Finite Group Databasebeta

← #29835464

Subgroups — #29835464

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840199C211yesmaximal, characteristic, complemented
2C2×#16840196C211yesmaximal, characteristic, complemented
3C2×#16840193C211yesmaximal, characteristic, complemented
4C2×D16×D61C211yesmaximal, characteristic, complemented
5#21456301C211yesmaximal, characteristic, complemented
6#21456299C211yesmaximal, characteristic, complemented
7#21448085C211yesmaximal, characteristic, complemented
8#21448084C211yesmaximal, characteristic, complemented
9#21448082C211yesmaximal, characteristic, complemented
10#21448079C211yesmaximal, characteristic, complemented
11#21448078C211yesmaximal, characteristic, complemented
12#21448068C211yesmaximal, characteristic, complemented
13#21448067C211yesmaximal, characteristic, complemented
14#21447417C211yesmaximal, characteristic, complemented
15C61×#2250C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17C2×#14519801—21no—
18C2×#14519800C2211yescharacteristic
19C2×#14519796C2211yescharacteristic
20C2×D8×D61C2211yescharacteristic
21C2×D8×D61C2211yescharacteristic
22#16840330—21no—
23#16840325—21no—
24#16840199—24no—
25#16840196—24no—
26#16840193—24no—
27C2×#14519723C2211yescharacteristic, complemented
28C2×#14519722C2211yescharacteristic, complemented
29C2×#14519722C2211yescharacteristic, complemented
30C2×#14519721C2211yescharacteristic, complemented
31C2×#14519720C2211yescharacteristic, complemented
32C2×D976C2211yescharacteristic
33C2×D976C2211yescharacteristic, complemented
34C2×#14519718C2211yescharacteristic
35C2×#14519718C2211yescharacteristic
36C2×#14519718C2211yescharacteristic, complemented
37C2×#14519718C2211yescharacteristic, complemented
38C2×#14519718C2211yescharacteristic, complemented
39D16×D61—24no—
40C2×#14519716C2211yescharacteristic
41C2×C16×D61C2211yescharacteristic
42#16840152—21no—
43#16840150—21no—
44#16840149—21no—
45#16840148—21no—
46#16840147C2211yescharacteristic, complemented
47#16840146C2211yescharacteristic, complemented
48#16840145C2211yescharacteristic, complemented
49#16840139C2211yescharacteristic, complemented
50#16840137C2211yescharacteristic, complemented
51#16840137C2211yescharacteristic, complemented
52#16839716C2211yescharacteristic
53#16839537C2211yescharacteristic, complemented
54C4×#11758185C2211yescharacteristic, complemented
55C16×Dic61C2211yescharacteristic, complemented
56C2×SD32×C61C2211yescharacteristic, complemented
57C2×SD32×C61C2211yescharacteristic, complemented
58C2×D16×C61C2211yescharacteristic
59C2×D16×C61C2211yescharacteristic, complemented
60C61×#495C2211yescharacteristic, complemented
61C61×#493C2211yescharacteristic, complemented
62C61×#346C2211yescharacteristic, complemented
63C2×#11758226—21no—
64C2×#11758224—21no—
65C2×D4×D61—21no—
66C2×D4×D61—21no—
67#14519804—44no—
68#14519801—44no—
69#14519800—22no—
70#14519800—22no—
71#14519796—22no—
72#14519796—22no—
73C2×Dic244C2311yescharacteristic
74C2×#11758190—21no—
75C2×#11758200—21no—
76C2×#11758200—21no—
77C2×#11758201—21no—
78C2×#11758201—21no—
79C2×D488—21no—
80C2×D488C2311yescharacteristic
81C2×D488C2311yescharacteristic
82C2×D488C2311yescharacteristic
83C2×#11758199—21no—
84C2×#11758199—21no—
85C2×#11758199—21no—
86C2×#11758199—21no—
87D8×D61—22no—
88D8×D61—22no—
89D8×D61—22no—
90D8×D61—22no—
91C2×#11758189D411yescharacteristic
92C2×C8×D61C2311yescharacteristic
93#14519774—21no—
94#14519773—21no—
95#14519773—21no—
96#14519766—21no—
97#14519765—21no—
98#14519764—21no—
99#14519750—42no—
100#14519749—42no—
101#14519747—42no—
102#14519745—42no—
103#14519734D411yescharacteristic
104#14519723—22no—
105#14519722—22no—
106#14519722—22no—
107#14519721—22no—
108#14519720—22no—
109D976—22no—
110D976—22no—
111#14519718—22no—
112#14519718—22no—
113#14519718—22no—
114#14519718—22no—
115#14519718—22no—
116#14519716D414yescomplemented
117C2×#11758185C2311yescharacteristic
118C2×#11758185C2311yescharacteristic
119C16×D61D414yescomplemented
120#14519688D411yescharacteristic
121#14519687D411yescharacteristic
122C8×Dic61C2311yescharacteristic
123C2×Q16×C61C2311yescharacteristic
124C2×SD16×C61—21no—
125C2×D8×C61—21no—
126C2×D8×C61C2311yescharacteristic
127C2×D8×C61C2311yescharacteristic
128C2×D8×C61C2311yescharacteristic
129#127×C61—21no—
130#124×C61—21no—
131SD32×C61—22no—
132SD32×C61—22no—
133D16×C61—22no—
134D16×C61—22no—
135C2×C16×C61C2311yescharacteristic
136C2×C16×C61C2311yescharacteristic
137C4×C8×C61C2311yescharacteristic
138C23×D61—41no—
139C23×D61—41no—
140C2×Q8×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—42no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C2×C22⋊D61—41no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C22×Dic61—41no—
151#11758226—42no—
152#11758226—42no—
153#11758224—42no—
154#11758224—42no—
155D4×D61—42no—
156D4×D61—42no—
157D4×D61—42no—
158D4×D61—42no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—41no—
163C2×D244—42no—
164C2×C4×D61—41no—
165C2×C4×D61C2×D411yescharacteristic
166C2×Dic122—21no—
167Q16×C61—21no—
168Q16×C61—21no—
169SD16×C61—42no—
170D8×C61—21no—
171D8×C61—21no—
172D8×C61—21no—
173D8×C61—21no—
174D8×C61—21no—
175D8×C61—21no—
176D8×C61—42no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2411yescharacteristic
179C22⋊C4×C61—42no—
180C42×C61C2×D411yescharacteristic
181C22⋊Dic61—42no—
182#11758201—42no—
183#11758201—42no—
184#11758200—42no—
185#11758200—42no—
186#11758199—42no—
187#11758199—42no—
188#11758199—42no—
189#11758199—42no—
190#11758198—42no—
191#11758198—42no—
192#11758198D812yes—
193#11758196D812yes—
194C4×Dic61C2×D411yescharacteristic
195C2×#92418C2×D411yescharacteristic
196C2×#92418C2×D411yescharacteristic
197Dic244—21no—
198Dic244—21no—
199D488—21no—
200D488—21no—
201D488—21no—
202D488—21no—
203D488—21no—
204D488—21no—
205D488—42no—
206#11758190—42no—
207#11758189D814yes—
208C8×D61C2×D412yes—
209C8×D61C2×D412yes—
210C16×C61C2×D412yes—
211C16×C61C2×D412yes—
212#11758185C2×D412yes—
213#11758185C2×D412yes—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—82no—
221C22×D61—82no—
222C22×D61—162no—
223C22×D61—162no—
224C22×D61C2×D811yescharacteristic
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—162no—
234C2×C4×C61—41no—
235C2×C4×C61C2×D812yes—
236C2×C4×C61C22×D411yescharacteristic
237C22⋊D61—82no—
238C22⋊D61—82no—
239C22⋊D61—82no—
240C22⋊D61—162no—
241C22⋊D61—162no—
242C2×Dic61—41no—
243C2×Dic61—82no—
244C2×Dic61C2×D812yes—
245C2×Dic61C2×D811yescharacteristic
246D244—41no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—82no—
253D244—162no—
254C4×D61—82no—
255C4×D61#12712yes—
256C4×D61C2×D812yes—
257Dic122—41no—
258Dic122—41no—
259C8×C61C2×D812yes—
260C8×C61C22×D411yescharacteristic
261C8×C61C22×D411yescharacteristic
262#92418#12712yes—
263#92418C2×D812yes—
264#33720—611nomaximal, Sylow
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—82no—
269C22×C61—82no—
270C22×C61—82no—
271C22×C61C22×D811yescharacteristic
272C2×D61—22no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—82no—
277C2×D61—82no—
278C2×D61—82no—
279C2×D61—164no—
280C2×D61—164no—
281C2×D61#49312yes—
282C4×C61—22no—
283C4×C61—81no—
284C4×C61C2×#12711yescharacteristic
285C4×C61C22×D811yescharacteristic
286Dic61—22no—
287Dic61—81no—
288Dic61#49312yes—
289C2×#509—611no—
290C2×#509—611no—
291C2×#508—611no—
292C22×D16—611no—
293#3155—611no—
294#3155—611no—
295#2250—611no—
296#2250—611no—
297#2250—611no—
298#2250D6111yescharacteristic, complemented
299#2249—611no—
300#2249—611no—
301#2246—611no—
302#2246—611no—
303#2108—611no—
304C2×C61—161no—
305C2×C61—161no—
306C2×C61—161no—
307C2×C61C2×#49311yescharacteristic
308C2×C61#341512yes—
309D61—22no—
310D61—161no—
311D61—161no—
312D61—161no—
313C2×AGL(1,ℤ/8)—1221no—
314C2×AGL(1,ℤ/8)—1221no—
315C2×#135—611no—
316C2×#135—611no—
317C22×D8—611no—
318C22×D8—611no—
319#535—1221no—
320#535—1221no—
321#509—1224no—
322#509—1224no—
323#508—1224no—
324C2×Q32—611no—
325C2×SD32—611no—
326C2×SD32—611no—
327C2×SD32—611no—
328C2×SD32—611no—
329C2×SD32C2×D6111yescharacteristic, complemented
330C2×SD32C2×D6111yescharacteristic, complemented
331C2×D16—611no—
332C2×D16—611no—
333C2×D16—611no—
334C2×D16—611no—
335C2×D16—611no—
336C2×D16—611no—
337C2×D16—611no—
338C2×D16—1224no—
339C2×D16C2×D6111yescharacteristic
340C2×D16C2×D6111yescharacteristic, complemented
341C2×M32—611no—
342C22×C16—611no—
343#496—1221no—
344#496—1221no—
345#496—1221no—
346#496—1221no—
347#495—611no—
348#495—611no—
349#495—611no—
350#495C2×D6111yescharacteristic, complemented
351#493—611no—
352#493—611no—
353#493—611no—
354#493C2×D6111yescharacteristic, complemented
355#405—611no—
356#346—611no—
357#346C2×D6111yescharacteristic, complemented
358C4×C16—611no—
359C4×C16—611no—
360C61#3372011yescharacteristic, complemented, Sylow
361C2×#41—1221no—
362C2×#41—1221no—
363C22×D4—1221no—
364C22×D4—1221no—
365AGL(1,ℤ/8)—2444no—
366AGL(1,ℤ/8)—2444no—
367#135—1222no—
368#135—1222no—
369#135—1222no—
370#135—1222no—
371C2×Q16—611no—
372C2×Q16C22×D6111yescharacteristic
373C2×SD16—21no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×D8—21no—
380C2×D8—611no—
381C2×D8—611no—
382C2×D8—611no—
383C2×D8—1222no—
384C2×D8—1222no—
385C2×D8—1222no—
386C2×D8—1222no—
387C2×D8—1221no—
388C2×D8—1221no—
389C2×D8—1221no—
390C2×D8—1221no—
391C2×D8—1221no—
392C2×D8C22×D6111yescharacteristic
393C2×D8C22×D6111yescharacteristic
394C2×D8C22×D6111yescharacteristic
395C2×M16—611no—
396C22×C8—611no—
397#127—21no—
398#127—1221no—
399#127—1221no—
400#127—1221no—
401#124—21no—
402#124—1221no—
403#124—1221no—
404#124—1221no—
405#121—2442no—
406#121—2442no—
407#120—2442no—
408#120—2442no—
409#117—611no—
410Q32—1222no—
411SD32—22no—
412SD32—22no—
413SD32—1222no—
414SD32—1222no—
415SD32—1222no—
416SD32—1222no—
417D16—22no—
418D16—22no—
419D16—1222no—
420D16—1222no—
421D16—1222no—
422D16—1222no—
423D16—1222no—
424D16—1222no—
425D16—1222no—
426M32—614no—
427C2×C16—611no—
428C2×C16—611no—
429C2×C16—614no—
430C2×C16C22×D6111yescharacteristic
431C2×C16C22×D6111yescharacteristic
432#97—611no—
433#97—611no—
434C4×C8—611no—
435C4×C8C22×D6111yescharacteristic
436C24—2441no—
437C24—2441no—
438#41—2442no—
439#41—2442no—
440#41—2442no—
441#41—2442no—
442C2×Q8—21no—
443C2×Q8—1221no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—2441no—
452C2×D4—2442no—
453C2×D4—2442no—
454C2×D4—2442no—
455C2×D4—2442no—
456C2×D4—2442no—
457C2×D4—2442no—
458C2×D4—2441no—
459C2×D4—2441no—
460C2×D4—2441no—
461C2×D4—2442no—
462C22×C4—611no—
463C22×C4—2441no—
464C22×C4—2441no—
465Q16—21no—
466Q16—21no—
467Q16—1221no—
468Q16—1221no—
469SD16—42no—
470SD16—2442no—
471SD16—2442no—
472SD16—2442no—
473SD16—2442no—
474SD16—2442no—
475D8—21no—
476D8—21no—
477D8—21no—
478D8—21no—
479D8—21no—
480D8—21no—
481D8—42no—
482D8—1221no—
483D8—1221no—
484D8—1221no—
485D8—1221no—
486D8—1221no—
487D8—1221no—
488D8—2442no—
489D8—2442no—
490D8—2442no—
491D8—2442no—
492D8—2442no—
493M16—614no—
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—612no—
497C2×C8—612no—
498C2×C8D4×D6111yescharacteristic
499C2×C8C23×D6111yescharacteristic
500C4⋊C4—612no—
501C22⋊C4—42no—
502C22⋊C4—612no—
503C22⋊C4—2442no—
504C22⋊C4—2442no—
505C22⋊C4—2442no—
506C42—611no—
507C42D4×D6111yescharacteristic
508C16—612no—
509C16—612no—
510C16D4×D6112yes—
511C16D4×D6112yes—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—611no—
516C23—9762no—
517C23—9762no—
518C23—4882no—
519C23—4882no—
520C23—2441no—
521C23—2441no—
522C23—2441no—
523Q8—41no—
524Q8—41no—
525Q8—2441no—
526Q8—2441no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—162no—
534D4—9762no—
535D4—9762no—
536D4—9762no—
537D4—4882no—
538D4—4882no—
539D4—4882no—
540D4—4882no—
541D4—2441no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547C2×C4—41no—
548C2×C4—611no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—4882no—
553C2×C4—4882no—
554C2×C4—2441no—
555C2×C4D8×D6112yes—
556C2×C4C2×D4×D6111yescharacteristic
557C8—612no—
558C8—612no—
559C8D8×D6112yes—
560C8C2×D4×D6111yescharacteristic
561C8C2×D4×D6111yescharacteristic
562C22—82no—
563C22—82no—
564C22—82no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—612no—
569C22—1222no—
570C22—9764no—
571C22—9764no—
572C22—4882no—
573C22—4882no—
574C22—4882no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22C2×D8×D6111yescharacteristic
579C4—22no—
580C4—81no—
581C4—612no—
582C4—1222no—
583C4—4881no—
584C4#127×D6111yescharacteristic
585C4C2×D8×D6111yescharacteristic
586C2—161no—
587C2—161no—
588C2—161no—
589C2—1222no—
590C2—9761no—
591C2—9761no—
592C2—9761no—
593C2#493×D6111yescharacteristic
594C2#2145895912yes—
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2250C2
socleC22×C61C22×D8
hypercenter#2250D61
hypocenterC61#33720
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2250

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. C2×#16840196C2semidirect
3. C2×#16840193C2semidirect
4. C2×D16×D61C2semidirect
5. #21456301C2semidirect
6. #21456299C2semidirect
7. #21448085C2semidirect
8. #21448084C2semidirect
9. #21448082C2semidirect
10. #21448079C2semidirect
11. #21448078C2semidirect
12. #21448068C2semidirect
13. #21448067C2semidirect
14. #21447417C2semidirect
15. C61×#2250C2semidirect
27. C2×#14519723C22semidirect
28. C2×#14519722C22semidirect
29. C2×#14519722C22semidirect
30. C2×#14519721C22semidirect
31. C2×#14519720C22semidirect
33. C2×D976C22semidirect
36. C2×#14519718C22semidirect
37. C2×#14519718C22semidirect
38. C2×#14519718C22semidirect
46. #16840147C22semidirect
47. #16840146C22semidirect
48. #16840145C22semidirect
49. #16840139C22semidirect
50. #16840137C22semidirect
51. #16840137C22semidirect
53. #16839537C22semidirect
54. C4×#11758185C22semidirect
55. C16×Dic61C22semidirect
56. C2×SD32×C61C22semidirect
57. C2×SD32×C61C22semidirect
59. C2×D16×C61C22semidirect
60. C61×#495C22semidirect
61. C61×#493C22semidirect
62. C61×#346C22semidirect
116. #14519716D4semidirect
119. C16×D61D4semidirect
298. #2250D61semidirect
329. C2×SD32C2×D61semidirect
330. C2×SD32C2×D61semidirect
340. C2×D16C2×D61semidirect
350. #495C2×D61semidirect
354. #493C2×D61semidirect
357. #346C2×D61semidirect
360. C61#33720semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
C2×#168401962
C2×#168401932
C2×D16×D612
#214563012
#214562992
#214480852
#214480842
#214480822
#214480792
#214480782
#214480682
#214480672
#214474172
C61×#22502
#3372061