Finite Group Databasebeta

← #29833998

Subgroups — #29833998

Order 15616.

1104 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840197C211yesmaximal, characteristic, complemented
2C2×#16840197C211yesmaximal, characteristic, complemented
3C2×#16840032C211yesmaximal, characteristic, complemented
4#21447783C214yesmaximal, complemented
5#21447776C214yesmaximal, complemented
6#21447568C211yesmaximal, characteristic, complemented
7#21447547C211yesmaximal, characteristic, complemented
8#21447447C211yesmaximal, characteristic, complemented
9C61×#3584C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C22×D488C2211yescharacteristic, complemented
12#16840197—22no—
13#16840197—22no—
14#16840197—22no—
15#16840197C2214yescomplemented
16C2×#14519720—21no—
17C2×#14519720C2212yescomplemented
18C2×#14519718—21no—
19C2×#14519718C2212yescomplemented
20C2×#14519717C2211yescharacteristic, complemented
21C2×#14519717C2211yescharacteristic, complemented
22#16840032—22no—
23#16840032C2214yescomplemented
24C2×#14519713C2211yescharacteristic, complemented
25C2×#14519693—21no—
26#16839578C2212yescomplemented
27#16839575C2212yescomplemented
28#16839569C2214yescomplemented
29#16839546C2214yescomplemented
30C61×#585—21no—
31C61×#578—21no—
32C61×#578C2212yescomplemented
33C61×#577—22no—
34C61×#577—22no—
35C61×#576—21no—
36C61×#576C2212yescomplemented
37C61×#575—22no—
38C61×#575—21no—
39C2×#135×C61—22no—
40C2×#135×C61C2211yescharacteristic
41C2×#135×C61C2211yescharacteristic
42C2×#131×C61C2211yescharacteristic
43C22×D244—21no—
44#14519802—24no—
45#14519802—42no—
46C2×#11758201—41no—
47C2×D488—21no—
48C2×D488—21no—
49C2×D488—21no—
50C2×D488—21no—
51C2×D488—21no—
52C2×D488C2312yes—
53C2×#11758199—41no—
54C2×#11758194—21no—
55#14519720—24no—
56#14519720—24no—
57#14519718—24no—
58#14519718—24no—
59#14519717—21no—
60#14519717—21no—
61#14519717—21no—
62#14519717C2312yes—
63C2×#11758185—21no—
64C2×#11758185C2312yes—
65#14519713—21no—
66#14519713C2312yes—
67#14519693—24no—
682−1+4×C61—22no—
692−1+4×C61—22no—
702−1+4×C61—41no—
71H(5,𝔽2)×C61—22no—
72H(5,𝔽2)×C61—22no—
73H(5,𝔽2)×C61—42no—
74H(5,𝔽2)×C61—41no—
75C2×#41×C61—24no—
76C2×#41×C61—21no—
77C2×#41×C61—21no—
78C2×#41×C61—42no—
79C2×#41×C61—42no—
80C2×#41×C61D411yescharacteristic
81#137×C61—22no—
82#137×C61—24no—
83#137×C61—24no—
84#137×C61—22no—
85AGL(1,ℤ/8)×C61—22no—
86AGL(1,ℤ/8)×C61—24no—
87AGL(1,ℤ/8)×C61—24no—
88AGL(1,ℤ/8)×C61—22no—
89#135×C61—24no—
90#135×C61—24no—
91#135×C61—22no—
92#135×C61—22no—
93#135×C61—21no—
94#135×C61—21no—
95#135×C61—21no—
96#135×C61—21no—
97#135×C61D412yes—
98#135×C61D412yes—
99#135×C61D412yes—
100#135×C61C2312yes—
101C2×Q16×C61—22no—
102C2×Q16×C61D411yescharacteristic
103C2×Q16×C61C2311yescharacteristic
104C2×SD16×C61—24no—
105C2×SD16×C61—21no—
106C2×SD16×C61—21no—
107C2×D8×C61—22no—
108C2×D8×C61D411yescharacteristic
109C2×D8×C61C2311yescharacteristic
110#131×C61—21no—
111#131×C61—21no—
112#131×C61D412yes—
113#131×C61C2312yes—
114C2×M16×C61—21no—
115C2×M16×C61D411yescharacteristic
116C22×C8×C61—21no—
117C22×C8×C61C2311yescharacteristic
118C23×D61—41no—
119#41×C61—21no—
120#41×C61—21no—
121#41×C61—24no—
122#41×C61—24no—
123#41×C61—22no—
124#41×C61—22no—
125#41×C61—22no—
126#41×C61—22no—
127#41×C61—44no—
128#41×C61—44no—
129#41×C61—44no—
130#41×C61—44no—
131#41×C61—42no—
132#41×C61—42no—
133#41×C61—42no—
134#41×C61—42no—
135#41×C61—41no—
136#41×C61—41no—
137#41×C61—41no—
138#41×C61—41no—
139#41×C61D812yes—
140#41×C61C2×D412yes—
141C2×Q8×C61—24no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—42no—
145C2×Q8×C61—42no—
146C2×Q8×C61D811yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—24no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—44no—
152C2×D4×C61—44no—
153C2×D4×C61—42no—
154C2×D4×C61—44no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61—42no—
158C2×D4×C61D811yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—44no—
161C22×C4×C61—41no—
162C22×C4×C61—41no—
163C22×C4×C61C2×D411yescharacteristic
164C2×D244—22no—
165C2×D244—41no—
166C2×D244—41no—
167C2×D244—41no—
168C2×D244—41no—
169C2×D244—41no—
170Q16×C61—24no—
171Q16×C61—21no—
172Q16×C61—21no—
173Q16×C61C2×D412yes—
174Q16×C61C2×D412yes—
175SD16×C61—24no—
176SD16×C61—24no—
177SD16×C61—22no—
178SD16×C61—22no—
179SD16×C61—22no—
180SD16×C61—22no—
181D8×C61—24no—
182D8×C61—21no—
183D8×C61—21no—
184D8×C61C2×D412yes—
185D8×C61C2×D412yes—
186M16×C61—21no—
187M16×C61—22no—
188M16×C61—22no—
189M16×C61C2×D412yes—
190C2×C8×C61—21no—
191C2×C8×C61—22no—
192C2×C8×C61—22no—
193C2×C8×C61C2×D411yescharacteristic
194C2×C8×C61C2×D411yescharacteristic
195C2×C8×C61C2×D411yescharacteristic
196C2×C8×C61C2×D411yescharacteristic
197C2×C8×C61C2×D411yescharacteristic
198C2×C8×C61C2411yescharacteristic
199#11758201—44no—
200#11758199—44no—
201#11758194—22no—
202#11758194—41no—
203C2×#92418—41no—
204D488—22no—
205D488—22no—
206D488—41no—
207D488—41no—
208#11758185—22no—
209#11758185—22no—
210C23×C61—21no—
211C23×C61—44no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61AGL(1,ℤ/8)11yescharacteristic
215C22×D61—42no—
216C22×D61—81no—
217C22×D61—81no—
218C22×D61—162no—
219Q8×C61—21no—
220Q8×C61—22no—
221Q8×C61—22no—
222Q8×C61—44no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61C2×D812yes—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—21no—
231D4×C61—44no—
232D4×C61—44no—
233D4×C61—44no—
234D4×C61—42no—
235D4×C61—42no—
236D4×C61—42no—
237D4×C61—42no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61C2×D812yes—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—44no—
245C2×C4×C61—44no—
246C2×C4×C61—44no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61C2×D811yescharacteristic
256C2×C4×C61C2×D811yescharacteristic
257C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259D244—42no—
260D244—42no—
261D244—81no—
262D244—81no—
263C8×C61—22no—
264C8×C61#12011yescharacteristic
265C8×C61#12011yescharacteristic
266C8×C61C22×D411yescharacteristic
267C8×C61C22×D411yescharacteristic
268#92418—42no—
269#33541—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—41no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—81no—
275C22×C61—161no—
276C22×C61—161no—
277C22×C61—164no—
278C22×C61#44711yescharacteristic
279C22×C61#44711yescharacteristic
280C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
281C2×D61—82no—
282C2×D61—162no—
283C2×D61—162no—
284C2×D61—162no—
285C4×C61—22no—
286C4×C61—42no—
287C4×C61—42no—
288C4×C61#44711yescharacteristic
289C4×C61#44711yescharacteristic
290C4×C61C2×#12011yescharacteristic
291C4×C61C22×D811yescharacteristic
292#3584D6111yescharacteristic, complemented
293C2×#509—611no—
294C2×#509—611no—
295C2×#472—611no—
296#2193—614no—
297#2192—614no—
298#2148—611no—
299#2143—611no—
300#2117—611no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—42no—
305C2×C61—82no—
306C2×C61—82no—
307C2×C61C2×#44711yescharacteristic
308D61—162no—
309#585—21no—
310#578—21no—
311#578C2×D6112yescomplemented
312#577—22no—
313#577—22no—
314#576—21no—
315#576C2×D6112yescomplemented
316#575—22no—
317#575—21no—
318C2×AGL(1,ℤ/8)—1221no—
319C2×#135—22no—
320C2×#135C2×D6111yescharacteristic
321C2×#135C2×D6111yescharacteristic
322C22×D8—611no—
323C2×#131C2×D6111yescharacteristic
324#509—614no—
325#509—1222no—
326#509—1222no—
327#509—1222no—
328C2×SD32—612no—
329C2×SD32—1221no—
330C2×D16—612no—
331C2×D16—1221no—
332C2×M32—611no—
333C2×M32—611no—
334#472—614no—
335#472—1222no—
336C2×#108—611no—
337C2×#100—1221no—
338#362—612no—
339#361—612no—
340#359—614no—
341#350—614no—
342C61#3354111yescharacteristic, complemented, Sylow
3432−1+4—22no—
3442−1+4—22no—
3452−1+4—41no—
346H(5,𝔽2)—22no—
347H(5,𝔽2)—22no—
348H(5,𝔽2)—42no—
349H(5,𝔽2)—41no—
350C2×#41—24no—
351C2×#41—21no—
352C2×#41—21no—
353C2×#41—42no—
354C2×#41—42no—
355C2×#41C22⋊D6111yescharacteristic
356C22×D4—1221no—
357#137—22no—
358#137—24no—
359#137—24no—
360#137—22no—
361AGL(1,ℤ/8)—22no—
362AGL(1,ℤ/8)—24no—
363AGL(1,ℤ/8)—24no—
364AGL(1,ℤ/8)—22no—
365AGL(1,ℤ/8)—1224no—
366AGL(1,ℤ/8)—2442no—
367#135—24no—
368#135—24no—
369#135—22no—
370#135—22no—
371#135—21no—
372#135—21no—
373#135—21no—
374#135—21no—
375#135C22⋊D6112yes—
376#135C22⋊D6112yes—
377#135C22⋊D6112yes—
378#135C22×D6112yes—
379C2×Q16—22no—
380C2×Q16C22⋊D6111yescharacteristic
381C2×Q16C22×D6111yescharacteristic
382C2×SD16—24no—
383C2×SD16—21no—
384C2×SD16—21no—
385C2×SD16—2441no—
386C2×D8—22no—
387C2×D8—612no—
388C2×D8—1221no—
389C2×D8—1221no—
390C2×D8—1221no—
391C2×D8—1221no—
392C2×D8—1221no—
393C2×D8—2441no—
394C2×D8C22⋊D6111yescharacteristic
395C2×D8C22×D6111yescharacteristic
396#131—21no—
397#131—21no—
398#131C22⋊D6112yes—
399#131C22×D6112yes—
400C2×M16—21no—
401C2×M16—1221no—
402C2×M16C22⋊D6111yescharacteristic
403C22×C8—21no—
404C22×C8C22×D6111yescharacteristic
405SD32—1224no—
406SD32—1224no—
407D16—1224no—
408D16—1224no—
409M32—612no—
410M32—1221no—
411M32—1221no—
412M32—1221no—
413C2×C16—612no—
414C2×C16—1221no—
415#108—612no—
416#108—1221no—
417#100—1224no—
418C24—2441no—
419#41—21no—
420#41—21no—
421#41—24no—
422#41—24no—
423#41—22no—
424#41—22no—
425#41—22no—
426#41—22no—
427#41—44no—
428#41—44no—
429#41—44no—
430#41—44no—
431#41—42no—
432#41—42no—
433#41—42no—
434#41—42no—
435#41—41no—
436#41—41no—
437#41—41no—
438#41—41no—
439#41#1175819912yes—
440#41C2×C22⋊D6112yes—
441C2×Q8—24no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8#1175819911yescharacteristic
447C2×D4—21no—
448C2×D4—24no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—42no—
458C2×D4—1222no—
459C2×D4—2441no—
460C2×D4—2441no—
461C2×D4—2441no—
462C2×D4—2441no—
463C2×D4—2441no—
464C2×D4#1175819911yescharacteristic
465C22×C4—21no—
466C22×C4—44no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4C2×C22⋊D6111yescharacteristic
470Q16—24no—
471Q16—21no—
472Q16—21no—
473Q16C2×C22⋊D6112yes—
474Q16C2×C22⋊D6112yes—
475SD16—24no—
476SD16—24no—
477SD16—22no—
478SD16—22no—
479SD16—22no—
480SD16—22no—
481SD16—2444no—
482D8—24no—
483D8—21no—
484D8—21no—
485D8—1222no—
486D8—1222no—
487D8—2444no—
488D8—2441no—
489D8—2441no—
490D8C2×C22⋊D6112yes—
491D8C2×C22⋊D6112yes—
492M16—21no—
493M16—22no—
494M16—22no—
495M16—1222no—
496M16—2441no—
497M16C2×C22⋊D6112yes—
498C2×C8—21no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—2441no—
502C2×C8C2×C22⋊D6111yescharacteristic
503C2×C8C2×C22⋊D6111yescharacteristic
504C2×C8C2×C22⋊D6111yescharacteristic
505C2×C8C2×C22⋊D6111yescharacteristic
506C2×C8C2×C22⋊D6111yescharacteristic
507C2×C8C23×D6111yescharacteristic
508C16—1222no—
509C16—1222no—
510C23—21no—
511C23—44no—
512C23—41no—
513C23—41no—
514C23—9762no—
515C23—4881no—
516C23—4881no—
517C23—2442no—
518C23#1451980311yescharacteristic
519Q8—21no—
520Q8—22no—
521Q8—22no—
522Q8—44no—
523Q8—41no—
524Q8—41no—
525Q8C2×#1175819912yes—
526D4—22no—
527D4—22no—
528D4—22no—
529D4—22no—
530D4—21no—
531D4—44no—
532D4—44no—
533D4—44no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—41no—
539D4—41no—
540D4—4881no—
541D4—4881no—
542D4—2442no—
543D4—2442no—
544D4C2×#1175819912yes—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4#1451974611yescharacteristic
558C2×C4#1451974611yescharacteristic
559C2×C4C2×#1175819911yescharacteristic
560C2×C4C2×#1175819911yescharacteristic
561C2×C4#1451980311yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C8—22no—
564C8—2442no—
565C8#1451974611yescharacteristic
566C8#1451974611yescharacteristic
567C8C22×C22⋊D6111yescharacteristic
568C8C22×C22⋊D6111yescharacteristic
569C22—22no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—161no—
575C22—161no—
576C22—164no—
577C22—9762no—
578C22—9762no—
579C22—9762no—
580C22—4882no—
581C22#1683989611yescharacteristic
582C22#1683989611yescharacteristic
583C22C2×#1451980311yescharacteristic
584C4—22no—
585C4—42no—
586C4—42no—
587C4#1683989611yescharacteristic
588C4#1683989611yescharacteristic
589C4C2×#1451974611yescharacteristic
590C4C22×#1175819911yescharacteristic
591C2—21no—
592C2—21no—
593C2—21no—
594C2—42no—
595C2—82no—
596C2—82no—
597C2—9762no—
598C2C2×#1683989611yescharacteristic
subgroupG/H
centerC2C2×#16839896
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3584C2
socleC2×C61C2×#447
hypercenter#3584D61
hypocenterC61#33541
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3584

Complements

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

NH ≅ G/Ntype
1. C2×#16840197C2semidirect
2. C2×#16840197C2semidirect
3. C2×#16840032C2semidirect
4. #21447783C2semidirect
5. #21447776C2semidirect
6. #21447568C2semidirect
7. #21447547C2semidirect
8. #21447447C2semidirect
9. C61×#3584C2semidirect
11. C22×D488C22semidirect
15. #16840197C22semidirect
17. C2×#14519720C22semidirect
19. C2×#14519718C22semidirect
20. C2×#14519717C22semidirect
21. C2×#14519717C22semidirect
23. #16840032C22semidirect
24. C2×#14519713C22semidirect
26. #16839578C22semidirect
27. #16839575C22semidirect
28. #16839569C22semidirect
29. #16839546C22semidirect
32. C61×#578C22semidirect
36. C61×#576C22semidirect
292. #3584D61semidirect
311. #578C2×D61semidirect
315. #576C2×D61semidirect
342. C61#33541semidirect

Maximal subgroups

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

subgroupindex
C2×#168401972
C2×#168400322
#214477832
#214477762
#214475682
#214475472
#214474472
C61×#35842
#3354161