Finite Group Databasebeta

← #29669609

Subgroups — #29669609

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457634C211yesmaximal, characteristic, complemented
2#21457626C211yesmaximal, characteristic, complemented
3#21457537C211yesmaximal, characteristic, complemented
4#21445476C211yesmaximal, characteristic, complemented
5#21445380C211yesmaximal, characteristic, complemented
6#21445258C211yesmaximal, characteristic, complemented
7#21445239C211yesmaximal, characteristic, complemented
8#21445182C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic, complemented
10#21445101C211yesmaximal, characteristic, complemented
11#21444997C211yesmaximal, characteristic, complemented
12#21444908C211yesmaximal, characteristic, complemented
13#21444646C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3330C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17#16840417C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840158C2211yescharacteristic, complemented
20#16840141C2211yescharacteristic, complemented
21#16840110C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840077—21no—
25#16840075—21no—
26#16840042C2211yescharacteristic
27#16840039C2211yescharacteristic
28#16840001—21no—
29#16839997—21no—
30#16839996—21no—
31#16839967C2211yescharacteristic, complemented
32#16839964C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839924—21no—
36#16839922—21no—
37#16839895—21no—
38#16839894—21no—
39C4×#11758199C2211yescharacteristic
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42D4×#92418C2211yescharacteristic, complemented
43C2×#14519697—21no—
44#102×D61—21no—
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51D4⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56C61×#547C2211yescharacteristic
57C61×#547C2211yescharacteristic, complemented
58C61×#487C2211yescharacteristic, complemented
59C61×#484—21no—
60C61×#474C2211yescharacteristic
61C61×#468—21no—
62C61×#461C2211yescharacteristic, complemented
63C61×#449—21no—
64C4×D8×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic, complemented
66C61×#417—21no—
67C2×#11758222—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519797—44no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77C2×#11758199D411yescharacteristic
78C22×#92418—21no—
79C22×#92418—21no—
80C2×C8×D61—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519759—41no—
84#14519756—21no—
85#14519752—21no—
86#14519751—21no—
87#14519748—21no—
88#14519747—41no—
89C4×D244C2311yescharacteristic
90C4⋊C4×D61—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710D411yescharacteristic
94#14519708—41no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519701—21no—
99#14519700—21no—
100#14519699—21no—
101#14519699—21no—
102#14519698—21no—
103#14519698D411yescharacteristic
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×#41×C61—21no—
118C2×SD16×C61—21no—
119C2×D8×C61D411yescharacteristic
120C2×M16×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#123×C61—41no—
124#123×C61—41no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—21no—
127#121×C61—21no—
128#121×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61—21no—
134#107×C61—21no—
135#107×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—42no—
148#41×C61—81no—
149#41×C61—81no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C2×D4×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C2×C22⋊D61—41no—
163C2×C22⋊D61—41no—
164D4×D61—42no—
165D4×D61—81no—
166#11758222—42no—
167#11758222—81no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—42no—
172C2×D244—41no—
173C2×D244—81no—
174C2×D244C2×D411yescharacteristic
175C2×C4×D61—21no—
176C2×C4×D61—41no—
177C2×Dic122—21no—
178SD16×C61—81no—
179D8×C61—22no—
180M16×C61—81no—
181C2×C8×C61—21no—
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#11758202—42no—
201#11758202—81no—
202#11758201—42no—
203#11758200—42no—
204#11758199—22no—
205#11758199—81no—
206#11758199—81no—
207#11758198—21no—
208#11758198—41no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61C2×D411yescharacteristic
212#11758196—41no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—42no—
218C2×#92418—81no—
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C8×D61—81no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C22×D61—21no—
226C22×D61—41no—
227C22×D61—82no—
228C22×D61—82no—
229C22×D61—161no—
230Q8×C61—41no—
231Q8×C61—41no—
232Q8×C61—41no—
233Q8×C61—41no—
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—21no—
237D4×C61—21no—
238D4×C61—21no—
239D4×C61—82no—
240D4×C61—82no—
241D4×C61#13512yescomplemented
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
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—
262D244—21no—
263D244—41no—
264D244—41no—
265D244—41no—
266D244—41no—
267D244—81no—
268D244—82no—
269D244AGL(1,ℤ/8)12yes—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—82no—
273Dic122—41no—
274Dic122—41no—
275C8×C61—21no—
276C8×C61—21no—
277C8×C61—41no—
278#92418—21no—
279#92418—21no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#19772—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61D4211yescharacteristic
295C2×D61—22no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—164no—
302C4×C61—21no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×#13511yescharacteristic
309C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C4×C61#57611yescharacteristic
311C4×C61#57611yescharacteristic
312Dic61—41no—
313Dic61—81no—
314#3330D6111yescharacteristic, complemented
315#3294—611no—
316#3294—611no—
317#3283—611no—
318#1720—611no—
319#1701—611no—
320#1681—611no—
321#1679—611no—
322#1669—611no—
323#1654—611no—
324#1654—611no—
325#1639—611no—
326#1625—611no—
327#1581—611no—
328#1580—611no—
329C2×C61—42no—
330C2×C61—81no—
331C2×C61—81no—
332C2×C61#326411yescharacteristic
333C2×C61#328011yescharacteristic
334C2×C61#328911yescharacteristic
335D61—42no—
336D61—161no—
337C2×#135—1221no—
338#547—611no—
339#547C2×D6111yescharacteristic
340#547C2×D6111yescharacteristic, complemented
341D42—611no—
342#498—611no—
343#494—611no—
344#487C2×D6111yescharacteristic, complemented
345#486—611no—
346#486—611no—
347#486—611no—
348#484—21no—
349#480—1221no—
350#480—1221no—
351#474—611no—
352#474—611no—
353#474C2×D6111yescharacteristic
354#468—21no—
355#467—1221no—
356#466—1221no—
357#466—1221no—
358#461—611no—
359#461—611no—
360#461C2×D6111yescharacteristic, complemented
361#459—611no—
362#459—611no—
363#452—1221no—
364#452—1221no—
365#449—21no—
366#447—1221no—
367#447—1221no—
368C4×D8—611no—
369C4×D8C2×D6111yescharacteristic
370#436C2×D6111yescharacteristic, complemented
371C8×D4—611no—
372C8×D4—611no—
373C8×D4—611no—
374#417—21no—
375C2×#102—1221no—
376C2×#102—1221no—
377#335—611no—
378#334—611no—
379#332—611no—
380#332—611no—
381#332—611no—
382#331—611no—
383D4⋊C8—611no—
384D4⋊C8—611no—
385D4⋊C8—611no—
386D4⋊C8—611no—
387#322—611no—
388C61#1977211yescharacteristic, complemented, Sylow
389C2×#41—21no—
390C2×#41—21no—
391C2×#41—1221no—
392C22×D4—1221no—
393C22×D4—1221no—
394#135—2444no—
395C2×Q16—1221no—
396C2×Q16—1221no—
397C2×SD16—21no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×D8—611no—
401C2×D8—1221no—
402C2×D8—1221no—
403C2×D8C22⋊D6111yescharacteristic
404C2×M16—21no—
405C22×C8—1221no—
406C22×C8—1221no—
407C22×C8—1221no—
408C4⋊Q8—611no—
409C4⋊Q8C22×D6111yescharacteristic
410C4⋊Q8C22×D6111yescharacteristic
411#127—611no—
412#123—41no—
413#123—41no—
414#123—2441no—
415C22⋊Q8—21no—
416C22⋊Q8—21no—
417C22⋊Q8—1221no—
418#121—21no—
419#121—21no—
420#121—1221no—
421#121—1221no—
422#121—1221no—
423#120—2441no—
424C4×D4—611no—
425C4×D4C22×D6111yescharacteristic
426C4×D4C22×D6111yescharacteristic
427C4×D4C22×D6111yescharacteristic
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—1221no—
431#107—21no—
432#107—611no—
433#107—1221no—
434#107—1221no—
435#107C22⋊D6111yescharacteristic
436#106—2441no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8C22×D6111yescharacteristic
441#103—21no—
442#103—21no—
443#103—21no—
444#103—1221no—
445#103—1221no—
446#102—21no—
447#102—611no—
448#102—611no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—1221no—
456#102—1221no—
457#102—2441no—
458#102C22⋊D6111yescharacteristic
459#102C22⋊D6111yescharacteristic
460C22⋊C8—21no—
461C22⋊C8—1221no—
462C22⋊C8—1221no—
463C22⋊C8—1221no—
464C4×C8—611no—
465C4×C8—611no—
466C4×C8C22×D6111yescharacteristic
467C24—2441no—
468#41—42no—
469#41—42no—
470#41—81no—
471#41—81no—
472#41—4881no—
473#41—2442no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—1221no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—611no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—4881no—
484C2×D4—4881no—
485C2×D4—2441no—
486C2×D4—2442no—
487C2×D4—2441no—
488C2×D4—2441no—
489C2×D4—2442no—
490C2×D4D4×D6111yescharacteristic
491C2×D4C2×C22⋊D6111yescharacteristic
492C2×D4C2×C22⋊D6111yescharacteristic
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—1221no—
499C22×C4—2441no—
500Q16—4881no—
501Q16—2442no—
502SD16—81no—
503SD16—2442no—
504SD16—2442no—
505D8—22no—
506D8—1222no—
507D8—4881no—
508D8—4881no—
509M16—81no—
510C2×C8—21no—
511C2×C8—611no—
512C2×C8—611no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—4881no—
518C2×C8—4881no—
519C2×C8—2442no—
520C2×C8C2×C22⋊D6111yescharacteristic
521C2×C8C2×C22⋊D6111yescharacteristic
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—611no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—2441no—
532C4⋊C4D4×D6111yescharacteristic
533C4⋊C4C2×C22⋊D6111yescharacteristic
534C4⋊C4C2×C22⋊D6111yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—1221no—
541C22⋊C4—2441no—
542C22⋊C4—2441no—
543C42C23×D6111yescharacteristic
544C23—21no—
545C23—21no—
546C23—21no—
547C23—1221no—
548C23—9761no—
549C23—4882no—
550C23—4882no—
551C23—2441no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8—2441no—
557Q8—2441no—
558D4—21no—
559D4—21no—
560D4—21no—
561D4—21no—
562D4—21no—
563D4—82no—
564D4—82no—
565D4—612no—
566D4—1221no—
567D4—4881no—
568D4—4882no—
569D4—4882no—
570D4—4882no—
571D4—2441no—
572D4—2441no—
573D4—2441no—
574D4—2441no—
575D4#1451979712yescomplemented
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—1221no—
590C2×C4—4882no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C22×C22⋊D6111yescharacteristic
596C2×C4#1451982211yescharacteristic
597C8—21no—
598C8—21no—
599C8—41no—
600C8—1221no—
601C8—1221no—
602C8—2441no—
603C8—2441no—
604C8—2441no—
605C8—2441no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—1222no—
616C22—9764no—
617C22—4881no—
618C22—4881no—
619C22—4881no—
620C22—2441no—
621C22—2441no—
622C22D4×C22⋊D6111yescharacteristic
623C4—21no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—4881no—
630C4—2441no—
631C4C2×#1451979711yescharacteristic
632C4AGL(1,ℤ/8)×D6111yescharacteristic
633C4#1684056511yescharacteristic
634C4#1684056611yescharacteristic
635C2—42no—
636C2—81no—
637C2—81no—
638C2—9761no—
639C2—2442no—
640C2#2145729311yescharacteristic
641C2#2145750011yescharacteristic
642C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3330C2
socleC22×C61D42
hypercenter#3330D61
hypocenterC61#19772
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3330

Complements

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

NH ≅ G/Ntype
1. #21457634C2semidirect
2. #21457626C2semidirect
3. #21457537C2semidirect
4. #21445476C2semidirect
5. #21445380C2semidirect
6. #21445258C2semidirect
7. #21445239C2semidirect
8. #21445182C2semidirect
9. #21445103C2semidirect
10. #21445101C2semidirect
11. #21444997C2semidirect
12. #21444908C2semidirect
13. #21444646C2semidirect
14. #21444643C2semidirect
15. C61×#3330C2semidirect
19. #16840158C22semidirect
20. #16840141C22semidirect
31. #16839967C22semidirect
42. D4×#92418C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839508C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
52. D4⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839483C22semidirect
57. C61×#547C22semidirect
58. C61×#487C22semidirect
62. C61×#461C22semidirect
65. C61×#436C22semidirect
241. D4×C61#135semidirect
314. #3330D61semidirect
340. #547C2×D61semidirect
344. #487C2×D61semidirect
360. #461C2×D61semidirect
370. #436C2×D61semidirect
388. C61#19772semidirect
575. D4#14519797semidirect

Maximal subgroups

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

subgroupindex
#214576342
#214576262
#214575372
#214454762
#214453802
#214452582
#214452392
#214451822
#214451032
#214451012
#214449972
#214449082
#214446462
#214446432
C61×#33302
#1977261