Finite Group Databasebeta

← #29836466

Subgroups — #29836466

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×#16840196C211yesmaximal, characteristic, complemented
2C2×D16×D61C211yesmaximal, characteristic, complemented
3#21457505C211yesmaximal, characteristic, complemented
4#21457494C211yesmaximal, characteristic, complemented
5#21448083C211yesmaximal, characteristic, complemented
6#21447921C211yesmaximal, characteristic, complemented
7#21447914C211yesmaximal, characteristic, complemented
8#21447875C211yesmaximal, characteristic, complemented
9#21447749C211yesmaximal, characteristic, complemented
10#21447739C211yesmaximal, characteristic, complemented
11#21447732C211yesmaximal, characteristic, complemented
12#21447729C211yesmaximal, characteristic, complemented
13#21447693C211yesmaximal, characteristic, complemented
14#21447643C211yesmaximal, characteristic, complemented
15C61×#2206C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic, complemented
17C2×D8×D61C2211yescharacteristic
18C2×D8×D61C2211yescharacteristic, complemented
19C2×D8×D61C2211yescharacteristic, complemented
20#16840407—21no—
21#16840397—21no—
22#16840196—24no—
23C2×#14519722C2211yescharacteristic, complemented
24C2×#14519721C2211yescharacteristic, complemented
25C2×#14519721C2211yescharacteristic, complemented
26C2×D976C2211yescharacteristic
27C2×#14519718C2211yescharacteristic
28C2×#14519718C2211yescharacteristic, complemented
29C2×#14519718C2211yescharacteristic, complemented
30D16×D61—24no—
31C2×#14519716C2211yescharacteristic, complemented
32C2×C16×D61C2211yescharacteristic, complemented
33#16840146C2211yescharacteristic, complemented
34#16840004C2211yescharacteristic, complemented
35#16839999C2211yescharacteristic, complemented
36#16839995C2211yescharacteristic
37#16839995C2211yescharacteristic, complemented
38#16839977—21no—
39#16839954—21no—
40#16839914—21no—
41#16839908—21no—
42#16839894—21no—
43#16839894—21no—
44C4×D488C2211yescharacteristic, complemented
45C8×D244C2211yescharacteristic
46#16839589C2211yescharacteristic, complemented
47#16839567C2211yescharacteristic, complemented
48#16839564C2211yescharacteristic, complemented
49#16839563C2211yescharacteristic, complemented
50#16839562C2211yescharacteristic, complemented
51#16839541C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839538C2211yescharacteristic, complemented
54C2×D16×C61C2211yescharacteristic, complemented
55C2×D16×C61C2211yescharacteristic, complemented
56C61×#493C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic, complemented
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C2×D4×D61—21no—
63C2×D4×D61—21no—
64C2×D4×D61—21no—
65C2×D4×D61—41no—
66#14519796—22no—
67#14519796—41no—
68C2×Dic244C2311yescharacteristic
69C2×#11758190—21no—
70C2×#11758200—21no—
71C2×D488C2311yescharacteristic
72C2×D488C2311yescharacteristic
73C2×#11758199—21no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76D8×D61—22no—
77D8×D61—22no—
78D8×D61—22no—
79D8×D61—41no—
80D8×D61—41no—
81D8×D61—41no—
82C2×C8×D61C2311yescharacteristic
83C2×C8×D61C2311yescharacteristic
84#14519765—21no—
85#14519761—41no—
86#14519757—21no—
87#14519752—21no—
88#14519751—41no—
89#14519748—21no—
90#14519748—21no—
91#14519748—41no—
92#14519747—41no—
93#14519745—41no—
94#14519745—41no—
95C4×D244—21no—
96C4×D244D411yescharacteristic
97C2×#11758198—41no—
98#14519722—22no—
99#14519721—22no—
100#14519721—41no—
101D976—22no—
102#14519718—22no—
103#14519718—41no—
104#14519718D414yescomplemented
105#14519716—22no—
106C2×#11758185C2311yescharacteristic
107C2×#11758185C2311yescharacteristic
108C16×D61—22no—
109#14519711C2311yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519700—21no—
112#14519699—21no—
113#14519698—21no—
114#14519698—21no—
115#14519697—21no—
116#14519689D411yescharacteristic
117#14519689D411yescharacteristic
118C2×D8×C61—21no—
119C2×D8×C61C2311yescharacteristic
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122#127×C61—21no—
123C4×D4×C61—21no—
124D16×C61—22no—
125D16×C61—22no—
126C2×C16×C61C2311yescharacteristic
127C2×C16×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C23×D61—41no—
133C23×D61—41no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—41no—
138C2×D4×C61—41no—
139C22×C4×C61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C22×Dic61—41no—
147#11758224—42no—
148#11758224—81no—
149D4×D61—42no—
150D4×D61—42no—
151D4×D61—42no—
152D4×D61—81no—
153D4×D61—81no—
154D4×D61—81no—
155D4×D61—84no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—41no—
159C2×D244D811yescharacteristic
160C2×C4×D61—41no—
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×Dic122—21no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61—21no—
167D8×C61—21no—
168D8×C61—21no—
169D8×C61—81no—
170D8×C61C2×D412yescomplemented
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2411yescharacteristic
173C4⋊C4×C61—21no—
174C22⋊C4×C61—41no—
175C42×C61C2×D411yescharacteristic
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178#11758200—42no—
179#11758199—42no—
180#11758199—42no—
181#11758199—42no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186#11758198—41no—
187#11758198D811yescharacteristic
188#11758198D811yescharacteristic
189C4⋊Dic61—21no—
190C4⋊Dic61—41no—
191C4⋊Dic61D811yescharacteristic
192C2×#92418C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194Dic244—21no—
195Dic244—21no—
196D488—21no—
197D488—21no—
198D488—21no—
199D488C2×D412yes—
200#11758190—81no—
201C8×D61—21no—
202C8×D61—21no—
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×C61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61—161no—
221C22×D61—161no—
222C22×D61—161no—
223C22×D61C2×D811yescharacteristic
224C22×D61C2×D811yescharacteristic
225D4×C61—22no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—82no—
232D4×C61—161no—
233C2×C4×C61—41no—
234C2×C4×C61—161no—
235C2×C4×C61C2×D811yescharacteristic
236C2×C4×C61C2×D811yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C22⋊D61—82no—
239C22⋊D61—82no—
240C22⋊D61—82no—
241C22⋊D61—82no—
242C22⋊D61—82no—
243C22⋊D61—82no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—82no—
247C2×Dic61C2×D811yescharacteristic
248C2×Dic61C2×D811yescharacteristic
249D244—22no—
250D244—22no—
251D244—41no—
252D244—41no—
253D244—41no—
254D244—161no—
255C4×D61—21no—
256C4×D61—21no—
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—82no—
260Dic122—41no—
261Dic122—41no—
262C8×C61—21no—
263C8×C61C22×D411yescharacteristic
264C8×C61H(5,𝔽2)11yescharacteristic
265#92418—21no—
266#92418—21no—
267#33830—611nomaximal, Sylow
268C22×C61—42no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
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—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—42no—
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—21no—
295C4×C61—81no—
296C4×C61D4211yescharacteristic
297C4×C61#57611yescharacteristic
298Dic61—21no—
299Dic61—21no—
300Dic61—81no—
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#2206D6111yescharacteristic, complemented
311#2187—611no—
312#2186—611no—
313#2184—611no—
314#2184—611no—
315#2177—611no—
316#2168—611no—
317C2×C61—82no—
318C2×C61—161no—
319C2×C61—161no—
320C2×C61D4×D811yescharacteristic
321C2×C61C2×#50911yescharacteristic
322C2×C61#341511yescharacteristic
323D61—22no—
324D61—41no—
325D61—82no—
326D61—161no—
327C2×#135—611no—
328C22×D8—611no—
329C22×D8—611no—
330C22×D8—611no—
331#546—1221no—
332D42—1221no—
333#509—1224no—
334C2×SD32—611no—
335C2×SD32—611no—
336C2×SD32—611no—
337C2×D16—611no—
338C2×D16—611no—
339C2×D16—611no—
340C2×D16—611no—
341C2×D16—1224no—
342C2×D16C2×D6111yescharacteristic, complemented
343C2×D16C2×D6111yescharacteristic, complemented
344C2×M32—611no—
345C22×C16—611no—
346#495—611no—
347#493C2×D6111yescharacteristic, complemented
348#467—611no—
349#466—611no—
350#466—611no—
351#466—611no—
352#463—1221no—
353#459—1221no—
354#450—1221no—
355#449—1221no—
356#447—1221no—
357#447—1221no—
358C4×D8—611no—
359C4×D8C2×D6111yescharacteristic, complemented
360C8×D4—611no—
361#367—611no—
362C4⋊C16C2×D6111yescharacteristic, complemented
363#358—611no—
364#357—611no—
365#357—611no—
366#357—611no—
367#357C2×D6111yescharacteristic, complemented
368#357C2×D6111yescharacteristic, complemented
369C22⋊C16—611no—
370C22⋊C16—611no—
371#346—611no—
372C61#3383011yescharacteristic, complemented, Sylow
373C2×#41—1221no—
374C22×D4—1221no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4—2441no—
378#135—1222no—
379#135—2441no—
380C2×Q16—611no—
381C2×SD16—1221no—
382C2×SD16—1221no—
383C2×D8—21no—
384C2×D8—611no—
385C2×D8—611no—
386C2×D8—1221no—
387C2×D8—1221no—
388C2×D8—1221no—
389C2×D8—1222no—
390C2×D8—1222no—
391C2×D8—1222no—
392C2×D8—2441no—
393C2×D8—2441no—
394C2×D8—2441no—
395C2×D8C22×D6111yescharacteristic
396C2×D8C22×D6111yescharacteristic
397C2×D8C22×D6111yescharacteristic
398C22×C8—611no—
399C22×C8—611no—
400#127—21no—
401#124—1221no—
402#123—2441no—
403C22⋊Q8—1221no—
404#121—1221no—
405#121—1221no—
406#121—1221no—
407#121—2441no—
408#121—2441no—
409#120—2441no—
410#120—2441no—
411#120—2441no—
412C4×D4—21no—
413C4×D4—611no—
414C4×D4—1221no—
415C2×C22⋊C4—2441no—
416SD32—1222no—
417SD32—1222no—
418SD32—2441no—
419D16—22no—
420D16—22no—
421D16—614no—
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×C8C22×D6111yescharacteristic
443C24—2441no—
444C24—2441no—
445C24—2441no—
446#41—4881no—
447#41—2442no—
448C2×Q8—1221no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4—611no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—4881no—
458C2×D4—4881no—
459C2×D4—4881no—
460C2×D4—4884no—
461C2×D4—2441no—
462C2×D4—2441no—
463C2×D4—2441no—
464C2×D4—2441no—
465C2×D4—2442no—
466C2×D4—2442no—
467C2×D4—2442no—
468C2×D4—2441no—
469C2×D4—2441no—
470C2×D4—2441no—
471C22×C4—41no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—2441no—
475C22×C4—2441no—
476Q16—1221no—
477Q16—1221no—
478SD16—4881no—
479SD16—2442no—
480D8—21no—
481D8—21no—
482D8—21no—
483D8—21no—
484D8—21no—
485D8—81no—
486D8—612no—
487D8—1221no—
488D8—1221no—
489D8—1221no—
490D8—2442no—
491D8—2442no—
492D8—2442no—
493D8D4×D6112yescomplemented
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—1221no—
497C2×C8—1221no—
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—
516C42D4×D6111yescharacteristic
517C16—21no—
518C16—21no—
519C16—612no—
520C16—1221no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—611no—
525C23—611no—
526C23—9761no—
527C23—9761no—
528C23—9761no—
529C23—4882no—
530C23—4882no—
531C23—4882no—
532C23—4882no—
533C23—4882no—
534C23—4882no—
535C23—2441no—
536C23—2441no—
537Q8—2441no—
538Q8—2441no—
539D4—22no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—82no—
546D4—161no—
547D4—1222no—
548D4—1222no—
549D4—9761no—
550D4—4882no—
551D4—4882no—
552D4—4882no—
553D4—4882no—
554D4—4882no—
555D4—4882no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559C2×C4—41no—
560C2×C4—161no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—4882no—
568C2×C4—4882no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4D8×D6111yescharacteristic
572C2×C4D8×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C8—21no—
575C8—1221no—
576C8—1221no—
577C8C2×D4×D6111yescharacteristic
578C8#1451982311yescharacteristic
579C22—42no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—612no—
589C22—1221no—
590C22—1221no—
591C22—1221no—
592C22—1221no—
593C22—1221no—
594C22—9764no—
595C22—9764no—
596C22—9764no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—4881no—
601C22—4881no—
602C22—2442no—
603C22C2×D8×D6111yescharacteristic
604C4—21no—
605C4—21no—
606C4—81no—
607C4—1221no—
608C4—1221no—
609C4—4881no—
610C4—4881no—
611C4#1684039611yescharacteristic
612C4#1684056611yescharacteristic
613C2—82no—
614C2—161no—
615C2—161no—
616C2—1222no—
617C2—9761no—
618C2—4882no—
619C2—2441no—
620C2#2145749111yescharacteristic
621C2#509×D6111yescharacteristic
622C2#2145895611yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2206C2
socleC22×C61C22×D8
hypercenter#2206D61
hypocenterC61#33830
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2206

Complements

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

NH ≅ G/Ntype
1. C2×#16840196C2semidirect
2. C2×D16×D61C2semidirect
3. #21457505C2semidirect
4. #21457494C2semidirect
5. #21448083C2semidirect
6. #21447921C2semidirect
7. #21447914C2semidirect
8. #21447875C2semidirect
9. #21447749C2semidirect
10. #21447739C2semidirect
11. #21447732C2semidirect
12. #21447729C2semidirect
13. #21447693C2semidirect
14. #21447643C2semidirect
15. C61×#2206C2semidirect
16. C2×#14519796C22semidirect
18. C2×D8×D61C22semidirect
19. C2×D8×D61C22semidirect
23. C2×#14519722C22semidirect
24. C2×#14519721C22semidirect
25. C2×#14519721C22semidirect
28. C2×#14519718C22semidirect
29. C2×#14519718C22semidirect
31. C2×#14519716C22semidirect
32. C2×C16×D61C22semidirect
33. #16840146C22semidirect
34. #16840004C22semidirect
35. #16839999C22semidirect
37. #16839995C22semidirect
44. C4×D488C22semidirect
46. #16839589C22semidirect
47. #16839567C22semidirect
48. #16839564C22semidirect
49. #16839563C22semidirect
50. #16839562C22semidirect
51. #16839541C22semidirect
52. #16839541C22semidirect
53. #16839538C22semidirect
54. C2×D16×C61C22semidirect
55. C2×D16×C61C22semidirect
56. C61×#493C22semidirect
57. C4×D8×C61C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#357C22semidirect
60. C61×#357C22semidirect
104. #14519718D4semidirect
170. D8×C61C2×D4semidirect
208. #11758185C2×D4semidirect
292. C2×D61#509semidirect
310. #2206D61semidirect
342. C2×D16C2×D61semidirect
343. C2×D16C2×D61semidirect
347. #493C2×D61semidirect
359. C4×D8C2×D61semidirect
362. C4⋊C16C2×D61semidirect
367. #357C2×D61semidirect
368. #357C2×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×#168401962
C2×D16×D612
#214575052
#214574942
#214480832
#214479212
#214479142
#214478752
#214477492
#214477392
#214477322
#214477292
#214476932
#214476432
C61×#22062
#3383061