Finite Group Databasebeta

← #29663896

Subgroups — #29663896

Order 15616.

786 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#492×D61C211yesmaximal, characteristic, complemented
2#21455629C211yesmaximal, characteristic, complemented
3#21455296C211yesmaximal, characteristic
4C8×#11758226C211yesmaximal, characteristic, complemented
5#21445169C211yesmaximal, characteristic, complemented
6#21445167C211yesmaximal, characteristic, complemented
7#21445165C211yesmaximal, characteristic, complemented
8#21445163C211yesmaximal, characteristic, complemented
9#21445161C211yesmaximal, characteristic, complemented
10#21445159C211yesmaximal, characteristic, complemented
11#21445157C211yesmaximal, characteristic, complemented
12#21444573C211yesmaximal, characteristic, complemented
13#21444502C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic, complemented
15C61×#1666C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17C2×SD16×D61—21no—
18C2×SD16×D61—21no—
19C4⋊Q8×D61C2211yescharacteristic
20#127×D61C2211yescharacteristic
21C4×#11758226C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840135C2211yescharacteristic
24#16840134C2211yescharacteristic
25#16840133—21no—
26#16840133—21no—
27#16840055—21no—
28#16840051—21no—
29#16840047C2211yescharacteristic, complemented
30#16840043C2211yescharacteristic, complemented
31#16840039C2211yescharacteristic, complemented
32#16839962C2211yescharacteristic
33#16839960—21no—
34#16839958C2211yescharacteristic
35#16839956—21no—
36#16839954C2211yescharacteristic
37Q8×#92418C2211yescharacteristic, complemented
38C8×D244—21no—
39C8×D244C2211yescharacteristic, complemented
40#16839827—21no—
41#16839827C2211yescharacteristic, complemented
42#16839814—21no—
43#16839814C2211yescharacteristic
44#16839801—21no—
45#16839784C2211yescharacteristic
46#16839784C2211yescharacteristic
47#16839765—21no—
48#16839762—21no—
49C4×C8×D61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839508C2211yescharacteristic, complemented
56#16839507C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#492C2211yescharacteristic
61C61×#475C2211yescharacteristic, complemented
62C61×#460C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×#11758226D411yescharacteristic
68C2×Q8×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61—21no—
72#14519804—44no—
73C2×#11758190—21no—
74C2×#11758190—21no—
75C2×#11758200—21no—
76C2×#11758200—21no—
77C2×#11758201—21no—
78C2×#11758201—21no—
79C2×#11758201—21no—
80SD16×D61—44no—
81SD16×D61—44no—
82C2×D488—21no—
83C2×#11758199—21no—
84C2×#11758189—21no—
85C2×#11758189—21no—
86C2×C8×D61—21no—
87C2×C8×D61—21no—
88C2×C8×D61D411yescharacteristic
89C2×C8×D61D411yescharacteristic
90#14519777C2311yescharacteristic
91#14519776—21no—
92#14519776—21no—
93#14519775—21no—
94#14519774C2311yescharacteristic
95#14519773—21no—
96#14519773—21no—
97#14519773—21no—
98Q8×Dic61D411yescharacteristic
99C4×D244—21no—
100C4×D244C2311yescharacteristic
101#14519739—21no—
102#14519739D411yescharacteristic
103#14519735—21no—
104#14519735D411yescharacteristic
105C4⋊C4×D61—21no—
106C4⋊C4×D61—21no—
107C42×D61—21no—
108C42×D61C2311yescharacteristic
109#14519711—21no—
110#14519710—21no—
111#14519708—21no—
112C4⋊#92418—21no—
113C4⋊#92418C2311yescharacteristic
114C4⋊#92418C2311yescharacteristic
115C4⋊#92418C2311yescharacteristic
116#14519706—21no—
117#14519706—21no—
118#14519701—21no—
119#14519699—21no—
120#14519699—21no—
121#14519698—21no—
122#14519697—21no—
123#14519697—21no—
124#14519689—21no—
125#14519689—21no—
126#14519689—21no—
127C4×#92418—21no—
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C8×Dic61D411yescharacteristic
131C8×Dic61D411yescharacteristic
132C2×SD16×C61—21no—
133C2×SD16×C61—21no—
134C2×SD16×C61—21no—
135C4⋊Q8×C61C2311yescharacteristic
136#127×C61C2311yescharacteristic
137C4×Q8×C61C2311yescharacteristic
138#106×C61—21no—
139C4⋊C8×C61—21no—
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142C4⋊C8×C61C2311yescharacteristic
143#103×C61—21no—
144#102×C61—21no—
145C4×C8×C61—21no—
146C4×C8×C61C2311yescharacteristic
147C23×D61—41no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×C22⋊D61—41no—
154C2×C22⋊D61—41no—
155C2×C22⋊D61—41no—
156#11758226—22no—
157#11758226SD1614yescomplemented
158Q8×D61—42no—
159Q8×D61—42no—
160D4×D61—42no—
161D4×D61—42no—
162D4×D61—42no—
163D4×D61—42no—
164D4×D61—82no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244C2×D411yescharacteristic
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—41no—
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61#4111yescharacteristic
177C2×Dic122—21no—
178C2×Dic122—41no—
179SD16×C61—42no—
180SD16×C61—42no—
181SD16×C61—42no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C2×D411yescharacteristic
186C2×C8×C61C2×D411yescharacteristic
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61C2×D411yescharacteristic
191C42×C61—21no—
192C42×C61C2411yescharacteristic
193#11758201—42no—
194#11758201—42no—
195#11758201—42no—
196#11758200—42no—
197#11758200—42no—
198#11758199—42no—
199#11758198—21no—
200#11758198—21no—
201#11758198—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—41no—
207#11758196—41no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61#4111yescharacteristic
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218D488—42no—
219#11758190—42no—
220#11758190—42no—
221#11758189—42no—
222#11758189—42no—
223C8×D61—22no—
224C8×D61—22no—
225C8×D61—42no—
226C8×D61—42no—
227C23×C61—41no—
228C22×D61—21no—
229C22×D61—41no—
230C22×D61—82no—
231C22×D61—82no—
232C22×D61—82no—
233C22×D61#12111yescharacteristic
234Q8×C61—21no—
235Q8×C61—41no—
236Q8×C61—41no—
237Q8×C61C2×SD1612yescomplemented
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—81no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C22⋊D61—82no—
250C22⋊D61—82no—
251C22⋊D61—82no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254C2×Dic61#12111yescharacteristic
255C2×Dic61#12111yescharacteristic
256C2×Dic61#12111yescharacteristic
257D244—24no—
258D244—21no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—41no—
263D244—81no—
264D244C2×SD1612yescomplemented
265C4×D61—21no—
266C4×D61—21no—
267C4×D61—24no—
268C4×D61—22no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—42no—
272C4×D61—82no—
273C4×D61C2×SD1612yes—
274C4×D61#13512yes—
275Dic122—41no—
276Dic122—41no—
277Dic122—82no—
278C8×C61—21no—
279C8×C61—41no—
280C8×C61—41no—
281C8×C61—41no—
282C8×C61AGL(1,ℤ/8)12yescomplemented
283#92418—21no—
284#92418—41no—
285#92418—41no—
286#92418—41no—
287#92418AGL(1,ℤ/8)12yescomplemented
288#19266—611nomaximal, Sylow
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61C2×#12111yescharacteristic
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—22no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—164no—
302C2×D61#46512yes—
303C4×C61—22no—
304C4×C61—21no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61C22×SD1611yescharacteristic
308C4×C61C2×#13511yescharacteristic
309C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
311Dic61—21no—
312Dic61—41no—
313Dic61—81no—
314Dic61#46512yes—
315Dic61#47112yes—
316C2×#492—611no—
317#3079—611no—
318#3038—611no—
319C8×#41—611no—
320#1666—611no—
321#1666—611no—
322#1666—611no—
323#1666D6111yescharacteristic, complemented
324#1665—611no—
325#1665—611no—
326#1665—611no—
327#1665—611no—
328#1566—611no—
329#1549—611no—
330#1545—611no—
331C2×C61—161no—
332C2×C61#303811yescharacteristic
333C2×C61C2×#46511yescharacteristic
334C2×C61C2×#47111yescharacteristic
335D61—22no—
336D61—42no—
337D61—161no—
338C2×AGL(1,ℤ/8)—1221no—
339C22×SD16—1221no—
340C22×SD16—1221no—
341C2×C4⋊Q8—611no—
342C2×#127—611no—
343C4×#41—611no—
344#492—611no—
345#492—611no—
346#492—611no—
347#492—1221no—
348#492—1221no—
349#492C2×D6111yescharacteristic
350#477—1221no—
351#476—1221no—
352#475—611no—
353#475C2×D6111yescharacteristic, complemented
354#474—611no—
355#474—611no—
356#460—611no—
357#460—1221no—
358#460C2×D6111yescharacteristic
359#459—611no—
360#459—611no—
361#459—1221no—
362C8×Q8—611no—
363C8×Q8C2×D6111yescharacteristic, complemented
364C8×D4—611no—
365C8×D4—611no—
366C8×D4—1221no—
367C8×D4—1221no—
368#431—611no—
369#431—1221no—
370#428—1221no—
371#423—611no—
372#423—611no—
373#418—1221no—
374#417—1221no—
375C2×C4×C8—611no—
376C2×C4×C8—1221no—
377#334—611no—
378#334—611no—
379#334—611no—
380#334C2×D6111yescharacteristic, complemented
381#332—611no—
382#332—611no—
383#332—611no—
384#332C2×D6111yescharacteristic, complemented
385#331—611no—
386#331—611no—
387#331—611no—
388#331C2×D6111yescharacteristic
389C61#1926611yescharacteristic, complemented, Sylow
390C2×#41—611no—
391C22×Q8—1221no—
392C22×D4—1221no—
393C22×D4—1221no—
394C22×D4—1221no—
395AGL(1,ℤ/8)—2444no—
396C2×SD16—21no—
397C2×SD16—21no—
398C2×SD16—21no—
399C2×SD16—1221no—
400C2×SD16—1221no—
401C2×SD16—1221no—
402C2×SD16—1221no—
403C2×SD16—1221no—
404C2×SD16—1221no—
405C2×SD16—1221no—
406C2×SD16—2444no—
407C2×SD16—2444no—
408C2×D8—1221no—
409C2×D8—1221no—
410C2×M16—1221no—
411C2×M16—1221no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8—1221no—
415C22×C8—1221no—
416C4⋊Q8—611no—
417C4⋊Q8—1221no—
418C4⋊Q8—1221no—
419C4⋊Q8—1221no—
420C4⋊Q8C22×D6111yescharacteristic
421#127—611no—
422#127—1221no—
423#127—1221no—
424#127—1221no—
425#127C22×D6111yescharacteristic
426C4×Q8—611no—
427C4×Q8C22×D6111yescharacteristic
428C4×D4—611no—
429C4×D4—611no—
430C4×D4—1221no—
431C4×D4—1221no—
432#117—611no—
433#117—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C42—611no—
437C2×C42—1221no—
438#107—1221no—
439#107—1221no—
440#106—21no—
441#106—1221no—
442C4⋊C8—21no—
443C4⋊C8—611no—
444C4⋊C8—611no—
445C4⋊C8—611no—
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C4⋊C8—1221no—
449C4⋊C8C22×D6111yescharacteristic
450C4⋊C8C22×D6111yescharacteristic
451C4⋊C8C22×D6111yescharacteristic
452#103—21no—
453#103—1221no—
454#102—21no—
455#102—1221no—
456#102—1221no—
457#102—1221no—
458#102—1221no—
459#102—1221no—
460C22⋊C8—1221no—
461C22⋊C8—1221no—
462C22⋊C8—1221no—
463C4×C8—21no—
464C4×C8—611no—
465C4×C8—611no—
466C4×C8—611no—
467C4×C8—1221no—
468C4×C8—1221no—
469C4×C8C22×D6111yescharacteristic
470C24—2441no—
471#41—614no—
472#41—1222no—
473C2×Q8—21no—
474C2×Q8—1221no—
475C2×Q8—2441no—
476C2×Q8—2442no—
477C2×Q8—2442no—
478C2×Q8D4×D6111yescharacteristic
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—611no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—4882no—
488C2×D4—2442no—
489C2×D4—2442no—
490C2×D4—2441no—
491C2×D4—2441no—
492C2×D4—2441no—
493C2×D4—2442no—
494C2×D4—2442no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—2441no—
502SD16—42no—
503SD16—42no—
504SD16—42no—
505SD16—2442no—
506SD16—2442no—
507SD16—2442no—
508SD16—2442no—
509SD16—2442no—
510SD16—2442no—
511SD16—2442no—
512D8—2442no—
513D8—2442no—
514M16—2442no—
515M16—2442no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—611no—
520C2×C8—611no—
521C2×C8—1222no—
522C2×C8—1222no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—2442no—
527C2×C8—2442no—
528C2×C8D4×D6111yescharacteristic
529C2×C8D4×D6111yescharacteristic
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—611no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—2441no—
538C4⋊C4—2441no—
539C4⋊C4D4×D6111yescharacteristic
540C22⋊C4—1221no—
541C22⋊C4—1221no—
542C22⋊C4—1221no—
543C42—21no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—1221no—
548C42—1221no—
549C42C23×D6111yescharacteristic
550C23—41no—
551C23—611no—
552C23—1221no—
553C23—4882no—
554C23—4882no—
555C23—4882no—
556C23—2441no—
557Q8—21no—
558Q8—41no—
559Q8—41no—
560Q8—4882no—
561Q8—2441no—
562Q8—2441no—
563Q8SD16×D6112yescomplemented
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—81no—
569D4—612no—
570D4—1224no—
571D4—1221no—
572D4—4881no—
573D4—4882no—
574D4—4882no—
575D4—4882no—
576D4—2441no—
577D4—2441no—
578D4—2441no—
579D4—2441no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—611no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—4882no—
594C2×C4—2442no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C8—21no—
602C8—41no—
603C8—41no—
604C8—41no—
605C8—612no—
606C8—1221no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C8#1451980412yescomplemented
611C22—81no—
612C22—81no—
613C22—81no—
614C22—612no—
615C22—1221no—
616C22—1221no—
617C22—1222no—
618C22—9764no—
619C22—4881no—
620C22—4881no—
621C22—4881no—
622C22—2441no—
623C22—2441no—
624C22#121×D6111yescharacteristic
625C4—22no—
626C4—21no—
627C4—41no—
628C4—81no—
629C4—612no—
630C4—612no—
631C4—1221no—
632C4—4881no—
633C4—2441no—
634C4C2×SD16×D6111yescharacteristic
635C4#135×D6111yescharacteristic
636C4AGL(1,ℤ/8)×D6111yescharacteristic
637C4C2×#1451980411yescharacteristic
638C2—161no—
639C2—1222no—
640C2—9761no—
641C2—2442no—
642C2#2145529611yescharacteristic
643C2#465×D6111yescharacteristic
644C2#471×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1666C2
socleC22×C61C2×#121
prime radical#21455296C2
hypercenter#1666D61
hypocenterC61#19266
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1666

Complements

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

NH ≅ G/Ntype
1. #492×D61C2semidirect
2. #21455629C2semidirect
4. C8×#11758226C2semidirect
5. #21445169C2semidirect
6. #21445167C2semidirect
7. #21445165C2semidirect
8. #21445163C2semidirect
9. #21445161C2semidirect
10. #21445159C2semidirect
11. #21445157C2semidirect
12. #21444573C2semidirect
13. #21444502C2semidirect
14. #21444485C2semidirect
15. C61×#1666C2semidirect
29. #16840047C22semidirect
30. #16840043C22semidirect
31. #16840039C22semidirect
37. Q8×#92418C22semidirect
39. C8×D244C22semidirect
41. #16839827C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839509C22semidirect
55. #16839508C22semidirect
56. #16839507C22semidirect
61. C61×#475C22semidirect
63. C8×Q8×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
157. #11758226SD16semidirect
237. Q8×C61C2×SD16semidirect
264. D244C2×SD16semidirect
282. C8×C61AGL(1,ℤ/8)semidirect
287. #92418AGL(1,ℤ/8)semidirect
323. #1666D61semidirect
353. #475C2×D61semidirect
363. C8×Q8C2×D61semidirect
380. #334C2×D61semidirect
384. #332C2×D61semidirect
389. C61#19266semidirect
563. Q8SD16×D61semidirect
610. C8#14519804semidirect

Maximal subgroups

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

subgroupindex
#492×D612
#214556292
#214552962
C8×#117582262
#214451692
#214451672
#214451652
#214451632
#214451612
#214451592
#214451572
#214445732
#214445022
#214444852
C61×#16662
#1926661