Finite Group Databasebeta

← #29836474

Subgroups — #29836474

Order 15616.

760 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840197C211yesmaximal, characteristic, complemented
2C2×D16×D61C211yesmaximal, characteristic, complemented
3#21457498C211yesmaximal, characteristic, complemented
4#21457493C211yesmaximal, characteristic, complemented
5#21448084C211yesmaximal, characteristic, complemented
6#21447920C211yesmaximal, characteristic, complemented
7#21447918C211yesmaximal, characteristic, complemented
8#21447877C211yesmaximal, characteristic, complemented
9#21447852C211yesmaximal, characteristic, complemented
10#21447746C211yesmaximal, characteristic, complemented
11#21447733C211yesmaximal, characteristic, complemented
12#21447732C211yesmaximal, characteristic, complemented
13#21447692C211yesmaximal, characteristic, complemented
14#21447647C211yesmaximal, characteristic, complemented
15C61×#2186C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic, complemented
17C2×D8×D61C2211yescharacteristic
18C2×D8×D61C2211yescharacteristic, complemented
19#16840400—21no—
20#16840396—21no—
21#16840197—24no—
22C2×#14519720C2211yescharacteristic, complemented
23C2×#14519720C2211yescharacteristic, complemented
24C2×D976C2211yescharacteristic
25C2×D976C2211yescharacteristic, complemented
26C2×#14519718C2211yescharacteristic
27C2×#14519718C2211yescharacteristic, complemented
28C2×#14519718C2211yescharacteristic, complemented
29D16×D61—24no—
30C2×#14519717C2211yescharacteristic, complemented
31C2×C16×D61C2211yescharacteristic, complemented
32#16840147C2211yescharacteristic, complemented
33#16840137C2211yescharacteristic, complemented
34#16840000C2211yescharacteristic, complemented
35#16839999C2211yescharacteristic
36#16839998C2211yescharacteristic, complemented
37#16839979—21no—
38#16839956—21no—
39#16839911—21no—
40#16839907—21no—
41#16839897—21no—
42#16839895—21no—
43D8×Dic61C2211yescharacteristic, complemented
44#16839844C2211yescharacteristic, complemented
45C8×C22⋊D61C2211yescharacteristic
46#16839589C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839564C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839563C2211yescharacteristic, complemented
51#16839562C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839537C2211yescharacteristic, complemented
54C2×#135×C61C2211yescharacteristic, complemented
55C2×SD32×C61C2211yescharacteristic, complemented
56C2×D16×C61C2211yescharacteristic, complemented
57C61×#466C2211yescharacteristic, complemented
58C61×#358C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic, complemented
60C61×C22⋊C16C2211yescharacteristic, complemented
61C22×D244—21no—
62C2×D4×D61—21no—
63C2×D4×D61—21no—
64C2×D4×D61—41no—
65C2×#11758201—21no—
66C2×D488—22no—
67C2×D488—21no—
68C2×D488—41no—
69C2×D488C2311yescharacteristic
70C2×D488C2311yescharacteristic
71C2×D488C2311yescharacteristic
72C2×#11758199—21no—
73C2×#11758199—21no—
74C2×#11758199—21no—
75D8×D61—22no—
76D8×D61—22no—
77D8×D61—41no—
78D8×D61—41no—
79C2×C8×D61C2311yescharacteristic
80#14519773—21no—
81#14519766—21no—
82#14519760—41no—
83#14519753—21no—
84#14519752—21no—
85#14519751—21no—
86#14519750—41no—
87#14519749—41no—
88#14519747—41no—
89#14519745—41no—
90#14519745—41no—
91D4×Dic61—21no—
92#14519739—21no—
93C4×C22⋊D61D411yescharacteristic
94C2×#11758198—41no—
95#14519720—22no—
96#14519720—22no—
97D976—22no—
98D976—41no—
99#14519718—22no—
100#14519718—22no—
101#14519718D414yescomplemented
102#14519717—22no—
103C2×#11758185C2311yescharacteristic
104C2×#11758185C2311yescharacteristic
105C16×D61—22no—
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519701—21no—
109#14519699—21no—
110#14519698—21no—
111#14519697—21no—
112#14519697—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×#41×C61—21no—
117#135×C61—22no—
118#135×C61—41no—
119C2×Q16×C61C2311yescharacteristic
120C2×SD16×C61—21no—
121C2×D8×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124#121×C61—21no—
125SD32×C61—41no—
126D16×C61—22no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#102×C61—21no—
131C23×D61—41no—
132C23×D61—41no—
133C23×D61—41no—
134#41×C61—42no—
135#41×C61—81no—
136C2×Q8×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C22×C4×C61—41no—
142C22×C4×C61C2×D411yescharacteristic
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C2×C22⋊D61—41no—
148C2×C22⋊D61D811yescharacteristic
149C22×Dic61—41no—
150D4×D61—42no—
151D4×D61—42no—
152D4×D61—81no—
153D4×D61—81no—
154D4×D61—84no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—42no—
159C2×D244—41no—
160C2×D244—41no—
161C2×D244—81no—
162C2×C4×D61—41no—
163C2×C4×D61C2×D411yescharacteristic
164Q16×C61—21no—
165Q16×C61—21no—
166SD16×C61—42no—
167D8×C61—21no—
168D8×C61—21no—
169D8×C61—21no—
170D8×C61C2×D412yescomplemented
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2411yescharacteristic
175C4⋊C4×C61—21no—
176C22⋊C4×C61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61D811yescharacteristic
179#11758201—81no—
180#11758199—42no—
181#11758199—42no—
182#11758199—81no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186#11758198—41no—
187#11758198—41no—
188#11758198—41no—
189#11758198D811yescharacteristic
190C4⋊Dic61—21no—
191#11758196—41no—
192#11758196D811yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195C2×#92418C2×D411yescharacteristic
196D488—21no—
197D488—21no—
198D488—21no—
199D488—21no—
200D488—21no—
201D488—42no—
202D488C2×D412yes—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61—21no—
207#11758185—21no—
208#11758185C2×D412yescomplemented
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C2×D811yescharacteristic
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61—82no—
221C22×D61—161no—
222C22×D61—161no—
223C22×D61—161no—
224C22×D61C2×D811yescharacteristic
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—22no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—82no—
232D4×C61—82no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—82no—
238C2×C4×C61C2×D811yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C22⋊D61—22no—
241C22⋊D61—82no—
242C22⋊D61—82no—
243C22⋊D61—82no—
244C22⋊D61—161no—
245C22⋊D61—161no—
246C2×Dic61—41no—
247C2×Dic61—161no—
248C2×Dic61C2×D811yescharacteristic
249C2×Dic61C2×D811yescharacteristic
250C2×Dic61C2×D811yescharacteristic
251D244—22no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—41no—
257D244—82no—
258D244—82no—
259C4×D61—21no—
260C4×D61—21no—
261C4×D61—82no—
262C8×C61—21no—
263C8×C61C22×D411yescharacteristic
264C8×C61H(5,𝔽2)11yescharacteristic
265#92418—21no—
266#92418—21no—
267#33830—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—42no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61C22×D811yescharacteristic
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—42no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—164no—
290C2×D61—164no—
291C2×D61—164no—
292C2×D61#50912yescomplemented
293C4×C61—21no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61D4211yescharacteristic
297C4×C61#57611yescharacteristic
298Dic61—21no—
299Dic61—21no—
300Dic61—21no—
301Dic61—81no—
302C2×#509—611no—
303C22×D16—611no—
304#3280—611no—
305D4×D8—611no—
306#2250—611no—
307#2218—611no—
308#2217—611no—
309#2210—611no—
310#2206—611no—
311#2187—611no—
312#2186D6111yescharacteristic, complemented
313#2184—611no—
314#2184—611no—
315#2177—611no—
316#2168—611no—
317C2×C61—41no—
318C2×C61—82no—
319C2×C61—161no—
320C2×C61D4×D811yescharacteristic
321C2×C61C2×#50911yescharacteristic
322C2×C61#341511yescharacteristic
323D61—22no—
324D61—82no—
325D61—161no—
326D61—161no—
327C2×#135C2×D6111yescharacteristic, complemented
328C22×D8—611no—
329C22×D8—611no—
330C22×D8—611no—
331#546—1221no—
332D42—1221no—
333#509—1224no—
334C2×SD32—611no—
335C2×SD32—611no—
336C2×SD32C2×D6111yescharacteristic, complemented
337C2×D16—611no—
338C2×D16—611no—
339C2×D16—611no—
340C2×D16—611no—
341C2×D16—611no—
342C2×D16—1224no—
343C2×D16C2×D6111yescharacteristic, complemented
344C2×M32—611no—
345C22×C16—611no—
346#495—611no—
347#493—611no—
348#467—611no—
349#466—611no—
350#466—611no—
351#466C2×D6111yescharacteristic, complemented
352#463—1221no—
353#459—1221no—
354#450—1221no—
355#449—1221no—
356#447—1221no—
357#447—1221no—
358C4×D8—611no—
359C4×D8—611no—
360C8×D4—611no—
361#367—611no—
362C4⋊C16—611no—
363#358C2×D6111yescharacteristic, complemented
364#357—611no—
365#357—611no—
366#357—611no—
367#357—611no—
368#357C2×D6111yescharacteristic, complemented
369C22⋊C16—611no—
370C22⋊C16C2×D6111yescharacteristic, complemented
371#346—611no—
372C61#3383011yescharacteristic, complemented, Sylow
373C2×#41—21no—
374C22×D4—1221no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4—2441no—
378#135—22no—
379#135—41no—
380C2×Q16C22×D6111yescharacteristic
381C2×SD16—21no—
382C2×SD16—1221no—
383C2×D8—611no—
384C2×D8—611no—
385C2×D8—611no—
386C2×D8—1222no—
387C2×D8—1221no—
388C2×D8—1221no—
389C2×D8—1221no—
390C2×D8—1222no—
391C2×D8—1222no—
392C2×D8—1221no—
393C2×D8—2441no—
394C2×D8—2441no—
395C2×D8—2441no—
396C2×D8C22×D6111yescharacteristic
397C2×D8C22×D6111yescharacteristic
398C22×C8—611no—
399C22×C8C22×D6111yescharacteristic
400#127—1221no—
401#124—1221no—
402#123—2441no—
403C22⋊Q8—1221no—
404#121—21no—
405#121—1221no—
406#121—1221no—
407#121—2441no—
408#121—2441no—
409#120—2441no—
410#120—2441no—
411#120—2441no—
412C4×D4—611no—
413C4×D4—1221no—
414C4×D4—1221no—
415C2×C22⋊C4—2441no—
416SD32—41no—
417SD32—1222no—
418SD32—1222no—
419D16—22no—
420D16—614no—
421D16—1222no—
422D16—1222no—
423D16—1222no—
424D16—2441no—
425M32—1222no—
426C2×C16—611no—
427C2×C16—611no—
428C2×C16—1222no—
429C2×C16C22×D6111yescharacteristic
430C2×C16C22×D6111yescharacteristic
431#107—611no—
432#107C22×D6111yescharacteristic
433C4⋊C8—611no—
434#103—1221no—
435#102—21no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440C22⋊C8—611no—
441C22⋊C8—611no—
442C4×C8—611no—
443C24—2441no—
444C24—2441no—
445C24—2441no—
446#41—42no—
447#41—81no—
448C2×Q8—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—611no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—4881no—
458C2×D4—4881no—
459C2×D4—4881no—
460C2×D4—4884no—
461C2×D4—2442no—
462C2×D4—2441no—
463C2×D4—2441no—
464C2×D4—2441no—
465C2×D4—2441no—
466C2×D4—2441no—
467C2×D4—2442no—
468C2×D4—2442no—
469C2×D4—2441no—
470C2×D4—2441no—
471C22×C4—41no—
472C22×C4—611no—
473C22×C4—2441no—
474C22×C4—2441no—
475C22×C4D4×D6111yescharacteristic
476Q16—21no—
477Q16—21no—
478SD16—42no—
479SD16—4881no—
480D8—21no—
481D8—21no—
482D8—21no—
483D8—612no—
484D8—1221no—
485D8—1221no—
486D8—1221no—
487D8—1221no—
488D8—1221no—
489D8—4881no—
490D8—2442no—
491D8—2442no—
492D8—2442no—
493D8D4×D6112yescomplemented
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8D4×D6111yescharacteristic
501C2×C8C23×D6111yescharacteristic
502C4⋊C4—21no—
503C4⋊C4—611no—
504C4⋊C4—1221no—
505C4⋊C4—2441no—
506C22⋊C4—41no—
507C22⋊C4—611no—
508C22⋊C4—611no—
509C22⋊C4—2441no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C42—611no—
517C16—21no—
518C16—21no—
519C16—612no—
520C16—1221no—
521C23—41no—
522C23—41no—
523C23—611no—
524C23—9761no—
525C23—9761no—
526C23—9761no—
527C23—4882no—
528C23—4882no—
529C23—4882no—
530C23—4882no—
531C23—4882no—
532C23—4882no—
533C23—2441no—
534C23—2441no—
535C23—2441no—
536C23D8×D6111yescharacteristic
537Q8—41no—
538Q8—41no—
539D4—22no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—82no—
544D4—82no—
545D4—1222no—
546D4—1222no—
547D4—9761no—
548D4—9761no—
549D4—4882no—
550D4—4882no—
551D4—4882no—
552D4—4882no—
553D4—4882no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—82no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—611no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—9761no—
570C2×C4—4882no—
571C2×C4—2441no—
572C2×C4D8×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C8—21no—
575C8—1221no—
576C8—1221no—
577C8C2×D4×D6111yescharacteristic
578C8#1451982211yescharacteristic
579C22—21no—
580C22—21no—
581C22—21no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—612no—
589C22—1221no—
590C22—1221no—
591C22—9764no—
592C22—9764no—
593C22—9764no—
594C22—4881no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—4881no—
601C22—4881no—
602C22—2442no—
603C22C2×D8×D6111yescharacteristic
604C4—21no—
605C4—81no—
606C4—81no—
607C4—1221no—
608C4—1221no—
609C4—1221no—
610C4—4881no—
611C4#1684039411yescharacteristic
612C4#1684056611yescharacteristic
613C2—41no—
614C2—82no—
615C2—161no—
616C2—1222no—
617C2—9761no—
618C2—9761no—
619C2—4882no—
620C2#2145748911yescharacteristic
621C2#509×D6111yescharacteristic
622C2#2145895711yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2186C2
socleC22×C61C22×D8
hypercenter#2186D61
hypocenterC61#33830
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2186

Complements

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

NH ≅ G/Ntype
1. C2×#16840197C2semidirect
2. C2×D16×D61C2semidirect
3. #21457498C2semidirect
4. #21457493C2semidirect
5. #21448084C2semidirect
6. #21447920C2semidirect
7. #21447918C2semidirect
8. #21447877C2semidirect
9. #21447852C2semidirect
10. #21447746C2semidirect
11. #21447733C2semidirect
12. #21447732C2semidirect
13. #21447692C2semidirect
14. #21447647C2semidirect
15. C61×#2186C2semidirect
16. C22×D488C22semidirect
18. C2×D8×D61C22semidirect
22. C2×#14519720C22semidirect
23. C2×#14519720C22semidirect
25. C2×D976C22semidirect
27. C2×#14519718C22semidirect
28. C2×#14519718C22semidirect
30. C2×#14519717C22semidirect
31. C2×C16×D61C22semidirect
32. #16840147C22semidirect
33. #16840137C22semidirect
34. #16840000C22semidirect
36. #16839998C22semidirect
43. D8×Dic61C22semidirect
44. #16839844C22semidirect
46. #16839589C22semidirect
47. #16839580C22semidirect
48. #16839564C22semidirect
49. #16839564C22semidirect
50. #16839563C22semidirect
51. #16839562C22semidirect
52. #16839541C22semidirect
53. #16839537C22semidirect
54. C2×#135×C61C22semidirect
55. C2×SD32×C61C22semidirect
56. C2×D16×C61C22semidirect
57. C61×#466C22semidirect
58. C61×#358C22semidirect
59. C61×#357C22semidirect
60. C61×C22⋊C16C22semidirect
101. #14519718D4semidirect
170. D8×C61C2×D4semidirect
208. #11758185C2×D4semidirect
292. C2×D61#509semidirect
312. #2186D61semidirect
327. C2×#135C2×D61semidirect
336. C2×SD32C2×D61semidirect
343. C2×D16C2×D61semidirect
351. #466C2×D61semidirect
363. #358C2×D61semidirect
368. #357C2×D61semidirect
370. C22⋊C16C2×D61semidirect
372. C61#33830semidirect
493. D8D4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401972
C2×D16×D612
#214574982
#214574932
#214480842
#214479202
#214479182
#214478772
#214478522
#214477462
#214477332
#214477322
#214476922
#214476472
C61×#21862
#3383061