Finite Group Databasebeta

← #29834409

Subgroups — #29834409

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458954C211yesmaximal, characteristic, complemented
2#509×D61C211yesmaximal, characteristic, complemented
3#21456301C211yesmaximal, characteristic, complemented
4#21454731C211yesmaximal, characteristic, complemented
5#21447896C211yesmaximal, characteristic, complemented
6#21447895C211yesmaximal, characteristic, complemented
7#21447894C211yesmaximal, characteristic, complemented
8#21447893C211yesmaximal, characteristic, complemented
9#21447891C211yesmaximal, characteristic, complemented
10#21447890C211yesmaximal, characteristic, complemented
11#21447885C211yesmaximal, characteristic, complemented
12#21447583C211yesmaximal, characteristic, complemented
13#21447567C211yesmaximal, characteristic, complemented
14#361×D61C211yesmaximal, characteristic, complemented
15C61×#2212C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17C2×#14519800C2211yescharacteristic
18#135×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840527—21no—
21#16840330—21no—
22#16840204—21no—
23#16840203C2211yescharacteristic, complemented
24#16840200C2211yescharacteristic
25#16840200C2211yescharacteristic, complemented
26#16840199—21no—
27#16840199—21no—
28#16840198C2211yescharacteristic
29#16840197C2211yescharacteristic
30#16840197C2211yescharacteristic, complemented
31#16840196—21no—
32#16840195—21no—
33#16840193—21no—
34SD32×D61—21no—
35D16×D61—21no—
36M32×D61C2211yescharacteristic
37M32×D61C2211yescharacteristic
38#16840152—21no—
39#16840148—21no—
40#16840147C2211yescharacteristic, complemented
41#16840139C2211yescharacteristic, complemented
42#16840137C2211yescharacteristic, complemented
43#16839882—21no—
44#16839881—21no—
45#16839880C2211yescharacteristic, complemented
46#16839879C2211yescharacteristic, complemented
47#16839878C2211yescharacteristic, complemented
48#108×D61C2211yescharacteristic
49#16839777—21no—
50#16839716C2211yescharacteristic
51#16839583C2211yescharacteristic, complemented
52#16839582C2211yescharacteristic, complemented
53#16839582C2211yescharacteristic, complemented
54#16839578C2211yescharacteristic, complemented
55#16839576C2211yescharacteristic, complemented
56#16839576C2211yescharacteristic, complemented
57#16839575C2211yescharacteristic, complemented
58#16839574C2211yescharacteristic, complemented
59#16839574C2211yescharacteristic, complemented
60C61×#510C2211yescharacteristic, complemented
61C61×#509C2211yescharacteristic
62C61×#495C2211yescharacteristic, complemented
63C61×#443C2211yescharacteristic, complemented
64C61×#364C2211yescharacteristic, complemented
65C61×#362C2211yescharacteristic, complemented
66C61×#361C2211yescharacteristic, complemented
67C2×#11758226—21no—
68#41×D61—21no—
69C2×D4×D61—21no—
70#14519804—41no—
71#14519804—41no—
72#14519804—41no—
73#14519804—41no—
74#14519800—21no—
75#14519800—21no—
76#14519800—21no—
77#14519800—21no—
78#14519800D411yescharacteristic, complemented
79#14519799C2311yescharacteristic
80#14519798—21no—
81#14519797—21no—
82#14519796D411yescharacteristic, complemented
83Q16×D61D411yescharacteristic, complemented
84C2×#11758201—21no—
85C2×#11758201—21no—
86SD16×D61—21no—
87C2×D488—21no—
88C2×D488C2311yescharacteristic
89C2×D488C2311yescharacteristic
90C2×#11758199—21no—
91C2×#11758199—21no—
92D8×D61—21no—
93D8×D61—21no—
94D8×D61—21no—
95D8×D61—21no—
96D8×D61D411yescharacteristic, complemented
97#14519785—21no—
98#14519785—41no—
99#14519784—21no—
100#14519783—21no—
101#14519783—41no—
102C2×#11758189—41no—
103C2×#11758189D411yescharacteristic
104M16×D61—21no—
105M16×D61—41no—
106C2×C8×D61C2311yescharacteristic
107#14519774—21no—
108#14519773—21no—
109#14519766—21no—
110#14519750—42no—
111#14519747—42no—
112#14519734D411yescharacteristic
113#14519723—21no—
114#14519722—21no—
115#14519722—21no—
116#14519721—21no—
117#14519720—21no—
118#14519720—21no—
119#14519720—21no—
120D976—21no—
121D976—21no—
122#14519718—21no—
123#14519718—21no—
124#14519718—21no—
125#14519717C2311yescharacteristic
126#14519717C2311yescharacteristic
127#14519716—21no—
128#14519716—21no—
129C16×D61—21no—
130C16×D61—21no—
131#14519713C2311yescharacteristic
132#14519712—21no—
133#14519705—21no—
134#14519704—21no—
135#14519703—21no—
136#14519688D411yescharacteristic
137#14519687D411yescharacteristic
138C8×Dic61C2311yescharacteristic
139#135×C61C2311yescharacteristic
140C2×Q16×C61C2311yescharacteristic
141C2×SD16×C61—21no—
142C2×D8×C61C2311yescharacteristic
143#131×C61—21no—
144#124×C61—21no—
145Q32×C61—21no—
146SD32×C61—21no—
147SD32×C61—21no—
148D16×C61—21no—
149M32×C61C2311yescharacteristic
150M32×C61C2311yescharacteristic
151#108×C61C2311yescharacteristic
152#104×C61—21no—
153C4×C8×C61C2311yescharacteristic
154C23×D61—41no—
155#41×C61—21no—
156C2×Q8×C61—21no—
157C2×D4×C61—21no—
158C2×C22⋊D61—42no—
159C2×C22⋊D61—41no—
160#11758226—21no—
161#11758226—41no—
162#11758226—41no—
163#11758226—41no—
164#11758226—41no—
165Q8×D61—21no—
166#11758224—21no—
167#11758224—41no—
168D4×D61—21no—
169D4×D61—41no—
170D4×D61—41no—
171D4×D61—41no—
172D4×D61—41no—
173D4×D61—41no—
174#11758222—21no—
175#11758222—41no—
176C2×D244—21no—
177C2×D244—21no—
178C2×D244—41no—
179C2×D244—42no—
180C2×C4×D61—41no—
181C2×C4×D61—41no—
182C2×C4×D61C2×D411yescharacteristic
183Q16×C61—21no—
184Q16×C61—21no—
185Q16×C61C2×D411yescharacteristic
186SD16×C61—21no—
187SD16×C61—41no—
188SD16×C61—41no—
189D8×C61—21no—
190D8×C61—21no—
191D8×C61C2×D411yescharacteristic
192M16×C61—21no—
193M16×C61—41no—
194C2×C8×C61—41no—
195C2×C8×C61C2×D411yescharacteristic
196C2×C8×C61C2411yescharacteristic
197C22⋊C4×C61—42no—
198C42×C61C2×D411yescharacteristic
199#11758202—21no—
200#11758201—21no—
201#11758201—41no—
202#11758201—41no—
203#11758201—41no—
204#11758201—41no—
205#11758200—21no—
206#11758199—21no—
207#11758199—41no—
208#11758199—41no—
209#11758199—41no—
210#11758199—41no—
211#11758198—21no—
212#11758198—42no—
213#11758196—21no—
214C4×Dic61C2×D411yescharacteristic
215#11758194—21no—
216#11758194—41no—
217C2×#92418—41no—
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220Dic244C2×D411yescharacteristic
221D488—21no—
222D488—21no—
223D488—21no—
224D488—21no—
225D488—41no—
226D488—41no—
227D488C2×D411yescharacteristic
228#11758190—21no—
229#11758189—41no—
230#11758189—41no—
231#11758189—41no—
232#11758189D811yescharacteristic
233#11758189D811yescharacteristic
234#11758189D811yescharacteristic
235#11758189D811yescharacteristic
236C8×D61—41no—
237C8×D61—41no—
238C8×D61—41no—
239C8×D61C2×D411yescharacteristic
240C8×D61C2×D411yescharacteristic
241C8×D61#4111yescharacteristic
242C8×D61#4111yescharacteristic
243C16×C61—21no—
244C16×C61—21no—
245#11758185—21no—
246#11758185—21no—
247C23×C61—41no—
248C22×D61—41no—
249C22×D61—41no—
250C22×D61—41no—
251C22×D61—81no—
252C22×D61—81no—
253C22×D61—162no—
254C22×D61AGL(1,ℤ/8)11yescharacteristic
255Q8×C61—21no—
256Q8×C61—41no—
257Q8×C61—41no—
258D4×C61—21no—
259D4×C61—41no—
260D4×C61—41no—
261D4×C61—41no—
262C2×C4×C61—21no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61C22×D411yescharacteristic
266C22⋊D61—41no—
267C22⋊D61—41no—
268C22⋊D61—41no—
269C22⋊D61—81no—
270C22⋊D61—81no—
271C22⋊D61—162no—
272C2×Dic61—21no—
273C2×Dic61—41no—
274C2×Dic61AGL(1,ℤ/8)11yescharacteristic
275D244—21no—
276D244—41no—
277D244—41no—
278D244—41no—
279D244—41no—
280D244—41no—
281D244—81no—
282D244—81no—
283D244—162no—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61—81no—
288C4×D61—81no—
289C4×D61#12111yescharacteristic
290C4×D61#12111yescharacteristic
291C4×D61C2×D811yescharacteristic
292C4×D61C2×D811yescharacteristic
293Dic122—21no—
294Dic122—41no—
295C8×C61—41no—
296C8×C61C2×D811yescharacteristic
297C8×C61C2×D811yescharacteristic
298C8×C61C22×D411yescharacteristic
299C8×C61C2×#4111yescharacteristic
300#92418—41no—
301#92418#12111yescharacteristic
302#92418#12111yescharacteristic
303#92418C2×D811yescharacteristic
304#92418C2×D811yescharacteristic
305#33595—611nomaximal, Sylow
306C22×C61—41no—
307C22×C61—82no—
308C22×C61—81no—
309C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—82no—
313C2×D61—82no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—161no—
317C2×D61—162no—
318C2×D61—162no—
319C2×D61#45911yescharacteristic
320C2×D61#45911yescharacteristic
321C4×C61—41no—
322C4×C61—41no—
323C4×C61—81no—
324C4×C61C2×#12111yescharacteristic
325C4×C61C22×D811yescharacteristic
326Dic61—41no—
327Dic61—41no—
328Dic61#45911yescharacteristic
329Dic61#45911yescharacteristic
330#3414—611no—
331C2×#509—611no—
332#3155—611no—
333#2957—611no—
334#2213—611no—
335#2213—611no—
336#2213—611no—
337#2213—611no—
338#2212—611no—
339#2212—611no—
340#2212—611no—
341#2212D6111yescharacteristic, complemented
342#2153—611no—
343#2148—611no—
344C2×#361—611no—
345C2×C61—21no—
346C2×C61—81no—
347C2×C61—161no—
348C2×C61C2×#45911yescharacteristic
349D61—21no—
350D61—21no—
351D61—81no—
352D61—161no—
353D61—161no—
354C2×AGL(1,ℤ/8)—1221no—
355C2×#135—611no—
356C2×#135—611no—
357C22×D8—611no—
358#568—1221no—
359#535—1221no—
360#510—611no—
361#510—1221no—
362#510C2×D6111yescharacteristic, complemented
363#509—611no—
364#509—611no—
365#509—611no—
366#509—611no—
367#509—611no—
368#509—1221no—
369#509—1221no—
370#509—1221no—
371#509C2×D6111yescharacteristic
372#508—1221no—
373#508—1221no—
374C2×SD32—1221no—
375C2×D16—1221no—
376C2×M32—611no—
377C2×M32—611no—
378#496—1221no—
379#496—1221no—
380#495—611no—
381#495C2×D6111yescharacteristic, complemented
382#493—611no—
383#493—611no—
384#444—1221no—
385#444—1221no—
386#443—611no—
387#443—611no—
388#443—611no—
389#443C2×D6111yescharacteristic, complemented
390C2×#108—611no—
391#421—1221no—
392#405—611no—
393#364—611no—
394#364—611no—
395#364—611no—
396#364C2×D6111yescharacteristic, complemented
397#362—611no—
398#362C2×D6111yescharacteristic, complemented
399#361—611no—
400#361—611no—
401#361—611no—
402#361—611no—
403#361—611no—
404#361C2×D6111yescharacteristic, complemented
405C61#3359511yescharacteristic, complemented, Sylow
406C2×#41—1221no—
407C2×#41—1221no—
408C22×D4—1221no—
409AGL(1,ℤ/8)—2441no—
410AGL(1,ℤ/8)—2441no—
411AGL(1,ℤ/8)—2441no—
412AGL(1,ℤ/8)—2441no—
413#135—611no—
414#135—611no—
415#135—611no—
416#135—1221no—
417#135—1221no—
418#135—1221no—
419#135—1221no—
420#135—1221no—
421#135—1221no—
422#135C22×D6111yescharacteristic
423C2×Q16—611no—
424C2×Q16C22×D6111yescharacteristic
425C2×SD16—21no—
426C2×SD16—1221no—
427C2×SD16—1221no—
428C2×SD16—1221no—
429C2×D8—611no—
430C2×D8—611no—
431C2×D8—611no—
432C2×D8—1221no—
433C2×D8—1221no—
434C2×D8—1221no—
435C2×D8—1221no—
436C2×D8—1221no—
437C2×D8—1221no—
438C2×D8—1221no—
439C2×D8C22×D6111yescharacteristic
440#131—21no—
441#131—1221no—
442#131—1221no—
443#131—1221no—
444#131—2441no—
445#131—2441no—
446C2×M16—611no—
447C2×M16—1221no—
448C2×M16—2441no—
449C2×M16—2441no—
450C22×C8—611no—
451#127—1221no—
452#127—1221no—
453#124—21no—
454#124—1221no—
455#121—2442no—
456#120—2442no—
457#117—611no—
458Q32—21no—
459Q32—1221no—
460SD32—21no—
461SD32—21no—
462SD32—1221no—
463SD32—1221no—
464SD32—1221no—
465SD32—1221no—
466SD32—1221no—
467SD32—1221no—
468D16—21no—
469D16—1221no—
470D16—1221no—
471D16—1221no—
472D16—1221no—
473D16—1221no—
474M32—611no—
475M32—611no—
476M32—1221no—
477M32—1221no—
478M32C22×D6111yescharacteristic
479M32C22×D6111yescharacteristic
480C2×C16—1221no—
481C2×C16—1221no—
482#108—611no—
483#108—1221no—
484#108C22×D6111yescharacteristic
485#104—21no—
486#104—1221no—
487#104—1221no—
488#104—1221no—
489#97—611no—
490#97—611no—
491C4×C8—611no—
492C4×C8C22×D6111yescharacteristic
493C24—2441no—
494#41—21no—
495#41—1221no—
496#41—1221no—
497#41—1221no—
498#41—2441no—
499#41—2441no—
500#41—2441no—
501#41—2441no—
502#41—2441no—
503#41—2441no—
504C2×Q8—21no—
505C2×Q8—1221no—
506C2×D4—21no—
507C2×D4—1221no—
508C2×D4—1221no—
509C2×D4—1221no—
510C2×D4—2442no—
511C2×D4—2441no—
512C2×D4—2441no—
513C2×D4—2441no—
514C2×D4—2441no—
515C2×D4—2441no—
516C2×D4—2441no—
517C2×D4—2442no—
518C2×D4—2441no—
519C22×C4—611no—
520C22×C4—2441no—
521C22×C4—2441no—
522Q16—21no—
523Q16—21no—
524Q16—611no—
525Q16—1221no—
526Q16D4×D6111yescharacteristic
527SD16—21no—
528SD16—41no—
529SD16—41no—
530SD16—1221no—
531SD16—1221no—
532SD16—1221no—
533SD16—2441no—
534SD16—2441no—
535SD16—2441no—
536SD16—2441no—
537D8—21no—
538D8—21no—
539D8—611no—
540D8—1221no—
541D8—1221no—
542D8—1221no—
543D8—1221no—
544D8—1221no—
545D8—2441no—
546D8—2441no—
547D8—2441no—
548D8—2441no—
549D8—2441no—
550D8—2441no—
551D8D4×D6111yescharacteristic
552M16—21no—
553M16—41no—
554M16—611no—
555M16—611no—
556M16—611no—
557M16—611no—
558M16—1221no—
559M16—2441no—
560M16—2441no—
561M16—2441no—
562M16—2441no—
563C2×C8—41no—
564C2×C8—611no—
565C2×C8—611no—
566C2×C8—611no—
567C2×C8—611no—
568C2×C8—611no—
569C2×C8—611no—
570C2×C8—2441no—
571C2×C8—2441no—
572C2×C8—2441no—
573C2×C8—2441no—
574C2×C8D4×D6111yescharacteristic
575C2×C8C23×D6111yescharacteristic
576C4⋊C4—1221no—
577C22⋊C4—42no—
578C22⋊C4—1221no—
579C22⋊C4—2442no—
580C42—611no—
581C42D4×D6111yescharacteristic
582C16—21no—
583C16—21no—
584C16—1221no—
585C16—1221no—
586C23—41no—
587C23—611no—
588C23—9762no—
589C23—4881no—
590C23—4881no—
591C23—2441no—
592C23—2441no—
593C23—2441no—
594Q8—21no—
595Q8—41no—
596Q8—41no—
597Q8—1221no—
598Q8—2441no—
599D4—21no—
600D4—41no—
601D4—41no—
602D4—41no—
603D4—1221no—
604D4—9762no—
605D4—9762no—
606D4—4881no—
607D4—4881no—
608D4—4881no—
609D4—4881no—
610D4—2441no—
611D4—2441no—
612D4—2441no—
613D4—2441no—
614D4—2441no—
615D4—2441no—
616D4—2441no—
617D4—2441no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—611no—
622C2×C4—611no—
623C2×C4—611no—
624C2×C4—611no—
625C2×C4—611no—
626C2×C4—1221no—
627C2×C4—4881no—
628C2×C4—4881no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4C2×D4×D6111yescharacteristic
634C8—41no—
635C8—611no—
636C8—611no—
637C8—611no—
638C8—611no—
639C8—2441no—
640C8D8×D6111yescharacteristic
641C8D8×D6111yescharacteristic
642C8C2×D4×D6111yescharacteristic
643C8#41×D6111yescharacteristic
644C22—41no—
645C22—82no—
646C22—81no—
647C22—611no—
648C22—611no—
649C22—9761no—
650C22—9762no—
651C22—9762no—
652C22—4882no—
653C22—4882no—
654C22—4881no—
655C22—4881no—
656C22—2441no—
657C22—2441no—
658C22AGL(1,ℤ/8)×D6111yescharacteristic
659C4—41no—
660C4—41no—
661C4—81no—
662C4—611no—
663C4—611no—
664C4—2441no—
665C4—2441no—
666C4#121×D6111yescharacteristic
667C4C2×D8×D6111yescharacteristic
668C2—21no—
669C2—81no—
670C2—161no—
671C2—1221no—
672C2—1221no—
673C2—9761no—
674C2—9761no—
675C2—4881no—
676C2#459×D6111yescharacteristic
subgroupG/H
centerC2#459×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2212C2
socleC2×C61C2×#459
hypercenter#2212D61
hypocenterC61#33595
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2212

Complements

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

NH ≅ G/Ntype
1. #21458954C2semidirect
2. #509×D61C2semidirect
3. #21456301C2semidirect
4. #21454731C2semidirect
5. #21447896C2semidirect
6. #21447895C2semidirect
7. #21447894C2semidirect
8. #21447893C2semidirect
9. #21447891C2semidirect
10. #21447890C2semidirect
11. #21447885C2semidirect
12. #21447583C2semidirect
13. #21447567C2semidirect
14. #361×D61C2semidirect
15. C61×#2212C2semidirect
23. #16840203C22semidirect
25. #16840200C22semidirect
30. #16840197C22semidirect
40. #16840147C22semidirect
41. #16840139C22semidirect
42. #16840137C22semidirect
45. #16839880C22semidirect
46. #16839879C22semidirect
47. #16839878C22semidirect
51. #16839583C22semidirect
52. #16839582C22semidirect
53. #16839582C22semidirect
54. #16839578C22semidirect
55. #16839576C22semidirect
56. #16839576C22semidirect
57. #16839575C22semidirect
58. #16839574C22semidirect
59. #16839574C22semidirect
60. C61×#510C22semidirect
62. C61×#495C22semidirect
63. C61×#443C22semidirect
64. C61×#364C22semidirect
65. C61×#362C22semidirect
66. C61×#361C22semidirect
78. #14519800D4semidirect
82. #14519796D4semidirect
83. Q16×D61D4semidirect
96. D8×D61D4semidirect
341. #2212D61semidirect
362. #510C2×D61semidirect
381. #495C2×D61semidirect
389. #443C2×D61semidirect
396. #364C2×D61semidirect
398. #362C2×D61semidirect
404. #361C2×D61semidirect
405. C61#33595semidirect

Maximal subgroups

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

subgroupindex
#214589542
#509×D612
#214563012
#214547312
#214478962
#214478952
#214478942
#214478932
#214478912
#214478902
#214478852
#214475832
#214475672
#361×D612
C61×#22122
#3359561