Finite Group Databasebeta

← #29836518

Subgroups — #29836518

Order 15616.

692 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840196C211yesmaximal, characteristic, complemented
2C2×#16840191C211yesmaximal, characteristic, complemented
3#21457540C211yesmaximal, characteristic, complemented
4#21457505C211yesmaximal, characteristic, complemented
5#21448090C211yesmaximal, characteristic, complemented
6#21447928C211yesmaximal, characteristic, complemented
7#21447916C211yesmaximal, characteristic, complemented
8#21447874C211yesmaximal, characteristic, complemented
9#21447864C211yesmaximal, characteristic, complemented
10#21447760C211yesmaximal, characteristic, complemented
11#21447760C211yesmaximal, characteristic, complemented
12#21447749C211yesmaximal, characteristic, complemented
13#21447743C211yesmaximal, characteristic, complemented
14#21447646C211yesmaximal, characteristic, complemented
15C61×#2177C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×#14519796C2211yescharacteristic
18C2×#14519796C2211yescharacteristic, complemented
19C2×D8×D61C2211yescharacteristic
20#16840416—21no—
21#16840407—21no—
22#16840196—24no—
23#16840191—24no—
24C2×Dic488C2211yescharacteristic, complemented
25C2×Dic488C2211yescharacteristic, complemented
26C2×#14519722C2211yescharacteristic
27C2×#14519722C2211yescharacteristic
28C2×#14519721C2211yescharacteristic
29C2×#14519721C2211yescharacteristic
30C2×#14519721C2211yescharacteristic, complemented
31C2×#14519718C2211yescharacteristic
32C2×#14519716C2211yescharacteristic
33C2×C16×D61C2211yescharacteristic, complemented
34#16840146C2211yescharacteristic
35#16840142C2211yescharacteristic, complemented
36#16840004C2211yescharacteristic
37#16840004C2211yescharacteristic
38#16840004C2211yescharacteristic, complemented
39#16839999C2211yescharacteristic
40#16839977—21no—
41#16839967—21no—
42#16839925—21no—
43#16839925—21no—
44#16839914—21no—
45#16839908—21no—
46C8×D244C2211yescharacteristic
47C4⋊#11758185C2211yescharacteristic, complemented
48#16839565C2211yescharacteristic, complemented
49#16839565C2211yescharacteristic, complemented
50#16839565C2211yescharacteristic, complemented
51#16839562C2211yescharacteristic
52#16839541C2211yescharacteristic
53#16839541C2211yescharacteristic, complemented
54C2×D16×C61C2211yescharacteristic
55C4×D8×C61C2211yescharacteristic
56C4×D8×C61C2211yescharacteristic, complemented
57C61×#367C2211yescharacteristic, complemented
58C61×#357C2211yescharacteristic
59C61×#357C2211yescharacteristic, complemented
60C61×#346C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C2×#11758224—21no—
63C2×#11758224—21no—
64C2×D4×D61—21no—
65#14519796—22no—
66#14519796—22no—
67#14519796—22no—
68#14519796—41no—
69#14519796—41no—
70#14519796—41no—
71C2×Dic244—21no—
72C2×Dic244C2311yescharacteristic
73C2×Dic244C2311yescharacteristic
74C2×Dic244C2311yescharacteristic
75C2×#11758190—21no—
76C2×#11758200—21no—
77C2×#11758200—21no—
78C2×#11758200—21no—
79C2×D488C2311yescharacteristic
80C2×#11758199—21no—
81D8×D61—22no—
82D8×D61—41no—
83C2×C8×D61C2311yescharacteristic
84C2×C8×D61C2311yescharacteristic
85#14519777—21no—
86#14519765—21no—
87#14519761—41no—
88#14519761—41no—
89#14519761—41no—
90#14519757—21no—
91#14519757—21no—
92#14519757—21no—
93#14519752—21no—
94#14519751—41no—
95#14519751—41no—
96#14519747—41no—
97C4×D244D411yescharacteristic
98C2×C4⋊Dic61—41no—
99C2×#11758198—41no—
100Dic488—22no—
101Dic488—41no—
102#14519722—22no—
103#14519722—41no—
104#14519721—22no—
105#14519721—22no—
106#14519721—22no—
107#14519718—22no—
108#14519716—22no—
109C2×#11758185C2311yescharacteristic
110C2×#11758185C2311yescharacteristic
111C16×D61—22no—
112C4⋊#92418D411yescharacteristic
113#14519700—21no—
114#14519700—21no—
115#14519700—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—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C22×C4×C61—41no—
135C22×C4×C61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C22×Dic61—41no—
141C22×Dic61—41no—
142C22×Dic61—41no—
143#11758224—42no—
144#11758224—42no—
145#11758224—42no—
146#11758224—81no—
147#11758224—81no—
148#11758224—81no—
149D4×D61—42no—
150D4×D61—81no—
151C2×D244—21no—
152C2×D244—41no—
153C2×D244D811yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159D8×C61—21no—
160D8×C61—21no—
161D8×C61C2×D412yes—
162D8×C61C2×D412yes—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2411yescharacteristic
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C22⋊C4×C61—41no—
168C22⋊C4×C61—41no—
169C42×C61C2×D411yescharacteristic
170#11758200—42no—
171#11758200—42no—
172#11758200—42no—
173#11758199—42no—
174#11758198—41no—
175#11758198—41no—
176#11758198—41no—
177#11758198—41no—
178#11758198—41no—
179#11758198—41no—
180#11758198D811yescharacteristic
181#11758198D811yescharacteristic
182C4⋊Dic61—41no—
183C4⋊Dic61—41no—
184C4⋊Dic61—41no—
185C4⋊Dic61—41no—
186C4⋊Dic61—41no—
187C4⋊Dic61D811yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244—21no—
192Dic244—21no—
193Dic244—21no—
194Dic244—21no—
195Dic244—21no—
196Dic244—81no—
197D488—21no—
198D488—21no—
199#11758190—81no—
200C8×D61—21no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61C2×D412yes—
206#11758185—21no—
207#11758185—21no—
208C23×C61—41no—
209C23×C61—41no—
210C22×D61—41no—
211C22×D61—82no—
212C22×D61—82no—
213C22×D61—161no—
214C22×D61C2×D811yescharacteristic
215C22×D61C2×D811yescharacteristic
216D4×C61—22no—
217D4×C61—22no—
218D4×C61—41no—
219D4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—82no—
223C2×C4×C61—161no—
224C2×C4×C61C2×D811yescharacteristic
225C2×C4×C61C2×D811yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C22⋊D61—82no—
228C22⋊D61—82no—
229C22⋊D61—82no—
230C22⋊D61—82no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—82no—
235C2×Dic61—82no—
236C2×Dic61—82no—
237C2×Dic61C2×D811yescharacteristic
238C2×Dic61C2×D811yescharacteristic
239D244—41no—
240D244—41no—
241D244—41no—
242D244—161no—
243C4×D61—21no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—21no—
247Dic122—41no—
248Dic122—41no—
249Dic122—41no—
250Dic122—41no—
251Dic122—41no—
252Dic122—41no—
253C8×C61—21no—
254C8×C61C22×D411yescharacteristic
255C8×C61H(5,𝔽2)11yescharacteristic
256#92418—21no—
257#92418—21no—
258#33833—611nomaximal, Sylow
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61C22×D811yescharacteristic
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—164no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61D4211yescharacteristic
281C4×C61#57611yescharacteristic
282Dic61—21no—
283Dic61—21no—
284Dic61—81no—
285Dic61—81no—
286Dic61—81no—
287C2×#509—611no—
288C2×#508—611no—
289#3283—611no—
290#3280—611no—
291#2251—611no—
292#2219—611no—
293#2217—611no—
294#2210—611no—
295#2208—611no—
296#2189—611no—
297#2189—611no—
298#2187—611no—
299#2186—611no—
300#2177D6111yescharacteristic, complemented
301#2168—611no—
302C2×C61—82no—
303C2×C61—82no—
304C2×C61D4×D811yescharacteristic
305C2×C61#341411yescharacteristic
306C2×C61#341611yescharacteristic
307D61—41no—
308D61—41no—
309D61—161no—
310C2×#135—611no—
311C2×#135—611no—
312C2×#135—611no—
313C22×D8—611no—
314#547—1221no—
315#546—1221no—
316#509—1224no—
317#508—1224no—
318C2×Q32—611no—
319C2×Q32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32—611no—
323C2×SD32—611no—
324C2×SD32—611no—
325C2×D16—611no—
326C2×D16C2×D6111yescharacteristic
327C2×M32—611no—
328C22×C16—611no—
329#495—611no—
330#494—611no—
331#467—611no—
332#467—611no—
333#467—611no—
334#466—611no—
335#463—1221no—
336#461—1221no—
337#452—1221no—
338#452—1221no—
339#450—1221no—
340#449—1221no—
341C4×D8C2×D6111yescharacteristic
342C4×D8C2×D6111yescharacteristic, complemented
343C8×D4—611no—
344#367C2×D6111yescharacteristic, complemented
345C4⋊C16—611no—
346#358—611no—
347#358—611no—
348#358—611no—
349#357—611no—
350#357C2×D6111yescharacteristic
351#357C2×D6111yescharacteristic, complemented
352C22⋊C16—611no—
353C22⋊C16—611no—
354#346C2×D6111yescharacteristic, complemented
355C61#3383311yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C2×#41—1221no—
358C2×#41—1221no—
359C22×D4—1221no—
360#135—1222no—
361#135—1222no—
362#135—1222no—
363#135—2441no—
364#135—2441no—
365#135—2441no—
366C2×Q16—611no—
367C2×Q16—611no—
368C2×Q16—611no—
369C2×Q16—1221no—
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×D8—611no—
375C2×D8—1221no—
376C2×D8—1222no—
377C2×D8—2441no—
378C2×D8C22×D6111yescharacteristic
379C2×D8C22×D6111yescharacteristic
380C22×C8—611no—
381C22×C8—611no—
382C4⋊Q8—1221no—
383#124—1221no—
384#123—2441no—
385#123—2441no—
386#123—2441no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389C22⋊Q8—1221no—
390#121—1221no—
391#121—2441no—
392#121—2441no—
393#120—2441no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—611no—
397C2×C4⋊C4—2441no—
398C2×C22⋊C4—2441no—
399Q32—1222no—
400Q32—2441no—
401SD32—1222no—
402SD32—1222no—
403SD32—1222no—
404SD32—1222no—
405SD32—2441no—
406D16—1222no—
407D16D24414yescomplemented
408M32—1222no—
409C2×C16—611no—
410C2×C16—611no—
411C2×C16—1222no—
412C2×C16C22×D6111yescharacteristic
413C2×C16C22×D6111yescharacteristic
414#107C22×D6111yescharacteristic
415#107C22×D6111yescharacteristic
416C4⋊C8—611no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#102—21no—
421#102—21no—
422#102—1221no—
423C22⋊C8—611no—
424C22⋊C8—611no—
425C4×C8C22×D6111yescharacteristic
426C24—2441no—
427#41—4881no—
428#41—4881no—
429#41—4881no—
430#41—2442no—
431#41—2442no—
432#41—2442no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—611no—
439C2×D4—1221no—
440C2×D4—4881no—
441C2×D4—2441no—
442C2×D4—2441no—
443C2×D4—2442no—
444C2×D4—2441no—
445C2×D4—2441no—
446C2×D4—2441no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—2441no—
452C22×C4—2441no—
453C22×C4—2441no—
454Q16—1221no—
455Q16—1221no—
456Q16—1221no—
457Q16—1221no—
458Q16—1221no—
459Q16—1221no—
460Q16—4881no—
461SD16—4881no—
462SD16—2442no—
463SD16—2442no—
464SD16—2442no—
465D8—21no—
466D8—21no—
467D8—1221no—
468D8—1221no—
469D8—2442no—
470D8C2×D24412yes—
471D8C2×D24412yes—
472C2×C8—611no—
473C2×C8—611no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8D4×D6111yescharacteristic
479C2×C8C23×D6111yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—611no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C4⋊C4—2441no—
487C4⋊C4—2441no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—611no—
491C22⋊C4—611no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C42D4×D6111yescharacteristic
499C16—21no—
500C16—1221no—
501C16—1221no—
502C16C2×D24412yes—
503C23—41no—
504C23—41no—
505C23—611no—
506C23—611no—
507C23—9761no—
508C23—4882no—
509C23—4882no—
510C23—2441no—
511Q8—2441no—
512Q8—2441no—
513Q8—2441no—
514Q8—2441no—
515Q8—2441no—
516Q8—2441no—
517D4—22no—
518D4—22no—
519D4—41no—
520D4—41no—
521D4—9761no—
522D4—4882no—
523D4—4882no—
524D4—4882no—
525D4—4882no—
526D4—2441no—
527D4—2441no—
528D4—2441no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—82no—
532C2×C4—161no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—4882no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4D8×D6111yescharacteristic
546C2×C4D8×D6111yescharacteristic
547C2×C4C2×D4×D6111yescharacteristic
548C8—21no—
549C8—1221no—
550C8—1221no—
551C8C22×D24411yescharacteristic
552C8#1451982311yescharacteristic
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—1221no—
560C22—1221no—
561C22—1221no—
562C22—1221no—
563C22—1221no—
564C22—1221no—
565C22—9764no—
566C22—4881no—
567C22—4881no—
568C22—4881no—
569C22C2×D8×D6111yescharacteristic
570C4—21no—
571C4—21no—
572C4—81no—
573C4—81no—
574C4—1221no—
575C4—1221no—
576C4—4881no—
577C4—4881no—
578C4—4881no—
579C4D4×D24411yescharacteristic
580C4#1684056411yescharacteristic
581C2—82no—
582C2—82no—
583C2—9761no—
584C2—2441no—
585C2—2441no—
586C2D8×D24411yescharacteristic
587C2#2145895111yescharacteristic
588C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2177C2
socleC22×C61C22×D8
hypercenter#2177D61
hypocenterC61#33833
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×#16840191C2semidirect
3. #21457540C2semidirect
4. #21457505C2semidirect
5. #21448090C2semidirect
6. #21447928C2semidirect
7. #21447916C2semidirect
8. #21447874C2semidirect
9. #21447864C2semidirect
10. #21447760C2semidirect
11. #21447760C2semidirect
12. #21447749C2semidirect
13. #21447743C2semidirect
14. #21447646C2semidirect
15. C61×#2177C2semidirect
18. C2×#14519796C22semidirect
24. C2×Dic488C22semidirect
25. C2×Dic488C22semidirect
30. C2×#14519721C22semidirect
33. C2×C16×D61C22semidirect
35. #16840142C22semidirect
38. #16840004C22semidirect
47. C4⋊#11758185C22semidirect
48. #16839565C22semidirect
49. #16839565C22semidirect
50. #16839565C22semidirect
53. #16839541C22semidirect
56. C4×D8×C61C22semidirect
57. C61×#367C22semidirect
59. C61×#357C22semidirect
60. C61×#346C22semidirect
123. D16×C61D4semidirect
300. #2177D61semidirect
342. C4×D8C2×D61semidirect
344. #367C2×D61semidirect
351. #357C2×D61semidirect
354. #346C2×D61semidirect
355. C61#33833semidirect
407. D16D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168401962
C2×#168401912
#214575402
#214575052
#214480902
#214479282
#214479162
#214478742
#214478642
#214477602
#214477492
#214477432
#214476462
C61×#21772
#3383361