Finite Group Databasebeta

← #29836473

Subgroups — #29836473

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
4D8×D244C211yesmaximal, characteristic, complemented
5#21448085C211yesmaximal, characteristic, complemented
6#21447923C211yesmaximal, characteristic, complemented
7#21447916C211yesmaximal, characteristic, complemented
8#21447874C211yesmaximal, characteristic, complemented
9#21447854C211yesmaximal, characteristic, complemented
10#21447749C211yesmaximal, characteristic, complemented
11#21447743C211yesmaximal, characteristic, complemented
12#21447732C211yesmaximal, characteristic, complemented
13#21447732C211yesmaximal, characteristic, complemented
14#21447646C211yesmaximal, characteristic, complemented
15C61×#2177C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic, complemented
17C2×D8×D61C2211yescharacteristic
18C2×D8×D61C2211yescharacteristic, complemented
19C2×D8×D61C2211yescharacteristic, complemented
20#16840407—21no—
21D4×D244—21no—
22#16840196—24no—
23C2×#14519722C2211yescharacteristic, complemented
24C2×#14519722C2211yescharacteristic, complemented
25C2×#14519721C2211yescharacteristic, complemented
26C2×D976C2211yescharacteristic, complemented
27C2×D976C2211yescharacteristic, complemented
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic, complemented
30C2×#14519718C2211yescharacteristic, complemented
31D16×D61—24no—
32C2×#14519716C2211yescharacteristic, complemented
33C2×C16×D61C2211yescharacteristic, complemented
34#16840146C2211yescharacteristic, complemented
35#16840139C2211yescharacteristic, complemented
36#16840004C2211yescharacteristic, complemented
37#16839999C2211yescharacteristic
38#16839999C2211yescharacteristic, complemented
39#16839999C2211yescharacteristic, complemented
40#16839977—21no—
41#16839957—21no—
42#16839914—21no—
43#16839908—21no—
44#16839897—21no—
45#16839897—21no—
46C8×D244C2211yescharacteristic
47C4⋊#11758185C2211yescharacteristic, complemented
48#16839565C2211yescharacteristic, complemented
49#16839562C2211yescharacteristic, complemented
50#16839562C2211yescharacteristic, complemented
51#16839562C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839541C2211yescharacteristic, complemented
54C2×D16×C61C2211yescharacteristic
55C4×D8×C61C2211yescharacteristic, complemented
56C4×D8×C61C2211yescharacteristic, complemented
57C61×#367C2211yescharacteristic, complemented
58C61×#357C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic, complemented
60C61×#346C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C22×D244—41no—
63C2×D4×D61—21no—
64C2×D4×D61—21no—
65C2×D4×D61—21no—
66#14519796—22no—
67#14519796—41no—
68C2×Dic244C2311yescharacteristic
69C2×#11758190—21no—
70C2×#11758200—21no—
71C2×D488—21no—
72C2×D488C2311yescharacteristic
73C2×D488C2311yescharacteristic
74C2×D488C2311yescharacteristic
75C2×#11758199—21no—
76C2×#11758199—21no—
77C2×#11758199—21no—
78D8×D61—22no—
79D8×D61—22no—
80D8×D61—22no—
81D8×D61—41no—
82D8×D61—41no—
83D8×D61—41no—
84C2×C8×D61C2311yescharacteristic
85C2×C8×D61C2311yescharacteristic
86#14519774—21no—
87#14519765—21no—
88#14519761—41no—
89#14519757—21no—
90#14519752—21no—
91#14519752—21no—
92#14519752—21no—
93#14519751—41no—
94#14519751—41no—
95#14519747—41no—
96#14519747—41no—
97#14519747—41no—
98C4×D244D411yescharacteristic
99C2×#11758198—41no—
100#14519722—22no—
101#14519722—41no—
102#14519721—22no—
103D976—22no—
104D976—41no—
105#14519718—22no—
106#14519718—22no—
107#14519718—22no—
108#14519716—22no—
109C2×#11758185C2311yescharacteristic
110C2×#11758185C2311yescharacteristic
111C16×D61—22no—
112C4⋊#92418D411yescharacteristic
113#14519700—21no—
114#14519697—21no—
115#14519697—21no—
116#14519697—21no—
117#14519689D411yescharacteristic
118#14519689D411yescharacteristic
119C2×D8×C61C2311yescharacteristic
120C2×D8×C61C2311yescharacteristic
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123D16×C61D414yescomplemented
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#107×C61C2311yescharacteristic
128#102×C61—21no—
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C23×D61—41no—
133C23×D61—41no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C22×C4×C61—41no—
137C22×C4×C61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C22×Dic61—41no—
143#11758224—42no—
144#11758224—81no—
145D4×D61—42no—
146D4×D61—42no—
147D4×D61—42no—
148D4×D61—81no—
149D4×D61—81no—
150D4×D61—81no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—41no—
155C2×D244—41no—
156C2×D244—41no—
157C2×D244—41no—
158C2×D244—41no—
159C2×D244—84no—
160C2×D244D811yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×Dic122—21no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61C2×D412yes—
167D8×C61C2×D412yescomplemented
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2411yescharacteristic
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C22⋊C4×C61—41no—
173C22⋊C4×C61—41no—
174C42×C61C2×D411yescharacteristic
175#11758200—42no—
176#11758199—42no—
177#11758199—42no—
178#11758199—42no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198D811yescharacteristic
186#11758198D811yescharacteristic
187C4⋊Dic61—41no—
188C4⋊Dic61D811yescharacteristic
189C2×#92418C2×D411yescharacteristic
190C2×#92418C2×D411yescharacteristic
191Dic244—21no—
192Dic244—21no—
193D488—21no—
194D488—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488—21no—
199D488—81no—
200#11758190—81no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61C2×D412yescomplemented
207#11758185—21no—
208#11758185—21no—
209C23×C61—41no—
210C23×C61—41no—
211C22×D61—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—22no—
227D4×C61—41no—
228D4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—82no—
232C2×C4×C61—161no—
233C2×C4×C61C2×D811yescharacteristic
234C2×C4×C61C2×D811yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C22⋊D61—82no—
237C22⋊D61—82no—
238C22⋊D61—82no—
239C22⋊D61—82no—
240C2×Dic61—41no—
241C2×Dic61—82no—
242C2×Dic61C2×D811yescharacteristic
243C2×Dic61C2×D811yescharacteristic
244D244—22no—
245D244—41no—
246D244—41no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—82no—
252D244—82no—
253D244—82no—
254D244—161no—
255D244—161no—
256C4×D61—21no—
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—21no—
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—42no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61C22×D811yescharacteristic
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—81no—
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—21no—
295C4×C61—81no—
296C4×C61—81no—
297C4×C61D4211yescharacteristic
298C4×C61#57611yescharacteristic
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#2186—611no—
313#2184—611no—
314#2184—611no—
315#2177D6111yescharacteristic, complemented
316#2168—611no—
317C2×C61—82no—
318C2×C61—82no—
319C2×C61D4×D811yescharacteristic
320C2×C61C2×#50911yescharacteristic
321C2×C61#341511yescharacteristic
322D61—22no—
323D61—41no—
324D61—161no—
325D61—161no—
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—611no—
342C2×D16—1224no—
343C2×D16C2×D6111yescharacteristic
344C2×M32—611no—
345C22×C16—611no—
346#495—611no—
347#493—611no—
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×D8C2×D6111yescharacteristic, complemented
359C4×D8C2×D6111yescharacteristic, complemented
360C8×D4—611no—
361#367C2×D6111yescharacteristic, complemented
362C4⋊C16—611no—
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#346C2×D6111yescharacteristic, complemented
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—611no—
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—1221no—
393C2×D8—2441no—
394C2×D8—2441no—
395C2×D8—2441no—
396C2×D8C22×D6111yescharacteristic
397C2×D8C22×D6111yescharacteristic
398C22×C8—611no—
399C22×C8—611no—
400#127—1221no—
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—21no—
414C4×D4—611no—
415C2×C22⋊C4—2441no—
416SD32—1222no—
417SD32—1222no—
418SD32—2441no—
419D16—1222no—
420D16—1222no—
421D16—1222no—
422D16—1222no—
423D16—2441no—
424D16D24414yescomplemented
425M32—1222no—
426C2×C16—611no—
427C2×C16—611no—
428C2×C16—1222no—
429C2×C16C22×D6111yescharacteristic
430C2×C16C22×D6111yescharacteristic
431#107C22×D6111yescharacteristic
432#107C22×D6111yescharacteristic
433C4⋊C8—611no—
434#103—1221no—
435#102—21no—
436#102—21no—
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—611no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—4884no—
456C2×D4—4881no—
457C2×D4—4881no—
458C2×D4—4881no—
459C2×D4—2441no—
460C2×D4—2441no—
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—41no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—2441no—
476Q16—1221no—
477Q16—1221no—
478SD16—4881no—
479SD16—2442no—
480D8—21no—
481D8—21no—
482D8—1221no—
483D8—1221no—
484D8—1221no—
485D8—1221no—
486D8—1221no—
487D8—1221no—
488D8—4881no—
489D8—2442no—
490D8—2442no—
491D8—2442no—
492D8C2×D24412yes—
493D8C2×D24412yescomplemented
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—21no—
504C4⋊C4—611no—
505C4⋊C4—2441no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—611no—
509C22⋊C4—611no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C42D4×D6111yescharacteristic
517C16—21no—
518C16—1221no—
519C16—1221no—
520C16C2×D24412yescomplemented
521C23—41no—
522C23—41no—
523C23—611no—
524C23—611no—
525C23—9761no—
526C23—9761no—
527C23—9761no—
528C23—4882no—
529C23—4882no—
530C23—4882no—
531C23—4882no—
532C23—4882no—
533C23—4882no—
534C23—2441no—
535C23—2441no—
536C23—2441no—
537Q8—2441no—
538Q8—2441no—
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—1222no—
544D4—9761no—
545D4—9761no—
546D4—4882no—
547D4—4882no—
548D4—4882no—
549D4—4882no—
550D4—4882no—
551D4—4882no—
552D4—4882no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—82no—
562C2×C4—161no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—4882no—
570C2×C4—2441no—
571C2×C4D8×D6111yescharacteristic
572C2×C4D8×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C8—21no—
575C8—1221no—
576C8—1221no—
577C8C22×D24411yescharacteristic
578C8#1451982311yescharacteristic
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—612no—
586C22—1221no—
587C22—1221no—
588C22—1221no—
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—4881no—
603C22C2×D8×D6111yescharacteristic
604C4—21no—
605C4—21no—
606C4—81no—
607C4—81no—
608C4—1221no—
609C4—1221no—
610C4—4881no—
611C4D4×D24411yescharacteristic
612C4#1684056411yescharacteristic
613C2—82no—
614C2—82no—
615C2—1222no—
616C2—9761no—
617C2—9761no—
618C2—9761no—
619C2—2441no—
620C2D8×D24411yescharacteristic
621C2#509×D6111yescharacteristic
622C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2177C2
socleC22×C61C22×D8
hypercenter#2177D61
hypocenterC61#33830
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2177

Complements

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

NH ≅ G/Ntype
1. C2×#16840196C2semidirect
2. C2×D16×D61C2semidirect
3. #21457505C2semidirect
4. D8×D244C2semidirect
5. #21448085C2semidirect
6. #21447923C2semidirect
7. #21447916C2semidirect
8. #21447874C2semidirect
9. #21447854C2semidirect
10. #21447749C2semidirect
11. #21447743C2semidirect
12. #21447732C2semidirect
13. #21447732C2semidirect
14. #21447646C2semidirect
15. C61×#2177C2semidirect
16. C2×#14519796C22semidirect
18. C2×D8×D61C22semidirect
19. C2×D8×D61C22semidirect
23. C2×#14519722C22semidirect
24. C2×#14519722C22semidirect
25. C2×#14519721C22semidirect
26. C2×D976C22semidirect
27. C2×D976C22semidirect
29. C2×#14519718C22semidirect
30. C2×#14519718C22semidirect
32. C2×#14519716C22semidirect
33. C2×C16×D61C22semidirect
34. #16840146C22semidirect
35. #16840139C22semidirect
36. #16840004C22semidirect
38. #16839999C22semidirect
39. #16839999C22semidirect
47. C4⋊#11758185C22semidirect
48. #16839565C22semidirect
49. #16839562C22semidirect
50. #16839562C22semidirect
51. #16839562C22semidirect
52. #16839541C22semidirect
53. #16839541C22semidirect
55. C4×D8×C61C22semidirect
56. C4×D8×C61C22semidirect
57. C61×#367C22semidirect
58. C61×#357C22semidirect
59. C61×#357C22semidirect
60. C61×#346C22semidirect
123. D16×C61D4semidirect
167. D8×C61C2×D4semidirect
206. C16×C61C2×D4semidirect
292. C2×D61#509semidirect
315. #2177D61semidirect
358. C4×D8C2×D61semidirect
359. C4×D8C2×D61semidirect
361. #367C2×D61semidirect
367. #357C2×D61semidirect
368. #357C2×D61semidirect
371. #346C2×D61semidirect
372. C61#33830semidirect
424. D16D244semidirect
493. D8C2×D244semidirect
520. C16C2×D244semidirect

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
D8×D2442
#214480852
#214479232
#214479162
#214478742
#214478542
#214477492
#214477432
#214477322
#214476462
C61×#21772
#3383061