Finite Group Databasebeta

← #29834007

Subgroups — #29834007

Order 15616.

1084 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×#16840202C211yesmaximal, characteristic, complemented
2C2×#16840202C211yesmaximal, characteristic, complemented
3C2×#16840037C211yesmaximal, characteristic, complemented
4#21447797C214yesmaximal, complemented
5#21447790C214yesmaximal, complemented
6#21447568C211yesmaximal, characteristic, complemented
7#21447547C211yesmaximal, characteristic, complemented
8#21447447C211yesmaximal, characteristic, complemented
9C61×#3584C211yesmaximal, characteristic
10C2×#14519807—21no—
11C22×Dic244C2211yescharacteristic, complemented
12#16840202—22no—
13#16840202—22no—
14#16840202—22no—
15#16840202C2214yescomplemented
16C2×#14519723—21no—
17C2×#14519723C2212yescomplemented
18C2×#14519721—21no—
19C2×#14519721C2212yescomplemented
20C2×#14519717C2211yescharacteristic, complemented
21C2×#14519717C2211yescharacteristic, complemented
22#16840037—22no—
23#16840037C2214yescomplemented
24C2×#14519713C2211yescharacteristic, complemented
25C2×#14519695—21no—
26#16839578C2212yescomplemented
27#16839575C2212yescomplemented
28#16839569C2214yescomplemented
29#16839546C2214yescomplemented
30C61×#585—21no—
31C61×#578—21no—
32C61×#578C2212yes—
33C61×#577—22no—
34C61×#577—22no—
35C61×#576—21no—
36C61×#576C2212yes—
37C61×#575—22no—
38C61×#575—21no—
39C2×#135×C61—22no—
40C2×#135×C61C2211yescharacteristic
41C2×#135×C61C2211yescharacteristic
42C2×#131×C61C2211yescharacteristic
43C22×Dic122—21no—
44#14519807—24no—
45#14519807—42no—
46C2×Dic244—21no—
47C2×Dic244—21no—
48C2×Dic244—21no—
49C2×Dic244—21no—
50C2×Dic244—21no—
51C2×Dic244C2312yes—
52C2×#11758202—41no—
53C2×#11758200—41no—
54C2×#11758194—21no—
55#14519723—24no—
56#14519723—24no—
57#14519721—24no—
58#14519721—24no—
59#14519717—21no—
60#14519717—21no—
61#14519717—21no—
62#14519717C2312yes—
63C2×#11758185—21no—
64C2×#11758185C2312yes—
65#14519713—21no—
66#14519713C2312yes—
67#14519695—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
118#41×C61—21no—
119#41×C61—21no—
120#41×C61—24no—
121#41×C61—24no—
122#41×C61—22no—
123#41×C61—22no—
124#41×C61—22no—
125#41×C61—22no—
126#41×C61—44no—
127#41×C61—44no—
128#41×C61—44no—
129#41×C61—44no—
130#41×C61—42no—
131#41×C61—42no—
132#41×C61—42no—
133#41×C61—42no—
134#41×C61—41no—
135#41×C61—41no—
136#41×C61—41no—
137#41×C61—41no—
138#41×C61D812yes—
139#41×C61C2×D412yes—
140C2×Q8×C61—24no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—42no—
144C2×Q8×C61—42no—
145C2×Q8×C61D811yescharacteristic
146C2×D4×C61—21no—
147C2×D4×C61—24no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—44no—
151C2×D4×C61—44no—
152C2×D4×C61—42no—
153C2×D4×C61—44no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—42no—
157C2×D4×C61D811yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—44no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C22×C4×C61C2×D411yescharacteristic
163C22×Dic61—41no—
164C2×Dic122—22no—
165C2×Dic122—41no—
166C2×Dic122—41no—
167C2×Dic122—41no—
168C2×Dic122—41no—
169C2×Dic122—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#11758202—44no—
200#11758200—44no—
201#11758194—22no—
202#11758194—41no—
203C2×#92418—41no—
204Dic244—22no—
205Dic244—22no—
206Dic244—41no—
207Dic244—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
215Q8×C61—21no—
216Q8×C61—22no—
217Q8×C61—22no—
218Q8×C61—44no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61C2×D812yes—
222D4×C61—22no—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—21no—
227D4×C61—44no—
228D4×C61—44no—
229D4×C61—44no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—42no—
233D4×C61—42no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61C2×D812yes—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—44no—
241C2×C4×C61—44no—
242C2×C4×C61—44no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61#12011yescharacteristic
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61C2×D811yescharacteristic
252C2×C4×C61C2×D811yescharacteristic
253C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×Dic61—42no—
256C2×Dic61—81no—
257C2×Dic61—81no—
258Dic122—42no—
259Dic122—42no—
260Dic122—81no—
261Dic122—81no—
262C8×C61—22no—
263C8×C61#12011yescharacteristic
264C8×C61#12011yescharacteristic
265C8×C61C22×D411yescharacteristic
266C8×C61C22×D411yescharacteristic
267#92418—42no—
268#33542—611nomaximal, Sylow
269C22×C61—22no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—81no—
274C22×C61—161no—
275C22×C61—161no—
276C22×C61—164no—
277C22×C61#44711yescharacteristic
278C22×C61#44711yescharacteristic
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C4×C61—22no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61#44711yescharacteristic
284C4×C61#44711yescharacteristic
285C4×C61C2×#12011yescharacteristic
286C4×C61C22×D811yescharacteristic
287Dic61—82no—
288#3584D6111yescharacteristic
289C2×#510—611no—
290C2×#510—611no—
291C2×#473—611no—
292#2195—614no—
293#2194—614no—
294#2148—611no—
295#2143—611no—
296#2117—611no—
297C2×C61—21no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—42no—
301C2×C61—82no—
302C2×C61—82no—
303C2×C61C2×#44711yescharacteristic
304#585—21no—
305#578—21no—
306#578C2×D6112yes—
307#577—22no—
308#577—22no—
309#576—21no—
310#576C2×D6112yes—
311#575—22no—
312#575—21no—
313C2×#137—1221no—
314C2×#135—22no—
315C2×#135C2×D6111yescharacteristic
316C2×#135C2×D6111yescharacteristic
317C22×Q16—611no—
318C2×#131C2×D6111yescharacteristic
319#510—614no—
320#510—1222no—
321#510—1222no—
322#510—1222no—
323C2×Q32—612no—
324C2×Q32—1221no—
325C2×SD32—612no—
326C2×SD32—1221no—
327C2×M32—611no—
328C2×M32—611no—
329#473—614no—
330#473—1222no—
331C2×#108—611no—
332C2×#101—1221no—
333#362—612no—
334#361—612no—
335#359—614no—
336#350—614no—
337C61#3354211yescharacteristic, complemented, Sylow
3382−1+4—22no—
3392−1+4—22no—
3402−1+4—41no—
341H(5,𝔽2)—22no—
342H(5,𝔽2)—22no—
343H(5,𝔽2)—42no—
344H(5,𝔽2)—41no—
345C2×#41—24no—
346C2×#41—21no—
347C2×#41—21no—
348C2×#41—42no—
349C2×#41—42no—
350C2×#41C22⋊D6111yescharacteristic
351C22×Q8—1221no—
352#137—22no—
353#137—24no—
354#137—24no—
355#137—22no—
356#137—1224no—
357#137—2442no—
358AGL(1,ℤ/8)—22no—
359AGL(1,ℤ/8)—24no—
360AGL(1,ℤ/8)—24no—
361AGL(1,ℤ/8)—22no—
362#135—24no—
363#135—24no—
364#135—22no—
365#135—22no—
366#135—21no—
367#135—21no—
368#135—21no—
369#135—21no—
370#135C22⋊D6112yes—
371#135C22⋊D6112yes—
372#135C22⋊D6112yes—
373#135C22×D6112yes—
374C2×Q16—22no—
375C2×Q16—612no—
376C2×Q16—1221no—
377C2×Q16—1221no—
378C2×Q16—1221no—
379C2×Q16—1221no—
380C2×Q16—1221no—
381C2×Q16—2441no—
382C2×Q16C22⋊D6111yescharacteristic
383C2×Q16C22×D6111yescharacteristic
384C2×SD16—24no—
385C2×SD16—21no—
386C2×SD16—21no—
387C2×SD16—2441no—
388C2×D8—22no—
389C2×D8C22⋊D6111yescharacteristic
390C2×D8C22×D6111yescharacteristic
391#131—21no—
392#131—21no—
393#131C22⋊D6112yes—
394#131C22×D6112yes—
395C2×M16—21no—
396C2×M16—1221no—
397C2×M16C22⋊D6111yescharacteristic
398C22×C8—21no—
399C22×C8C22×D6111yescharacteristic
400Q32—1224no—
401Q32—1224no—
402SD32—1224no—
403SD32—1224no—
404M32—612no—
405M32—1221no—
406M32—1221no—
407M32—1221no—
408C2×C16—612no—
409C2×C16—1221no—
410#108—612no—
411#108—1221no—
412#101—1224no—
413#41—21no—
414#41—21no—
415#41—24no—
416#41—24no—
417#41—22no—
418#41—22no—
419#41—22no—
420#41—22no—
421#41—44no—
422#41—44no—
423#41—44no—
424#41—44no—
425#41—42no—
426#41—42no—
427#41—42no—
428#41—42no—
429#41—41no—
430#41—41no—
431#41—41no—
432#41—41no—
433#41#1175819912yes—
434#41C2×C22⋊D6112yes—
435C2×Q8—24no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—1222no—
441C2×Q8—2441no—
442C2×Q8—2441no—
443C2×Q8—2441no—
444C2×Q8—2441no—
445C2×Q8—2441no—
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#1175819911yescharacteristic
459C22×C4—21no—
460C22×C4—44no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—2441no—
464C22×C4C2×C22⋊D6111yescharacteristic
465Q16—24no—
466Q16—21no—
467Q16—21no—
468Q16—1222no—
469Q16—1222no—
470Q16—2444no—
471Q16—2441no—
472Q16—2441no—
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—
485D8C2×C22⋊D6112yes—
486D8C2×C22⋊D6112yes—
487M16—21no—
488M16—22no—
489M16—22no—
490M16—1222no—
491M16—2441no—
492M16C2×C22⋊D6112yes—
493C2×C8—21no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—2441no—
497C2×C8C2×C22⋊D6111yescharacteristic
498C2×C8C2×C22⋊D6111yescharacteristic
499C2×C8C2×C22⋊D6111yescharacteristic
500C2×C8C2×C22⋊D6111yescharacteristic
501C2×C8C2×C22⋊D6111yescharacteristic
502C2×C8C23×D6111yescharacteristic
503C16—1222no—
504C16—1222no—
505C23—21no—
506C23—44no—
507C23—41no—
508C23—41no—
509C23#1451980311yescharacteristic
510Q8—21no—
511Q8—22no—
512Q8—22no—
513Q8—44no—
514Q8—41no—
515Q8—41no—
516Q8—4881no—
517Q8—4881no—
518Q8—2442no—
519Q8—2442no—
520Q8C2×#1175819912yes—
521D4—22no—
522D4—22no—
523D4—22no—
524D4—22no—
525D4—21no—
526D4—44no—
527D4—44no—
528D4—44no—
529D4—42no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—41no—
534D4—41no—
535D4C2×#1175819912yes—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—4881no—
549C2×C4—4881no—
550C2×C4—2442no—
551C2×C4#1451974611yescharacteristic
552C2×C4#1451974611yescharacteristic
553C2×C4C2×#1175819911yescharacteristic
554C2×C4C2×#1175819911yescharacteristic
555C2×C4#1451980311yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C8—22no—
558C8—2442no—
559C8#1451974611yescharacteristic
560C8#1451974611yescharacteristic
561C8C22×C22⋊D6111yescharacteristic
562C8C22×C22⋊D6111yescharacteristic
563C22—22no—
564C22—41no—
565C22—42no—
566C22—42no—
567C22—81no—
568C22—161no—
569C22—161no—
570C22—164no—
571C22#1683989611yescharacteristic
572C22#1683989611yescharacteristic
573C22C2×#1451980311yescharacteristic
574C4—22no—
575C4—42no—
576C4—42no—
577C4—4882no—
578C4#1683989611yescharacteristic
579C4#1683989611yescharacteristic
580C4C2×#1451974611yescharacteristic
581C4C22×#1175819911yescharacteristic
582C2—21no—
583C2—21no—
584C2—21no—
585C2—42no—
586C2—82no—
587C2—82no—
588C2C2×#1683989611yescharacteristic
subgroupG/H
centerC2C2×#16839896
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
Fitting, prime radicalC61×#3584C2
socleC2×C61C2×#447
hypercenter#3584D61
hypocenterC61#33542
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×#16840202C2semidirect
2. C2×#16840202C2semidirect
3. C2×#16840037C2semidirect
4. #21447797C2semidirect
5. #21447790C2semidirect
6. #21447568C2semidirect
7. #21447547C2semidirect
8. #21447447C2semidirect
11. C22×Dic244C22semidirect
15. #16840202C22semidirect
17. C2×#14519723C22semidirect
19. C2×#14519721C22semidirect
20. C2×#14519717C22semidirect
21. C2×#14519717C22semidirect
23. #16840037C22semidirect
24. C2×#14519713C22semidirect
26. #16839578C22semidirect
27. #16839575C22semidirect
28. #16839569C22semidirect
29. #16839546C22semidirect
337. C61#33542semidirect

Maximal subgroups

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

subgroupindex
C2×#168402022
C2×#168400372
#214477972
#214477902
#214475682
#214475472
#214474472
C61×#35842
#3354261