Finite Group Databasebeta

← #29834489

Subgroups — #29834489

Order 15616.

738 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×#16840199C211yesmaximal, characteristic, complemented
2C2×M32×D61C211yesmaximal, characteristic, complemented
3#21455402C211yesmaximal, characteristic, complemented
4#466×D61C211yesmaximal, characteristic, complemented
5#21447934C211yesmaximal, characteristic, complemented
6#21447922C211yesmaximal, characteristic, complemented
7#21447920C211yesmaximal, characteristic, complemented
8#21447918C211yesmaximal, characteristic, complemented
9#21447912C211yesmaximal, characteristic, complemented
10#21447906C211yesmaximal, characteristic, complemented
11#21447904C211yesmaximal, characteristic, complemented
12#21447604C211yesmaximal, characteristic, complemented
13#21447533C211yesmaximal, characteristic, complemented
14#21447530C211yesmaximal, characteristic, complemented
15C61×#2220C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19#16840275—21no—
20#121×D61—21no—
21#16840199—24no—
22C2×#14519720C2211yescharacteristic, complemented
23C2×D976C2211yescharacteristic, complemented
24C2×#14519718C2211yescharacteristic, complemented
25C2×#14519717C2211yescharacteristic, complemented
26C2×#14519716C2211yescharacteristic
27C2×#14519716C2211yescharacteristic
28M32×D61—24no—
29C2×C16×D61C2211yescharacteristic, complemented
30C2×C16×D61C2211yescharacteristic, complemented
31#16840000C2211yescharacteristic, complemented
32#16839999C2211yescharacteristic
33#16839999C2211yescharacteristic, complemented
34#16839998C2211yescharacteristic
35#16839998C2211yescharacteristic, complemented
36#16839997—21no—
37#16839996—21no—
38#16839995C2211yescharacteristic
39#16839994—21no—
40#16839993—21no—
41#16839799C2211yescharacteristic
42#107×D61C2211yescharacteristic
43#16839757—21no—
44#102×D61—21no—
45#16839590C2211yescharacteristic, complemented
46#16839589C2211yescharacteristic, complemented
47#16839587C2211yescharacteristic, complemented
48#16839566C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic, complemented
51#16839563C2211yescharacteristic, complemented
52#16839562C2211yescharacteristic, complemented
53#16839562C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic, complemented
55C2×M32×C61C2211yescharacteristic, complemented
56C61×#467C2211yescharacteristic, complemented
57C61×#466C2211yescharacteristic
58C61×#366C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×D4×D61—21no—
63C2×D4×D61—41no—
64C22×C4×D61D411yescharacteristic
65#14519800—22no—
66#14519800—22no—
67C2×#11758201—21no—
68C2×D488C2311yescharacteristic
69C2×D488C2311yescharacteristic
70C2×#11758199—21no—
71D8×D61—22no—
72D8×D61—22no—
73C22×#92418D411yescharacteristic
74C2×C8×D61—22no—
75C2×C8×D61—22no—
76C2×C8×D61D411yescharacteristic
77C2×C8×D61D411yescharacteristic
78C2×C8×D61C2311yescharacteristic
79C2×C8×D61C2311yescharacteristic
80#14519762—41no—
81#14519753—21no—
82#14519752—21no—
83#14519752—21no—
84#14519751—21no—
85#14519751—21no—
86#14519750—41no—
87#14519749—41no—
88#14519748—21no—
89#14519747—41no—
90#14519739—41no—
91#14519737—41no—
92#14519735—21no—
93C4⋊C4×D61—21no—
94C22⋊C4×D61—41no—
95#14519720—22no—
96D976—22no—
97#14519718—22no—
98#14519717—22no—
99#14519716—22no—
100#14519716D414yescomplemented
101C2×#11758185C2311yescharacteristic
102C2×#11758185C2311yescharacteristic
103C16×D61—22no—
104C16×D61—22no—
105#14519711C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519710—21no—
108#14519708—21no—
109#14519701—21no—
110#14519699—21no—
111#14519699—21no—
112#14519698—21no—
113#14519697—21no—
114#14519697—21no—
115C2×Q16×C61C2311yescharacteristic
116C2×D8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22⋊Q8×C61—21no—
119#121×C61—21no—
120SD32×C61—22no—
121M32×C61—22no—
122C2×C16×C61C2311yescharacteristic
123C2×C16×C61C2311yescharacteristic
124#107×C61C2311yescharacteristic
125#107×C61C2311yescharacteristic
126#103×C61—21no—
127#102×C61—21no—
128C23×D61—41no—
129C23×D61D811yescharacteristic
130C2×Q8×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—41no—
133C22×C4×C61C2×D411yescharacteristic
134C2×C22⋊D61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C22×Dic61D811yescharacteristic
139#11758226—42no—
140#11758226—42no—
141D4×D61—42no—
142D4×D61—42no—
143D4×D61—84no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—41no—
147C2×D244—41no—
148C2×D244—41no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—41no—
152C2×C4×D61—41no—
153C2×C4×D61D811yescharacteristic
154C2×C4×D61D811yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157Q16×C61—21no—
158Q16×C61—21no—
159D8×C61—21no—
160D8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2411yescharacteristic
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—41no—
168C22⋊C4×C61—41no—
169C22⋊C4×C61—41no—
170C22⋊Dic61—41no—
171#11758201—42no—
172#11758199—42no—
173#11758198—41no—
174#11758198—41no—
175#11758198—41no—
176#11758198—41no—
177#11758198—41no—
178#11758198—41no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181#11758196—41no—
182#11758196—41no—
183C4×Dic61—41no—
184C2×#92418—22no—
185C2×#92418C2×D411yescharacteristic
186C2×#92418C2×D411yescharacteristic
187D488—21no—
188D488—21no—
189D488—21no—
190D488—21no—
191C8×D61—22no—
192C8×D61—21no—
193C8×D61—21no—
194C8×D61—22no—
195C8×D61C2×D412yes—
196C8×D61#4112yes—
197C16×C61—21no—
198C16×C61C2×D412yescomplemented
199#11758185—21no—
200#11758185C2×D412yescomplemented
201C23×C61—41no—
202C23×C61C2×D811yescharacteristic
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—82no—
209C22×D61—82no—
210C22×D61—82no—
211C22×D61C2×D811yescharacteristic
212C22×D61C2×D811yescharacteristic
213Q8×C61—41no—
214Q8×C61—41no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—82no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61C2×D811yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C22⋊D61—82no—
226C22⋊D61—82no—
227C22⋊D61—82no—
228C22⋊D61—82no—
229C2×Dic61—22no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61C2×D811yescharacteristic
233C2×Dic61C2×D811yescharacteristic
234D244—41no—
235D244—41no—
236D244—41no—
237D244—41no—
238D244—82no—
239D244—82no—
240D244—82no—
241C4×D61—22no—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—22no—
245C4×D61—82no—
246C4×D61—82no—
247C4×D61#12112yes—
248C4×D61#13512yes—
249C8×C61—21no—
250C8×C61C22×D411yescharacteristic
251C8×C61C2×#4111yescharacteristic
252#92418—21no—
253#92418#12112yes—
254#33605—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61C22×D811yescharacteristic
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—44no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—164no—
274C2×D61#46612yes—
275C2×D61#50912yescomplemented
276C4×C61—21no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61C2×#12111yescharacteristic
281C4×C61C2×#13511yescharacteristic
282Dic61—21no—
283Dic61—81no—
284Dic61—81no—
285Dic61#46612yes—
286C2×#509—611no—
287C22×M32—611no—
288#3050—611no—
289C2×#466—611no—
290#2220—611no—
291#2220D6111yescharacteristic, complemented
292#2218—611no—
293#2218—611no—
294#2217—611no—
295#2217—611no—
296#2215—611no—
297#2215—611no—
298#2159—611no—
299#2140—611no—
300#2139—611no—
301C2×C61—41no—
302C2×C61—161no—
303C2×C61C2×#46611yescharacteristic
304C2×C61C2×#50911yescharacteristic
305C2×C61#341411yescharacteristic
306D61—22no—
307D61—22no—
308D61—161no—
309D61—161no—
310C2×#135—611no—
311C22×D8—611no—
312C23×C8—611no—
313#525—1221no—
314C2×#121—1221no—
315#509—1224no—
316C2×SD32—611no—
317C2×SD32C2×D6111yescharacteristic, complemented
318C2×D16—611no—
319C2×D16—611no—
320C2×M32—611no—
321C2×M32—611no—
322C2×M32—611no—
323C2×M32—1224no—
324C2×M32C2×D6111yescharacteristic, complemented
325C22×C16—611no—
326C22×C16—611no—
327#467—611no—
328#467C2×D6111yescharacteristic, complemented
329#466—611no—
330#466—611no—
331#466—611no—
332#466—611no—
333#466—611no—
334#466—1221no—
335#466—1221no—
336#466C2×D6111yescharacteristic
337#465—1221no—
338#465—1221no—
339#427—611no—
340C2×#107—611no—
341#416—1221no—
342C2×#102—1221no—
343#367—611no—
344#367—611no—
345#366—611no—
346#366C2×D6111yescharacteristic, complemented
347#358—611no—
348#358C2×D6111yescharacteristic, complemented
349#357—611no—
350#357—611no—
351#357—611no—
352#357—611no—
353#357—611no—
354#357C2×D6111yescharacteristic, complemented
355C61#3360511yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C22×D4—1221no—
358C22×D4—2441no—
359C23×C4—611no—
360#135—1222no—
361#135—1222no—
362C2×Q16C22×D6111yescharacteristic
363C2×SD16—1221no—
364C2×D8—611no—
365C2×D8—611no—
366C2×D8—1222no—
367C2×D8—1222no—
368C2×D8—1221no—
369C2×D8C22×D6111yescharacteristic
370C22×C8—611no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8—1222no—
376C22×C8—1222no—
377C22×C8C22×D6111yescharacteristic
378#123—2441no—
379C22⋊Q8—21no—
380C22⋊Q8—1221no—
381#121—21no—
382#121—1221no—
383#121—1221no—
384#121—1221no—
385#121—1221no—
386#121—1221no—
387#121—2441no—
388#121—2441no—
389#120—2441no—
390C4×D4—2441no—
391C4×D4—2441no—
392#117—1221no—
393C2×C4⋊C4—1221no—
394C2×C22⋊C4—2441no—
395SD32—22no—
396SD32—1222no—
397D16—1222no—
398D16—1222no—
399M32—22no—
400M32—614no—
401M32—1222no—
402M32—1222no—
403C2×C16—611no—
404C2×C16—611no—
405C2×C16—1222no—
406C2×C16—1222no—
407C2×C16C22×D6111yescharacteristic
408C2×C16C22×D6111yescharacteristic
409#107—611no—
410#107—611no—
411#107—1221no—
412#107C22×D6111yescharacteristic
413#107C22×D6111yescharacteristic
414#106—1221no—
415#103—21no—
416#103—1221no—
417#102—21no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422#102—1221no—
423C24—611no—
424C24—2441no—
425#41—2442no—
426#41—2442no—
427C2×Q8—21no—
428C2×D4—21no—
429C2×D4—41no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—4884no—
433C2×D4—2441no—
434C2×D4—2441no—
435C2×D4—2441no—
436C2×D4—2442no—
437C2×D4—2442no—
438C2×D4—2441no—
439C2×D4—2441no—
440C2×D4—2441no—
441C2×D4—2441no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—2441no—
450C22×C4—2441no—
451C22×C4D4×D6111yescharacteristic
452Q16—21no—
453Q16—21no—
454SD16—2442no—
455D8—21no—
456D8—21no—
457D8—1221no—
458D8—1221no—
459D8—1221no—
460D8—1221no—
461D8—2442no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—611no—
465C2×C8—612no—
466C2×C8—612no—
467C2×C8—611no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8D4×D6111yescharacteristic
474C2×C8C23×D6111yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—41no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C42—2441no—
492C16—21no—
493C16—612no—
494C16—1221no—
495C16D4×D6112yescomplemented
496C23—41no—
497C23—611no—
498C23—611no—
499C23—1222no—
500C23—1222no—
501C23—4882no—
502C23—4882no—
503C23—4882no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23D8×D6111yescharacteristic
508Q8—41no—
509Q8—41no—
510D4—41no—
511D4—41no—
512D4—82no—
513D4—4882no—
514D4—4882no—
515D4—4882no—
516D4—4882no—
517D4—4882no—
518D4—4882no—
519D4—4882no—
520D4—2441no—
521D4—2441no—
522D4—2441no—
523D4—2441no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—611no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—611no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—4882no—
539C2×C4—4882no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4D8×D6111yescharacteristic
543C2×C4C2×D4×D6111yescharacteristic
544C8—21no—
545C8—612no—
546C8—1221no—
547C8C2×D4×D6111yescharacteristic
548C8#41×D6111yescharacteristic
549C22—21no—
550C22—21no—
551C22—21no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—612no—
556C22—612no—
557C22—1221no—
558C22—1221no—
559C22—1221no—
560C22—1221no—
561C22—9764no—
562C22—4881no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—4881no—
567C22—4881no—
568C22—2444no—
569C22C2×D8×D6111yescharacteristic
570C4—21no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—612no—
575C4—1221no—
576C4—4881no—
577C4—4881no—
578C4#121×D6111yescharacteristic
579C4#135×D6111yescharacteristic
580C2—41no—
581C2—161no—
582C2—1222no—
583C2—1222no—
584C2—9761no—
585C2—9761no—
586C2#466×D6111yescharacteristic
587C2#509×D6111yescharacteristic
588C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2220C2
socleC22×C61C22×D8
hypercenter#2220D61
hypocenterC61#33605
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2220

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. C2×M32×D61C2semidirect
3. #21455402C2semidirect
4. #466×D61C2semidirect
5. #21447934C2semidirect
6. #21447922C2semidirect
7. #21447920C2semidirect
8. #21447918C2semidirect
9. #21447912C2semidirect
10. #21447906C2semidirect
11. #21447904C2semidirect
12. #21447604C2semidirect
13. #21447533C2semidirect
14. #21447530C2semidirect
15. C61×#2220C2semidirect
22. C2×#14519720C22semidirect
23. C2×D976C22semidirect
24. C2×#14519718C22semidirect
25. C2×#14519717C22semidirect
29. C2×C16×D61C22semidirect
30. C2×C16×D61C22semidirect
31. #16840000C22semidirect
33. #16839999C22semidirect
35. #16839998C22semidirect
45. #16839590C22semidirect
46. #16839589C22semidirect
47. #16839587C22semidirect
48. #16839566C22semidirect
49. #16839564C22semidirect
50. #16839564C22semidirect
51. #16839563C22semidirect
52. #16839562C22semidirect
53. #16839562C22semidirect
54. C2×SD32×C61C22semidirect
55. C2×M32×C61C22semidirect
56. C61×#467C22semidirect
58. C61×#366C22semidirect
59. C61×#358C22semidirect
60. C61×#357C22semidirect
100. #14519716D4semidirect
198. C16×C61C2×D4semidirect
200. #11758185C2×D4semidirect
275. C2×D61#509semidirect
291. #2220D61semidirect
317. C2×SD32C2×D61semidirect
324. C2×M32C2×D61semidirect
328. #467C2×D61semidirect
346. #366C2×D61semidirect
348. #358C2×D61semidirect
354. #357C2×D61semidirect
355. C61#33605semidirect
495. C16D4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
C2×M32×D612
#214554022
#466×D612
#214479342
#214479222
#214479202
#214479182
#214479122
#214479062
#214479042
#214476042
#214475332
#214475302
C61×#22202
#3360561