Finite Group Databasebeta

← #29663151

Subgroups — #29663151

Order 15616.

714 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456946C211yesmaximal, characteristic, complemented
2#21456736C211yesmaximal, characteristic
3#486×D61C211yesmaximal, characteristic, complemented
4#21454528C211yesmaximal, characteristic, complemented
5#21445353C211yesmaximal, characteristic, complemented
6#21445352C211yesmaximal, characteristic, complemented
7#21445349C211yesmaximal, characteristic, complemented
8#21445231C211yesmaximal, characteristic, complemented
9#21445230C211yesmaximal, characteristic, complemented
10#21445227C211yesmaximal, characteristic, complemented
11#21445226C211yesmaximal, characteristic, complemented
12#21444566C211yesmaximal, characteristic, complemented
13#21444498C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic, complemented
15C61×#1697C211yesmaximal, characteristic, complemented
16C2×#14519785—21no—
17#16840309C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19#127×D61C2211yescharacteristic
20#16840159—21no—
21#16840137—21no—
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840107—21no—
26#16840107—21no—
27#16840107—21no—
28#16840107—21no—
29#16840082—21no—
30#16840079—21no—
31#16840053C2211yescharacteristic, complemented
32#16840052C2211yescharacteristic, complemented
33#16840049C2211yescharacteristic, complemented
34#16839958C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839954C2211yescharacteristic
37#16839897—21no—
38#16839894—21no—
39#16839841C2211yescharacteristic, complemented
40#16839838—21no—
41D4×#92418C2211yescharacteristic, complemented
42#16839827C2211yescharacteristic, complemented
43#16839825—21no—
44#16839814C2211yescharacteristic
45#16839814C2211yescharacteristic
46#16839784C2211yescharacteristic
47#102×D61—21no—
48#102×D61—21no—
49#16839727—21no—
50C4×C8×D61C2211yescharacteristic
51#16839515C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839510C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#476C2211yescharacteristic, complemented
62C61×#459C2211yescharacteristic
63C61×#436C2211yescharacteristic, complemented
64C61×#335C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×#11758222—21no—
68C2×Q8×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61—21no—
72C2×D488—21no—
73C2×#11758199—21no—
74C2×#11758199—21no—
75#14519785—44no—
76C2×#11758189—21no—
77C2×#11758189—21no—
78C22×#92418—21no—
79C2×C8×D61—21no—
80C2×C8×D61D411yescharacteristic
81C2×C8×D61D411yescharacteristic
82#14519777C2311yescharacteristic
83#14519776—21no—
84#14519776—21no—
85#14519776D411yescharacteristic
86#14519775—21no—
87#14519774C2311yescharacteristic
88#14519773—21no—
89#14519773—21no—
90#14519773—21no—
91#14519773D411yescharacteristic
92#14519763—21no—
93#14519752D411yescharacteristic
94#14519750—21no—
95#14519748D411yescharacteristic
96C4×D244C2311yescharacteristic
97C4×C22⋊D61—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61—21no—
100C42×D61C2311yescharacteristic
101#14519709—21no—
102#14519708—21no—
103#14519708—21no—
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107#14519706—21no—
108#14519699—21no—
109#14519699—21no—
110#14519699—21no—
111#14519699—21no—
112#14519698—21no—
113#14519698—21no—
114#14519698—21no—
115#14519698—21no—
116#14519697—21no—
117#14519697—21no—
118#14519697—21no—
119#14519697—21no—
120C22⋊#92418—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×D8×C61—21no—
131C2×M16×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133#127×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—21no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158Q8×D61—82no—
159D4×D61—42no—
160D4×D61—42no—
161D4×D61—81no—
162D4×D61—81no—
163D4×D61—82no—
164#11758222—42no—
165#11758222—42no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244C2×D411yescharacteristic
171C2×C4×D61—21no—
172C2×C4×D61—41no—
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61#4111yescharacteristic
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—41no—
179D8×C61—81no—
180M16×C61—42no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C42×C61C2411yescharacteristic
191C22⋊Dic61—21no—
192#11758199—81no—
193#11758199—81no—
194#11758198—21no—
195#11758198—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—21no—
200#11758196—41no—
201#11758196—41no—
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61#4111yescharacteristic
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—42no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211D488—81no—
212#11758189—42no—
213#11758189—42no—
214C8×D61—22no—
215C8×D61—22no—
216C8×D61—42no—
217C23×C61—21no—
218C23×C61—41no—
219C22×D61—21no—
220C22×D61—41no—
221C22×D61—82no—
222C22×D61—82no—
223C22×D61—82no—
224C22×D61#12111yescharacteristic
225Q8×C61—81no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—81no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C22⋊D61—42no—
241C22⋊D61—42no—
242C22⋊D61—42no—
243C22⋊D61—82no—
244C22⋊D61—82no—
245C22⋊D61—82no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61#12111yescharacteristic
249C2×Dic61#12111yescharacteristic
250C2×Dic61#12111yescharacteristic
251D244—21no—
252D244—21no—
253D244—42no—
254D244—41no—
255D244—41no—
256D244—41no—
257D244—41no—
258D244—81no—
259C4×D61—21no—
260C4×D61—21no—
261C4×D61—22no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—82no—
265C4×D61#13512yes—
266C4×D61AGL(1,ℤ/8)12yes—
267Dic122—42no—
268Dic122—82no—
269Dic122—81no—
270C8×C61—21no—
271C8×C61—21no—
272C8×C61—41no—
273C8×C61—41no—
274C8×C61—41no—
275#92418—21no—
276#92418—21no—
277#92418—41no—
278#92418—41no—
279#92418—41no—
280#19196—611nomaximal, Sylow
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61C2×#12111yescharacteristic
288C2×D61—21no—
289C2×D61—21no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—164no—
297C2×D61#46312yes—
298C4×C61—21no—
299C4×C61—41no—
300C4×C61—81no—
301C4×C61C2×#13511yescharacteristic
302C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
303C4×C61#57711yescharacteristic
304C4×C61#57711yescharacteristic
305Dic61—21no—
306Dic61—41no—
307Dic61—81no—
308Dic61#46312yes—
309Dic61#47212yes—
310#3227—611no—
311#3205—611no—
312C2×#486—611no—
313#2931—611no—
314#1697—611no—
315#1697—611no—
316#1697—611no—
317#1697D6111yescharacteristic, complemented
318#1677—611no—
319#1677—611no—
320#1677—611no—
321#1677—611no—
322#1564—611no—
323#1548—611no—
324#1545—611no—
325C2×C61—81no—
326C2×C61—161no—
327C2×C61C2×#46311yescharacteristic
328C2×C61#305711yescharacteristic
329C2×C61C2×#47211yescharacteristic
330D61—22no—
331D61—81no—
332D61—161no—
333C2×#131—1221no—
334#532—611no—
335C2×C4⋊Q8—611no—
336C2×#127—611no—
337#498—1221no—
338#493—1221no—
339#486—611no—
340#486—611no—
341#486—611no—
342#486—1221no—
343#486—1221no—
344#486—1221no—
345#486—1221no—
346#486C2×D6111yescharacteristic
347#481—1221no—
348#481—1221no—
349#476—611no—
350#476—611no—
351#476—611no—
352#476C2×D6111yescharacteristic, complemented
353#459—611no—
354#459—611no—
355#459—611no—
356#459C2×D6111yescharacteristic
357#447—1221no—
358#447—1221no—
359#436—611no—
360#436—1221no—
361#436C2×D6111yescharacteristic, complemented
362C8×D4—611no—
363C8×D4—611no—
364C8×D4—1221no—
365#431—611no—
366#431—611no—
367#423—611no—
368C2×#102—1221no—
369C2×#102—1221no—
370#408—1221no—
371C2×C4×C8—611no—
372#335—611no—
373#335C2×D6111yescharacteristic, complemented
374#334—611no—
375#334—611no—
376#333—611no—
377#333—611no—
378#333—611no—
379#333C2×D6111yescharacteristic, complemented
380#331—611no—
381#331—611no—
382#331—611no—
383#331C2×D6111yescharacteristic
384C61#1919611yescharacteristic, complemented, Sylow
385C2×#41—1221no—
386C22×Q8—1221no—
387C22×D4—1221no—
388C22×D4—1221no—
389C22×D4—1221no—
390C2×D8—21no—
391C2×D8—1221no—
392C2×D8—1221no—
393C2×D8—1221no—
394#131—2444no—
395C2×M16—21no—
396C2×M16—1221no—
397C2×M16—1221no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—1221no—
401C22×C8—1221no—
402C4⋊Q8—611no—
403C4⋊Q8—611no—
404C4⋊Q8—1221no—
405C4⋊Q8—1221no—
406C4⋊Q8—1221no—
407C4⋊Q8C22×D6111yescharacteristic
408#127—611no—
409#127—611no—
410#127—1221no—
411#127—1221no—
412#127—1221no—
413#127C22×D6111yescharacteristic
414#124—1221no—
415#121—611no—
416#121—611no—
417#121—1221no—
418C4×D4—611no—
419C4×D4—1221no—
420C4×D4C22×D6111yescharacteristic
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C42—611no—
424#106—21no—
425#106—1221no—
426#106—1221no—
427#106—1221no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—1221no—
432C4⋊C8C22×D6111yescharacteristic
433C4⋊C8C22×D6111yescharacteristic
434C4⋊C8C22×D6111yescharacteristic
435#102—21no—
436#102—21no—
437#102—21no—
438#102—21no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450#102—1221no—
451C22⋊C8—21no—
452C22⋊C8—1221no—
453C22⋊C8—1221no—
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457C4×C8—611no—
458C4×C8—611no—
459C4×C8—611no—
460C4×C8—1221no—
461C4×C8—1221no—
462C4×C8C22×D6111yescharacteristic
463C24—2441no—
464#41—2442no—
465#41—2442no—
466C2×Q8—21no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—4882no—
470C2×Q8—2441no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—611no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—4881no—
483C2×D4—4881no—
484C2×D4—4882no—
485C2×D4—2442no—
486C2×D4—2442no—
487C2×D4—2441no—
488C2×D4—2441no—
489C2×D4—2441no—
490C2×D4D4×D6111yescharacteristic
491C22×C4—21no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1221no—
496C22×C4—2441no—
497D8—81no—
498D8—4881no—
499D8—4881no—
500D8—4881no—
501M16—42no—
502M16—2442no—
503M16—2442no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—1222no—
510C2×C8—1222no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—2442no—
515C2×C8—2442no—
516C2×C8D4×D6111yescharacteristic
517C2×C8D4×D6111yescharacteristic
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—611no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C4⋊C4D4×D6111yescharacteristic
527C22⋊C4—21no—
528C22⋊C4—1221no—
529C22⋊C4—1221no—
530C22⋊C4—1221no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42C23×D6111yescharacteristic
535C23—21no—
536C23—41no—
537C23—611no—
538C23—1221no—
539C23—4882no—
540C23—4882no—
541C23—4882no—
542C23—2441no—
543Q8—81no—
544Q8—4881no—
545Q8—4882no—
546Q8—2442no—
547D4—21no—
548D4—21no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—81no—
554D4—1221no—
555D4—1221no—
556D4—4881no—
557D4—4882no—
558D4—4882no—
559D4—4882no—
560D4—2442no—
561D4—2442no—
562D4—2442no—
563D4—2442no—
564D4—2441no—
565D4—2441no—
566D4—2441no—
567D4—2441no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—611no—
577C2×C4—1222no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—4882no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4C2×D4×D6111yescharacteristic
586C2×C4C2×D4×D6111yescharacteristic
587C2×C4#41×D6111yescharacteristic
588C8—21no—
589C8—21no—
590C8—41no—
591C8—41no—
592C8—41no—
593C8—1221no—
594C8—1221no—
595C8—2441no—
596C8—2441no—
597C8—2441no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—612no—
605C22—1221no—
606C22—1221no—
607C22—9764no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—2441no—
612C22—2441no—
613C22—2441no—
614C22#121×D6111yescharacteristic
615C4—21no—
616C4—41no—
617C4—81no—
618C4—612no—
619C4—612no—
620C4—1221no—
621C4—4881no—
622C4—2441no—
623C4#135×D6111yescharacteristic
624C4AGL(1,ℤ/8)×D6111yescharacteristic
625C4#1684057111yescharacteristic
626C4#1684057211yescharacteristic
627C2—81no—
628C2—161no—
629C2—1222no—
630C2—9761no—
631C2—4881no—
632C2#463×D6111yescharacteristic
633C2#2145545611yescharacteristic
634C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1697C2
socleC22×C61C2×#121
prime radical#21456736C2
hypercenter#1697D61
hypocenterC61#19196
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1697

Complements

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

NH ≅ G/Ntype
1. #21456946C2semidirect
3. #486×D61C2semidirect
4. #21454528C2semidirect
5. #21445353C2semidirect
6. #21445352C2semidirect
7. #21445349C2semidirect
8. #21445231C2semidirect
9. #21445230C2semidirect
10. #21445227C2semidirect
11. #21445226C2semidirect
12. #21444566C2semidirect
13. #21444498C2semidirect
14. #21444485C2semidirect
15. C61×#1697C2semidirect
31. #16840053C22semidirect
32. #16840052C22semidirect
33. #16840049C22semidirect
39. #16839841C22semidirect
41. D4×#92418C22semidirect
42. #16839827C22semidirect
51. #16839515C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839510C22semidirect
56. #16839510C22semidirect
61. C61×#476C22semidirect
63. C61×#436C22semidirect
64. C61×#335C22semidirect
65. C61×#333C22semidirect
317. #1697D61semidirect
352. #476C2×D61semidirect
361. #436C2×D61semidirect
373. #335C2×D61semidirect
379. #333C2×D61semidirect
384. C61#19196semidirect

Maximal subgroups

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

subgroupindex
#214569462
#214567362
#486×D612
#214545282
#214453532
#214453522
#214453492
#214452312
#214452302
#214452272
#214452262
#214445662
#214444982
#214444852
C61×#16972
#1919661