×C22⋊D6111yescharacteristic
533C4⋊C4C2×C22⋊D6111yescharacteristic
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—1221no—
539C22⋊C4—2441no—
540C22⋊C4—2441no—
541C42—21no—
542C42C23×D6111yescharacteristic
543C23—21no—
544C23—21no—
545C23—1221no—
546C23—9761no—
547C23—4882no—
548C23—4882no—
549C23—2441no—
550Q8—21no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—82no—
556Q8—82no—
557Q8—2441no—
558Q8—2441no—
559Q8#1451980112yescomplemented
560D4—21no—
561D4—21no—
562D4—1221no—
563D4—1221no—
564D4—4881no—
565D4—4882no—
566D4—4882no—
567D4—4882no—
568D4—2441no—
569D4—2441no—
570D4—2441no—
571D4—2441no—
572D4C2×#1175820112yes—
573D4C2×#1175820112yescomplemented
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—1221no—
587C2×C4—4882no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4C22×C22⋊D6111yescharacteristic
593C2×C4#1451982211yescharacteristic
594C8—21no—
595C8—41no—
596C8—1221no—
597C8—1221no—
598C8—2441no—
599C8—2441no—
600C8—2441no—
601C8—2441no—
602C8C2×#1175820112yescomplemented
603C22—22no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—9764no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—2441no—
614C22—2441no—
615C22—2441no—
616C22D4×C22⋊D6111yescharacteristic
617C4—22no—
618C4—21no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—4881no—
625C4—2441no—
626C4C22×#1175820111yescharacteristic
627C4C2×#1451980111yescharacteristic
628C4#1684056311yescharacteristic
629C4#1684057011yescharacteristic
630C2—42no—
631C2—42no—
632C2—9761no—
633C2—4881no—
634C2#2145729911yescharacteristic
635C2D4×#1175820111yescharacteristic
636C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×D8×C61C2
socleC22×C61D42
prime radical#21457420C2
hypercenterQ8×D8D61
hypocenterC61#19803
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×D8

Complements

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

NH ≅ G/Ntype
1. #21457684C2semidirect
2. D4×#11758201C2semidirect
4. #21445442C2semidirect
5. #21445329C2semidirect
6. #21445240C2semidirect
7. #21445167C2semidirect
8. #21445162C2semidirect
9. #21445102C2semidirect
10. #21445082C2semidirect
11. #21444997C2semidirect
12. #21444922C2semidirect
13. #21444662C2semidirect
14. #21444643C2semidirect
15. Q8×D8×C61C2semidirect
20. #16840158C22semidirect
23. #16840112C22semidirect
26. #16840047C22semidirect
43. D4×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839483C22semidirect
57. D4×Q8×C61C22semidirect
59. C61×#500C22semidirect
63. C61×#474C22semidirect
64. C8×Q8×C61C22semidirect
178. D8×C61SD16semidirect
238. Q8×C61AGL(1,ℤ/8)semidirect
242. D4×C61C2×SD16semidirect
277. C8×C61C2×SD16semidirect
312. Q8×D8D61semidirect
337. D4×Q8C2×D61semidirect
341. #500C2×D61semidirect
352. #474C2×D61semidirect
366. C8×Q8C2×D61semidirect
385. C61#19803semidirect
507. D8#11758201semidirect
559. Q8#14519801semidirect
573. D4C2×#11758201semidirect
602. C8C2×#11758201semidirect

Maximal subgroups

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

subgroupindex
#214576842
D4×#117582012
#214574202
#214454422
#214453292
#214452402
#214451672
#214451622
#214451022
#214450822
#214449972
#214449222
#214446622
#214446432
Q8×D8×C612
#1980361