Finite Group Databasebeta

← #30624326

Subgroups — #30624326

Order 15616.

652 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#444C211yesmaximal, characteristic, complemented
2#21462573C211yesmaximal, characteristic, complemented
3#21462231C212yesmaximal, complemented
4#21462229C212yesmaximal, complemented
5#21462227C212yesmaximal, complemented
6#21462225C212yesmaximal, complemented
7#21462166C211yesmaximal, characteristic, complemented
8#21462146C211yesmaximal, characteristic, complemented
9#21462139C211yesmaximal, characteristic, complemented
10C2×#16840653C211yesmaximal, characteristic, complemented
11#21458952C211yesmaximal, characteristic
12D61⋊#131—21no—
13#41×#6923—21no—
14C2×#14519852C2211yescharacteristic
15#16840809C2212yes—
16#16840808C2212yes—
17#16840807—22no—
18#16840803C2212yescomplemented
19#16840802C2212yescomplemented
20#16840801—22no—
21D61⋊#108C2211yescharacteristic
22#16840772C2211yescharacteristic
23D61⋊#104—21no—
24#16840752—21no—
25C2×#14519831C2211yescharacteristic
26#16840687C2212yescomplemented
27#16840686C2212yescomplemented
28#16840681C2212yescomplemented
29#16840680C2212yescomplemented
30#16840679C2212yes—
31#16840678C2212yes—
32#16840653C2212yescomplemented
33#16840653C2212yescomplemented
34#135×D61C2211yescharacteristic
35#135×D61C2211yescharacteristic
36#16840204C411yescharacteristic, complemented
37#16840204C2211yescharacteristic
38#16840201C411yescharacteristic, complemented
39#16840201C2211yescharacteristic
40#16840199C411yescharacteristic, complemented
41#16840199C2211yescharacteristic
42#16840196C411yescharacteristic, complemented
43#16840196C2211yescharacteristic
44#16840194C411yescharacteristic, complemented
45#16840192C411yescharacteristic, complemented
46#16840192C411yescharacteristic, complemented
47C2×#14519716C2211yescharacteristic
48C61×#508C411yescharacteristic, complemented
49#14519865—22no—
50#14519864—22no—
51#14519864—42no—
52D61⋊M16—21no—
53D61⋊M16—41no—
54C2×D61⋊C8—41no—
55C2×D61⋊C8D411yescharacteristic
56Q8×#6923—22no—
57D4×#6923—22no—
58D4×#6923—42no—
59#14519856—21no—
60#14519856—41no—
61C2×C4×#6923—41no—
62C2×C4×#6923D411yescharacteristic
63#14519852—21no—
64#14519852C2312yes—
65C2×#11758187—21no—
66#14519848—21no—
67#14519847C2312yes—
68#14519846—21no—
69#14519845C2312yes—
70#14519842—22no—
71#14519841—22no—
72#14519840—22no—
73#14519839—22no—
74#14519831D412yes—
75#14519831C2312yes—
76#41×D61—21no—
77#41×D61—21no—
78#14519800C2×C411yescharacteristic
79#14519800C2311yescharacteristic
80#14519799C2×C411yescharacteristic
81#14519799C2×C411yescharacteristic, complemented
82#14519798—21no—
83#14519798—21no—
84#14519797—21no—
85#14519797—21no—
86#14519796C2×C411yescharacteristic
87#14519796C2311yescharacteristic
88Q16×D61C2×C411yescharacteristic, complemented
89Q16×D61C2311yescharacteristic
90SD16×D61—21no—
91SD16×D61—21no—
92D8×D61C2×C411yescharacteristic, complemented
93D8×D61C2311yescharacteristic
94C2×C8×D61C2311yescharacteristic
95Dic488C2×C411yescharacteristic, complemented
96#14519723C2×C411yescharacteristic, complemented
97#14519723C2×C411yescharacteristic, complemented
98#14519722C2×C411yescharacteristic, complemented
99#14519722C2×C411yescharacteristic, complemented
100#14519721C2×C411yescharacteristic, complemented
101#14519721C2×C411yescharacteristic, complemented
102#14519720C2×C411yescharacteristic, complemented
103#14519720C2×C411yescharacteristic, complemented
104D976C2×C411yescharacteristic, complemented
105#14519718C2×C411yescharacteristic, complemented
106#14519718C2×C411yescharacteristic, complemented
107#14519716C2×C411yescharacteristic, complemented
108#14519716C2×C411yescharacteristic, complemented
109#14519716C2311yescharacteristic
110#14519716C2311yescharacteristic
111C2×#11758185C2×C411yescharacteristic, complemented
112#135×C61C2×C411yescharacteristic
113#135×C61C2×C411yescharacteristic, complemented
114Q32×C61C2×C411yescharacteristic, complemented
115SD32×C61C2×C411yescharacteristic, complemented
116SD32×C61C2×C411yescharacteristic, complemented
117D16×C61C2×C411yescharacteristic, complemented
118C2×C16×C61C2×C411yescharacteristic, complemented
119C22×#6923—21no—
120C22×#6923—42no—
121#41×C61—21no—
122#41×C61—21no—
123#11758226—21no—
124#11758226—21no—
125Q8×D61—21no—
126Q8×D61—21no—
127#11758224—21no—
128#11758224—21no—
129#11758224—41no—
130#11758224—41no—
131D4×D61—21no—
132D4×D61—21no—
133D4×D61—41no—
134D4×D61—41no—
135#11758222—21no—
136#11758222—21no—
137#11758222—41no—
138#11758222—41no—
139C2×C4×D61—41no—
140C2×C4×D61—41no—
141C2×C4×D61C2×D411yescharacteristic
142C22⋊#6923—42no—
143C22⋊#6923—41no—
144#11758217—42no—
145#11758217—41no—
146C2×#92420—21no—
147C2×#92420—42no—
148C4⋊#6923—22no—
149C4⋊#6923—42no—
150C4×#6923—21no—
151C4×#6923—42no—
152C4×#6923—41no—
153C4×#6923C2×D412yes—
154#11758213—22no—
155#11758213—42no—
156D61⋊C8—21no—
157D61⋊C8—42no—
158D61⋊C8—41no—
159D61⋊C8C2×D412yes—
160Q16×C61C22×C411yescharacteristic
161Q16×C61C22×C411yescharacteristic
162SD16×C61—21no—
163SD16×C61—21no—
164D8×C61C22×C411yescharacteristic
165D8×C61C22×C411yescharacteristic
166C2×C8×C61C22×C411yescharacteristic
167#11758202—21no—
168#11758202—21no—
169#11758201—21no—
170#11758201—21no—
171#11758200—21no—
172#11758200—21no—
173#11758199—21no—
174#11758199—21no—
175C2×#92418#4111yescharacteristic
176Dic244C22×C411yescharacteristic
177Dic244C22×C411yescharacteristic
178D488C22×C411yescharacteristic
179D488C22×C411yescharacteristic
180#11758190—21no—
181#11758190—21no—
182C8×D61C22×C411yescharacteristic
183C8×D61C2×D411yescharacteristic
184C8×D61#4111yescharacteristic
185C8×D61C2411yescharacteristic
186#11758187—22no—
187C16×C61C22×C411yescharacteristic
188C16×C61C22×C411yescharacteristic
189#11758185C22×C411yescharacteristic
190#11758185C22×C411yescharacteristic
191C22×D61—41no—
192C22×D61—41no—
193C22×D61AGL(1,ℤ/8)11yescharacteristic
194C2×#6923—22no—
195C2×#6923—42no—
196C2×#6923—81no—
197C2×#6923—84no—
198Q8×C61—21no—
199Q8×C61—21no—
200D4×C61—21no—
201D4×C61—21no—
202D4×C61—41no—
203D4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61C4×D411yescharacteristic
207C22⋊D61—41no—
208C22⋊D61—41no—
209C22⋊D61—41no—
210C22⋊D61—41no—
211C22⋊D61—41no—
212C22⋊D61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61AGL(1,ℤ/8)11yescharacteristic
216D244—21no—
217D244—21no—
218D244—41no—
219D244—41no—
220C4×D61—41no—
221C4×D61—41no—
222C4×D61—41no—
223C4×D61—41no—
224C4×D61—41no—
225C4×D61—41no—
226C4×D61C4×D411yescharacteristic
227C4×D61AGL(1,ℤ/8)11yescharacteristic
228C4×D61AGL(1,ℤ/8)11yescharacteristic
229C4×D61C22×D411yescharacteristic
230Dic122—21no—
231Dic122—21no—
232Dic122—41no—
233Dic122—41no—
234#92420—22no—
235#92420—42no—
236C8×C61C4×D411yescharacteristic
237C8×C61C23×C411yescharacteristic
238#92418C4×D411yescharacteristic
239#92418C2×#4111yescharacteristic
240#33468—611nomaximal, Sylow
241C22×C61—41no—
242C22×C61—41no—
243C22×C61#44211yescharacteristic
244C2×D61—41no—
245C2×D61—41no—
246C2×D61—41no—
247C2×D61—82no—
248C2×D61—82no—
249C2×D61#44211yescharacteristic
250C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
251#6923—42no—
252#6923—82no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61#44211yescharacteristic
256C4×C61C2×C4×D411yescharacteristic
257Dic61—41no—
258Dic61—41no—
259Dic61#44211yescharacteristic
260Dic61C2×AGL(1,ℤ/8)11yescharacteristic
261#3414—611no—
262C2×#444—611no—
263#2943—611no—
264AGL(1,ℤ/16)—612no—
265AGL(1,ℤ/16)—612no—
266#2180—612no—
267#2180—612no—
268#2155—611no—
269#2148—611no—
270#2145—611no—
271C2×#347—611no—
272C2×C61—21no—
273C2×C61—81no—
274C2×C61—81no—
275C2×C61C2×#44211yescharacteristic
276D61—21no—
277D61—21no—
278D61—81no—
279D61—81no—
280C2×#135—611no—
281C2×#135—611no—
282C2×#131—1221no—
283C4×#41—1221no—
284#510—611no—
285#510—611no—
286#510—611no—
287#510—611no—
288#509—611no—
289#509—611no—
290#509—611no—
291#509—611no—
292#508—611no—
293#508—611no—
294#508—611no—
295#508#692311yescharacteristic, complemented
296C2×M32—611no—
297C2×M32—611no—
298#444—612no—
299#444—612no—
300#444—1222no—
301#442—612no—
302#441—612no—
303#440—1222no—
304C2×#108—611no—
305#427—611no—
306C2×#104—1221no—
307#416—1221no—
308C2×#97—611no—
309#365—612no—
310#365—612no—
311#362—612no—
312#361—612no—
313#360—612no—
314#360—612no—
315#347—612no—
316#347—612no—
317C61#3346811yescharacteristic, complemented, Sylow
318C2×#41—1221no—
319C2×#41—1221no—
320#135—611no—
321#135—611no—
322#135—611no—
323#135—611no—
324#135—611no—
325#135—611no—
326#135—1221no—
327#135—1221no—
328#135—1221no—
329#135—1221no—
330#135C2×#692311yescharacteristic
331#135C2×#692311yescharacteristic, complemented
332C2×Q16—611no—
333C2×Q16—611no—
334C2×SD16—1221no—
335C2×SD16—1221no—
336C2×D8—611no—
337C2×D8—611no—
338#131—1222no—
339#131—1222no—
340#131—2442no—
341C2×M16—1221no—
342C2×M16—2441no—
343C22×C8—611no—
344C22×C8—611no—
345C22×C8—2441no—
346C4×Q8—1222no—
347C4×D4—1222no—
348C4×D4—2442no—
349#117—1221no—
350#117—2441no—
351C2×C42—611no—
352C2×C42—2441no—
353Q32—611no—
354Q32—611no—
355Q32—611no—
356Q32C2×#692311yescharacteristic, complemented
357SD32—611no—
358SD32—611no—
359SD32—611no—
360SD32—611no—
361SD32—611no—
362SD32—611no—
363SD32C2×#692311yescharacteristic, complemented
364SD32C2×#692311yescharacteristic, complemented
365D16—611no—
366D16—611no—
367D16—611no—
368D16C2×#692311yescharacteristic, complemented
369M32—611no—
370M32—611no—
371M32—611no—
372M32—611no—
373M32—612no—
374M32—1221no—
375C2×C16—611no—
376C2×C16—1221no—
377C2×C16C2×#692311yescharacteristic, complemented
378#108—612no—
379#108—1221no—
380#107—1221no—
381#106—612no—
382#104—1222no—
383#104—1222no—
384#103—1222no—
385#102—1222no—
386#97—612no—
387#97—612no—
388#41—21no—
389#41—21no—
390#41—1221no—
391#41—1221no—
392#41—1221no—
393#41—1221no—
394#41—1221no—
395#41—1221no—
396#41—2441no—
397#41—2441no—
398#41—2441no—
399#41—2441no—
400C2×Q8—1221no—
401C2×Q8—1221no—
402C2×D4—1221no—
403C2×D4—1221no—
404C2×D4—2441no—
405C2×D4—2441no—
406C22×C4—611no—
407C22×C4—1221no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4—2442no—
411Q16—611no—
412Q16—611no—
413Q16—1221no—
414Q16—1221no—
415Q16C22×#692311yescharacteristic
416Q16C22×#692311yescharacteristic
417SD16—21no—
418SD16—21no—
419SD16—1221no—
420SD16—1221no—
421SD16—1221no—
422SD16—1221no—
423SD16—1221no—
424SD16—1221no—
425D8—611no—
426D8—611no—
427D8—1221no—
428D8—1221no—
429D8C22×#692311yescharacteristic
430D8C22×#692311yescharacteristic
431M16—1222no—
432M16—2442no—
433M16—2442no—
434M16—2441no—
435C2×C8—611no—
436C2×C8—611no—
437C2×C8—611no—
438C2×C8—611no—
439C2×C8—611no—
440C2×C8—612no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—2442no—
444C2×C8—2442no—
445C2×C8—2441no—
446C2×C8C22×#692311yescharacteristic
447C4⋊C4—1222no—
448C4⋊C4—2442no—
449C22⋊C4—2442no—
450C22⋊C4—2441no—
451C42—612no—
452C42—1221no—
453C42—2441no—
454C42—2442no—
455C16—611no—
456C16—611no—
457C16—1222no—
458C16C22×#692311yescharacteristic
459C16C22×#692311yescharacteristic
460C23—611no—
461C23—2441no—
462C23—2441no—
463Q8—21no—
464Q8—21no—
465Q8—1221no—
466Q8—1221no—
467Q8—2441no—
468Q8—2441no—
469D4—21no—
470D4—21no—
471D4—41no—
472D4—41no—
473D4—1221no—
474D4—1221no—
475D4—2441no—
476D4—2441no—
477D4—2441no—
478D4—2441no—
479D4—2441no—
480D4—2441no—
481D4—2441no—
482D4—2441no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—611no—
486C2×C4—611no—
487C2×C4—611no—
488C2×C4—611no—
489C2×C4—611no—
490C2×C4—1222no—
491C2×C4—4884no—
492C2×C4—4881no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2442no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4D4×#692311yescharacteristic
503C8—611no—
504C8—611no—
505C8—1222no—
506C8—2442no—
507C8D4×#692311yescharacteristic
508C8C23×#692311yescharacteristic
509C22—41no—
510C22—41no—
511C22—611no—
512C22—611no—
513C22—4882no—
514C22—4882no—
515C22—2441no—
516C22—2441no—
517C22—2441no—
518C22#1684080311yescharacteristic
519C4—41no—
520C4—41no—
521C4—611no—
522C4—611no—
523C4—4882no—
524C4—2442no—
525C4—2441no—
526C4—2441no—
527C4#1684080311yescharacteristic
528C4C2×D4×#692311yescharacteristic
529C2—21no—
530C2—81no—
531C2—81no—
532C2—1221no—
533C2—1221no—
534C2—4881no—
535C2—4881no—
536C2C2×#1684080311yescharacteristic
subgroupG/H
centerC4#16840803
commutatorC8×C61C23×C4
FrattiniC8C23×#6923
FittingC61×#508C4
socleC2×C61C2×#442
prime radical#21458952C2
cosocleC8×D61C24
hypercenter#508#6923
hypocenterC61#33468
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. D61⋊#444C2semidirect
2. #21462573C2semidirect
3. #21462231C2semidirect
4. #21462229C2semidirect
5. #21462227C2semidirect
6. #21462225C2semidirect
7. #21462166C2semidirect
8. #21462146C2semidirect
9. #21462139C2semidirect
10. C2×#16840653C2semidirect
18. #16840803C22semidirect
19. #16840802C22semidirect
26. #16840687C22semidirect
27. #16840686C22semidirect
28. #16840681C22semidirect
29. #16840680C22semidirect
32. #16840653C22semidirect
33. #16840653C22semidirect
36. #16840204C4semidirect
38. #16840201C4semidirect
40. #16840199C4semidirect
42. #16840196C4semidirect
44. #16840194C4semidirect
45. #16840192C4semidirect
46. #16840192C4semidirect
48. C61×#508C4semidirect
81. #14519799C2×C4semidirect
88. Q16×D61C2×C4semidirect
92. D8×D61C2×C4semidirect
95. Dic488C2×C4semidirect
96. #14519723C2×C4semidirect
97. #14519723C2×C4semidirect
98. #14519722C2×C4semidirect
99. #14519722C2×C4semidirect
100. #14519721C2×C4semidirect
101. #14519721C2×C4semidirect
102. #14519720C2×C4semidirect
103. #14519720C2×C4semidirect
104. D976C2×C4semidirect
105. #14519718C2×C4semidirect
106. #14519718C2×C4semidirect
107. #14519716C2×C4semidirect
108. #14519716C2×C4semidirect
111. C2×#11758185C2×C4semidirect
113. #135×C61C2×C4semidirect
114. Q32×C61C2×C4semidirect
115. SD32×C61C2×C4semidirect
116. SD32×C61C2×C4semidirect
117. D16×C61C2×C4semidirect
118. C2×C16×C61C2×C4semidirect
295. #508#6923semidirect
317. C61#33468semidirect
331. #135C2×#6923semidirect
356. Q32C2×#6923semidirect
363. SD32C2×#6923semidirect
364. SD32C2×#6923semidirect
368. D16C2×#6923semidirect
377. C2×C16C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4442
#214625732
#214622312
#214622292
#214622272
#214622252
#214621662
#214621462
#214621392
C2×#168406532
#214589522
#3346861