Finite Group Databasebeta

← #29836478

Subgroups — #29836478

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×#16840200C211yesmaximal, characteristic, complemented
2C2×D16×D61C211yesmaximal, characteristic, complemented
3#21457507C211yesmaximal, characteristic, complemented
4#21457493C211yesmaximal, characteristic, complemented
5#21448082C211yesmaximal, characteristic, complemented
6#21447916C211yesmaximal, characteristic, complemented
7#21447873C211yesmaximal, characteristic, complemented
8#21447853C211yesmaximal, characteristic, complemented
9#21447749C211yesmaximal, characteristic, complemented
10#21447740C211yesmaximal, characteristic, complemented
11#21447732C211yesmaximal, characteristic, complemented
12#21447730C211yesmaximal, characteristic, complemented
13#21447694C211yesmaximal, characteristic, complemented
14#21447642C211yesmaximal, characteristic, complemented
15C61×#2218C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic, complemented
17C22×D488C2211yescharacteristic, complemented
18C2×D8×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic, complemented
20#16840409—21no—
21#16840396—21no—
22#16840200—24no—
23C2×#14519722C2211yescharacteristic, complemented
24C2×#14519722C2211yescharacteristic, complemented
25C2×#14519721C2211yescharacteristic, complemented
26C2×D976C2211yescharacteristic
27C2×D976C2211yescharacteristic, complemented
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic, complemented
30C2×#14519718C2211yescharacteristic, complemented
31D16×D61—24no—
32C2×C16×D61C2211yescharacteristic, complemented
33#16840145C2211yescharacteristic, complemented
34#16840137C2211yescharacteristic, complemented
35#16840004C2211yescharacteristic, complemented
36#16839999C2211yescharacteristic
37#16839976—21no—
38#16839956—21no—
39#16839914—21no—
40#16839905—21no—
41#16839897—21no—
42#16839895—21no—
43#16839844C2211yescharacteristic, complemented
44#16839844C2211yescharacteristic, complemented
45C8×C22⋊D61C2211yescharacteristic
46#16839580C2211yescharacteristic, complemented
47#16839565C2211yescharacteristic, complemented
48#16839562C2211yescharacteristic, complemented
49#16839562C2211yescharacteristic, complemented
50#16839562C2211yescharacteristic, complemented
51C22⋊#11758185C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839537C2211yescharacteristic, complemented
54C2×D16×C61C2211yescharacteristic, complemented
55C2×M32×C61C2211yescharacteristic, complemented
56C61×#466C2211yescharacteristic, complemented
57C61×#466C2211yescharacteristic, complemented
58C61×#367C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758222—21no—
62C22×D244—21no—
63C2×D4×D61—21no—
64C2×D4×D61—21no—
65C2×D4×D61—41no—
66#14519799—22no—
67#14519799—41no—
68C2×Dic244C2311yescharacteristic
69C2×#11758190—21no—
70C2×#11758200—21no—
71C2×D488—22no—
72C2×D488—21no—
73C2×D488—41no—
74C2×D488C2311yescharacteristic
75C2×D488C2311yescharacteristic
76C2×D488C2311yescharacteristic
77C2×#11758199—21no—
78C2×#11758199—21no—
79C2×#11758199—21no—
80D8×D61—22no—
81D8×D61—22no—
82D8×D61—41no—
83D8×D61—41no—
84C2×C8×D61C2311yescharacteristic
85#14519773—21no—
86#14519764—21no—
87#14519759—41no—
88#14519757—21no—
89#14519752—21no—
90#14519750—41no—
91#14519748—41no—
92#14519747—41no—
93#14519745—41no—
94#14519745—41no—
95#14519739—21no—
96#14519739—21no—
97C4×C22⋊D61D411yescharacteristic
98C22⋊C4×D61—41no—
99#14519722—22no—
100#14519722—22no—
101#14519721—41no—
102D976—22no—
103D976D414yescomplemented
104#14519718—22no—
105#14519718—22no—
106#14519718—41no—
107C2×#11758185C2311yescharacteristic
108C2×#11758185C2311yescharacteristic
109C16×D61—22no—
110#14519706D411yescharacteristic
111#14519700—21no—
112#14519697—21no—
113#14519697—21no—
114#14519697—21no—
115C22⋊#92418D411yescharacteristic
116#14519689D411yescharacteristic
117C8×Dic61C2311yescharacteristic
118C2×D8×C61C2311yescharacteristic
119C2×D8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121#121×C61—21no—
122#121×C61—21no—
123D16×C61—22no—
124M32×C61—22no—
125C2×C16×C61C2311yescharacteristic
126C2×C16×C61C2311yescharacteristic
127#107×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#102×C61—21no—
130#102×C61—21no—
131C23×D61—41no—
132C23×D61—41no—
133C23×D61—41no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—41no—
137C2×D4×C61—41no—
138C22×C4×C61C2×D411yescharacteristic
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61D811yescharacteristic
145D4×D61—42no—
146D4×D61—42no—
147D4×D61—81no—
148D4×D61—81no—
149D4×D61—84no—
150#11758222—42no—
151#11758222—81no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—42no—
156C2×D244—41no—
157C2×D244—41no—
158C2×D244—81no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×C4×D61C2×D411yescharacteristic
163C2×Dic122—21no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61—21no—
167D8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C22⋊C4×C61—41no—
175C22⋊C4×C61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61D811yescharacteristic
179#11758200—81no—
180#11758199—42no—
181#11758199—42no—
182#11758199—81no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186#11758198—41no—
187#11758198D811yescharacteristic
188C4⋊Dic61—41no—
189#11758196D811yescharacteristic
190C4×Dic61C2×D411yescharacteristic
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193Dic244—21no—
194Dic244—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488—21no—
199D488—42no—
200D488C2×D412yes—
201D488C2×D412yescomplemented
202#11758190—42no—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61C2×D412yescomplemented
207#11758185—21no—
208#11758185—21no—
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
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—82no—
230D4×C61—161no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C2×D811yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C22⋊D61—22no—
238C22⋊D61—82no—
239C22⋊D61—82no—
240C22⋊D61—82no—
241C22⋊D61—82no—
242C22⋊D61—161no—
243C2×Dic61—41no—
244C2×Dic61C2×D811yescharacteristic
245C2×Dic61C2×D811yescharacteristic
246C2×Dic61C2×D811yescharacteristic
247D244—22no—
248D244—22no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—41no—
253D244—82no—
254D244—82no—
255C4×D61—21no—
256C4×D61—21no—
257C4×D61—82no—
258C4×D61—82no—
259C4×D61—161no—
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—21no—
269C22×C61—21no—
270C22×C61—21no—
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—42no—
281C2×D61—42no—
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#2218D6111yescharacteristic, complemented
308#2217—611no—
309#2210—611no—
310#2206—611no—
311#2187—611no—
312#2186—611no—
313#2184—611no—
314#2184—611no—
315#2177—611no—
316#2168—611no—
317C2×C61—41no—
318C2×C61—161no—
319C2×C61—161no—
320C2×C61D4×D811yescharacteristic
321C2×C61C2×#50911yescharacteristic
322C2×C61#341511yescharacteristic
323D61—22no—
324D61—82no—
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—611no—
342C2×D16—1224no—
343C2×D16C2×D6111yescharacteristic, complemented
344C2×M32C2×D6111yescharacteristic, complemented
345C22×C16—611no—
346#495—611no—
347#493—611no—
348#467—611no—
349#466—611no—
350#466C2×D6111yescharacteristic, complemented
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#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#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—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—21no—
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—1222no—
417SD32—1222no—
418SD32—2441no—
419D16—22no—
420D16—614no—
421D16—1222no—
422D16—1222no—
423D16—1222no—
424D16—2441no—
425M32—22no—
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×C8—611no—
443C24—2441no—
444C24—2441no—
445C24—2441no—
446#41—4881no—
447#41—2442no—
448C2×Q8—1221no—
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—611no—
472C22×C4—2441no—
473C22×C4—2441no—
474C22×C4—2441no—
475C22×C4D4×D6111yescharacteristic
476Q16—1221no—
477Q16—1221no—
478SD16—4881no—
479SD16—2442no—
480D8—21no—
481D8—21no—
482D8—21no—
483D8—21no—
484D8—612no—
485D8—612no—
486D8—1221no—
487D8—1221no—
488D8—1221no—
489D8—1221no—
490D8—4881no—
491D8—2442no—
492D8—2442no—
493D8—2442no—
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—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—
516C42—611no—
517C16—21no—
518C16—1221no—
519C16—1221no—
520C16D4×D6112yescomplemented
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—2441no—
538Q8—2441no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—82no—
544D4—161no—
545D4—1222no—
546D4—1222no—
547D4—1222no—
548D4—9761no—
549D4—4882no—
550D4—4882no—
551D4—4882no—
552D4—4882no—
553D4—4882no—
554D4—4882no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—9761no—
569C2×C4—4882no—
570C2×C4—4882no—
571C2×C4—2441no—
572C2×C4D8×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C8—21no—
575C8—1221no—
576C8—1221no—
577C8C2×D4×D6111yescharacteristic
578C8#1451982311yescharacteristic
579C22—21no—
580C22—21no—
581C22—21no—
582C22—81no—
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—2442no—
602C22—2442no—
603C22C2×D8×D6111yescharacteristic
604C4—21no—
605C4—81no—
606C4—81no—
607C4—1221no—
608C4—1221no—
609C4—1221no—
610C4—4881no—
611C4#1684039611yescharacteristic
612C4#1684056411yescharacteristic
613C2—41no—
614C2—161no—
615C2—161no—
616C2—1222no—
617C2—9761no—
618C2—4882no—
619C2—4882no—
620C2#2145749311yescharacteristic
621C2#509×D6111yescharacteristic
622C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2218C2
socleC22×C61C22×D8
hypercenter#2218D61
hypocenterC61#33830
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2218

Complements

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

NH ≅ G/Ntype
1. C2×#16840200C2semidirect
2. C2×D16×D61C2semidirect
3. #21457507C2semidirect
4. #21457493C2semidirect
5. #21448082C2semidirect
6. #21447916C2semidirect
7. #21447873C2semidirect
8. #21447853C2semidirect
9. #21447749C2semidirect
10. #21447740C2semidirect
11. #21447732C2semidirect
12. #21447730C2semidirect
13. #21447694C2semidirect
14. #21447642C2semidirect
15. C61×#2218C2semidirect
16. C2×#14519799C22semidirect
17. C22×D488C22semidirect
19. C2×D8×D61C22semidirect
23. C2×#14519722C22semidirect
24. C2×#14519722C22semidirect
25. C2×#14519721C22semidirect
27. C2×D976C22semidirect
29. C2×#14519718C22semidirect
30. C2×#14519718C22semidirect
32. C2×C16×D61C22semidirect
33. #16840145C22semidirect
34. #16840137C22semidirect
35. #16840004C22semidirect
43. #16839844C22semidirect
44. #16839844C22semidirect
46. #16839580C22semidirect
47. #16839565C22semidirect
48. #16839562C22semidirect
49. #16839562C22semidirect
50. #16839562C22semidirect
51. C22⋊#11758185C22semidirect
52. #16839541C22semidirect
53. #16839537C22semidirect
54. C2×D16×C61C22semidirect
55. C2×M32×C61C22semidirect
56. C61×#466C22semidirect
57. C61×#466C22semidirect
58. C61×#367C22semidirect
59. C61×#357C22semidirect
60. C61×#357C22semidirect
103. D976D4semidirect
201. D488C2×D4semidirect
206. C16×C61C2×D4semidirect
292. C2×D61#509semidirect
307. #2218D61semidirect
343. C2×D16C2×D61semidirect
344. C2×M32C2×D61semidirect
350. #466C2×D61semidirect
351. #466C2×D61semidirect
361. #367C2×D61semidirect
367. #357C2×D61semidirect
368. #357C2×D61semidirect
372. C61#33830semidirect
520. C16D4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402002
C2×D16×D612
#214575072
#214574932
#214480822
#214479162
#214478732
#214478532
#214477492
#214477402
#214477322
#214477302
#214476942
#214476422
C61×#22182
#3383061