Finite Group Databasebeta

← #30624306

Subgroups — #30624306

Order 15616.

652 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#443C211yesmaximal, characteristic, complemented
2C2×#16840804C211yesmaximal, characteristic, complemented
3#21462223C212yesmaximal, complemented
4#21462221C212yesmaximal, complemented
5#21462219C212yesmaximal, complemented
6#21462218C212yesmaximal, complemented
7#21462175C211yesmaximal, characteristic, complemented
8#21462134C211yesmaximal, characteristic, complemented
9#21462133C211yesmaximal, characteristic, complemented
10M32×#6923C211yesmaximal, characteristic, complemented
11#21458951C211yesmaximal, characteristic
12D61⋊#131—21no—
13C2×#14519864—21no—
14D61⋊M32C2211yescharacteristic
15#16840806C2211yescharacteristic, complemented
16#16840806C2211yescharacteristic, complemented
17#16840805—21no—
18#16840805—21no—
19#16840804—21no—
20#16840804—21no—
21#16840804C2212yescomplemented
22#16840804C2212yescomplemented
23#16840804C2211yescharacteristic, complemented
24#16840804C2211yescharacteristic, complemented
25C2×#14519848C2211yescharacteristic
26D61⋊#108C2211yescharacteristic
27D61⋊#104—21no—
28C2×#14519841—21no—
29C2×C8×#6923C2211yescharacteristic
30#16840691C2212yescomplemented
31#16840690C2212yescomplemented
32#16840677C2212yescomplemented
33#16840676C2212yescomplemented
34#16840676C2212yescomplemented
35#16840676C2212yescomplemented
36#16840649C2212yescomplemented
37C16×#6923C2212yescomplemented
38C2×#14519796C2211yescharacteristic
39#135×D61C2211yescharacteristic
40#16840205C411yescharacteristic, complemented
41#16840202C411yescharacteristic, complemented
42#16840201C412yescomplemented
43#16840198C411yescharacteristic, complemented
44#16840196C412yescomplemented
45#16840193C2212yes—
46#16840191C2212yes—
47M32×D61C2211yescharacteristic
48C61×#509C411yescharacteristic, complemented
49#14519865—21no—
50#14519865—21no—
51#14519864—22no—
52#14519864—22no—
53#14519864—21no—
54#14519864—21no—
55#14519864—41no—
56#14519864—41no—
57#14519864—41no—
58#14519864—41no—
59C2×#11758213—21no—
60D61⋊M16—21no—
61D61⋊M16—41no—
62C2×#11758217—41no—
63C2×D61⋊C8—41no—
64C2×D61⋊C8D411yescharacteristic
65C22×#92420—41no—
66C2×C4×#6923D411yescharacteristic
67#14519852—21no—
68#14519851—21no—
69D61⋊C16C2312yes—
70#14519848—21no—
71#14519848C2311yescharacteristic
72#14519848C2311yescharacteristic
73#14519848C2312yes—
74#14519847—21no—
75#14519842—21no—
76#14519842—21no—
77#14519841—22no—
78#14519841—22no—
79#14519841—21no—
80#14519841—21no—
81C8×#6923D411yescharacteristic
82C8×#6923D411yescharacteristic
83C8×#6923C2311yescharacteristic
84C8×#6923C2311yescharacteristic
85C2×#11758224—21no—
86#41×D61—21no—
87#14519800C2311yescharacteristic
88#14519799C2×C411yescharacteristic
89#14519798—21no—
90#14519797—21no—
91#14519796—21no—
92#14519796—21no—
93#14519796C2×C412yes—
94#14519796C2312yes—
95#14519796C2311yescharacteristic
96C2×Dic244C2×C411yescharacteristic
97Q16×D61C2×C411yescharacteristic
98C2×#11758200—21no—
99SD16×D61—21no—
100D8×D61C2×C411yescharacteristic
101C2×C8×D61C2311yescharacteristic
102Dic488C2×C412yescomplemented
103#14519723C2×C412yescomplemented
104#14519722C2×C412yescomplemented
105#14519721C2×C412yescomplemented
106#14519721C2×C412yescomplemented
107#14519718C2×C412yescomplemented
108#14519717C2×C411yescharacteristic
109#14519716C2×C412yes—
110C16×D61C2312yes—
111#135×C61C2×C411yescharacteristic
112C2×D8×C61C2×C411yescharacteristic
113SD32×C61C2×C412yescomplemented
114D16×C61C2×C412yescomplemented
115M32×C61C2×C411yescharacteristic
116C22×#6923—21no—
117#41×C61—21no—
118C2×D4×C61—21no—
119C2×C22⋊D61—41no—
120C22×Dic61—41no—
121#11758226—21no—
122Q8×D61—21no—
123#11758224—22no—
124#11758224—21no—
125#11758224—22no—
126#11758224—41no—
127#11758224—41no—
128#11758224—41no—
129D4×D61—21no—
130D4×D61—41no—
131#11758222—21no—
132#11758222—41no—
133C2×C4×D61—41no—
134C2×C4×D61C2×D411yescharacteristic
135C2×Dic122—21no—
136#11758217—41no—
137#11758217—42no—
138#11758217—42no—
139#11758217—41no—
140#11758217—41no—
141C2×#92420—41no—
142C2×#92420—42no—
143C2×#92420—42no—
144C2×#92420—41no—
145C2×#92420—41no—
146C2×#92420D811yescharacteristic
147C2×#92420D811yescharacteristic
148C4×#6923—21no—
149C4×#6923C2×D411yescharacteristic
150C4×#6923C2×D411yescharacteristic
151#11758213—22no—
152#11758213—21no—
153#11758213—21no—
154#11758213—41no—
155#11758213—41no—
156#11758213—41no—
157D61⋊C8—41no—
158D61⋊C8—41no—
159D61⋊C8—41no—
160D61⋊C8D811yescharacteristic
161D61⋊C8D811yescharacteristic
162D61⋊C8C2×D411yescharacteristic
163D61⋊C8C2×D411yescharacteristic
164Q16×C61C22×C411yescharacteristic
165SD16×C61—21no—
166D8×C61—21no—
167D8×C61C22×C411yescharacteristic
168D8×C61C22×C412yes—
169C2×C8×C61C22×C411yescharacteristic
170#11758202—21no—
171#11758201—21no—
172#11758200—22no—
173#11758200—21no—
174#11758200—22no—
175#11758199—21no—
176C2×#92418#4111yescharacteristic
177Dic244—21no—
178Dic244C22×C411yescharacteristic
179Dic244C22×C412yes—
180D488C22×C411yescharacteristic
181#11758190—21no—
182C8×D61C22×C411yescharacteristic
183C8×D61C2×D411yescharacteristic
184C8×D61#4111yescharacteristic
185C8×D61C2411yescharacteristic
186#11758187—22no—
187C16×C61C22×C412yes—
188#11758185C22×C412yes—
189C23×C61—41no—
190C22×D61—41no—
191C22×D61#13511yescharacteristic
192C2×#6923—21no—
193C2×#6923—21no—
194C2×#6923—42no—
195Q8×C61—21no—
196D4×C61—21no—
197D4×C61—22no—
198D4×C61—41no—
199D4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61C4×D411yescharacteristic
202C22⋊D61—41no—
203C22⋊D61—41no—
204C22⋊D61—42no—
205C22⋊D61—42no—
206C22⋊D61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61C2×D811yescharacteristic
212D244—21no—
213D244—41no—
214C4×D61—41no—
215C4×D61—41no—
216C4×D61—41no—
217C4×D61C4×D411yescharacteristic
218C4×D61C2×D811yescharacteristic
219C4×D61#13511yescharacteristic
220C4×D61C22×D411yescharacteristic
221Dic122—21no—
222Dic122—22no—
223Dic122—41no—
224Dic122—41no—
225#92420—41no—
226#92420—41no—
227#92420—42no—
228#92420C2×D811yescharacteristic
229#92420C2×D811yescharacteristic
230#92420C2×D811yescharacteristic
231#92420C2×D811yescharacteristic
232C8×C61C4×D411yescharacteristic
233C8×C61C23×C411yescharacteristic
234#92418C4×D411yescharacteristic
235#92418C2×#4111yescharacteristic
236#33463—611nomaximal, Sylow
237C22×C61—42no—
238C22×C61—41no—
239C22×C61—82no—
240C22×C61C4×D811yescharacteristic
241C2×D61—41no—
242C2×D61—41no—
243C2×D61—82no—
244C2×D61C4×D811yescharacteristic
245C2×D61C2×#13511yescharacteristic
246#6923—22no—
247#6923—41no—
248C4×C61—41no—
249C4×C61C4×D811yescharacteristic
250C4×C61C2×C4×D411yescharacteristic
251Dic61—41no—
252Dic61—42no—
253Dic61C4×D811yescharacteristic
254Dic61C22×D811yescharacteristic
255#3414—611no—
256C2×#443—611no—
257C2×#443—611no—
258#2179—612no—
259#2179—612no—
260#2178—612no—
261#2178—612no—
262#2161—611no—
263#2143—611no—
264#2143—611no—
265C4×M32—611no—
266C2×C61—21no—
267C2×C61—81no—
268C2×C61—82no—
269C2×C61C2×C4×D811yescharacteristic
270D61—21no—
271D61—21no—
272D61—81no—
273C2×#135—611no—
274C2×#135—611no—
275C2×#131—1221no—
276C2×#131—1221no—
277#510—611no—
278#510—612no—
279#510—611no—
280#509—611no—
281#509—612no—
282#509#692311yescharacteristic, complemented
283#508—612no—
284#508—612no—
285C2×M32—611no—
286C2×M32—611no—
287#443—612no—
288#443—611no—
289#443—612no—
290#443—611no—
291#443—611no—
292#443—611no—
293#443—1221no—
294#443—1221no—
295#443—1221no—
296#443—1221no—
297C2×#108—611no—
298C2×#108—611no—
299C2×#104—1221no—
300C2×#104—1221no—
301C2×C4×C8—611no—
302#368—612no—
303#368—612no—
304#359—612no—
305#359—612no—
306#359—612no—
307#359—612no—
308#346—612no—
309C4×C16—612no—
310C61#3346311yescharacteristic, complemented, Sylow
311C2×#41—1221no—
312C2×#41—1221no—
313#135—611no—
314#135—612no—
315#135—611no—
316#135—612no—
317#135—611no—
318#135—1221no—
319#135—1221no—
320#135—1221no—
321#135—1221no—
322#135C2×#692311yescharacteristic
323C2×Q16—611no—
324C2×Q16—611no—
325C2×SD16—1221no—
326C2×SD16—1221no—
327C2×D8—611no—
328C2×D8C2×#692311yescharacteristic
329#131—1222no—
330#131—1222no—
331#131—1221no—
332#131—1221no—
333#131—1221no—
334#131—1221no—
335#131—2441no—
336#131—2441no—
337#131—2441no—
338#131—2441no—
339C2×M16—1221no—
340C2×M16—1221no—
341C2×M16—2441no—
342C2×M16—2441no—
343C22×C8—611no—
344C22×C8—611no—
345C22×C8—2441no—
346C22×C8—2441no—
347C2×C42—611no—
348Q32—612no—
349Q32—612no—
350SD32—612no—
351SD32—612no—
352SD32—612no—
353SD32C2×#692312yescomplemented
354D16—612no—
355D16C2×#692312yescomplemented
356M32—611no—
357M32—612no—
358M32—1221no—
359M32—1221no—
360M32C2×#692311yescharacteristic
361C2×C16—612no—
362C2×C16—612no—
363#108—611no—
364#108—611no—
365#108—612no—
366#108—1221no—
367#108—1221no—
368#104—1222no—
369#104—1222no—
370#104—1221no—
371#104—1221no—
372#104—1221no—
373#104—1221no—
374C4×C8—611no—
375C4×C8—611no—
376C4×C8—611no—
377C4×C8—611no—
378#41—21no—
379#41—1222no—
380#41—1221no—
381#41—1222no—
382#41—1221no—
383#41—1221no—
384#41—2441no—
385#41—2441no—
386#41—2441no—
387#41—2441no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×D4—21no—
391C2×D4—1221no—
392C2×D4—2441no—
393C2×D4—2441no—
394C22×C4—611no—
395C22×C4—1221no—
396C22×C4—2441no—
397C22×C4—2441no—
398Q16—611no—
399Q16—612no—
400Q16—1221no—
401Q16—1221no—
402Q16C22×#692311yescharacteristic
403SD16—21no—
404SD16—1222no—
405SD16—1222no—
406SD16—1221no—
407SD16—1221no—
408SD16—1221no—
409D8—21no—
410D8—611no—
411D8—1221no—
412D8C22×#692312yes—
413D8C22×#692311yescharacteristic
414M16—1222no—
415M16—1221no—
416M16—1221no—
417M16—2441no—
418M16—2441no—
419M16—2442no—
420M16—2441no—
421M16—2442no—
422M16—2441no—
423M16—2441no—
424M16—2441no—
425C2×C8—611no—
426C2×C8—611no—
427C2×C8—611no—
428C2×C8—611no—
429C2×C8—611no—
430C2×C8—611no—
431C2×C8—611no—
432C2×C8—611no—
433C2×C8—611no—
434C2×C8—611no—
435C2×C8—611no—
436C2×C8—2441no—
437C2×C8—2441no—
438C2×C8—2442no—
439C2×C8—2441no—
440C2×C8—2442no—
441C2×C8—2441no—
442C2×C8—2441no—
443C2×C8—2441no—
444C2×C8C22×#692311yescharacteristic
445C42—611no—
446C42—611no—
447C42—1221no—
448C16—612no—
449C16—1222no—
450C16C22×#692312yes—
451C23—41no—
452C23—611no—
453C23—2441no—
454Q8—21no—
455Q8—1221no—
456Q8—1222no—
457Q8—2441no—
458Q8—2441no—
459D4—22no—
460D4—21no—
461D4—41no—
462D4—41no—
463D4—1221no—
464D4—2441no—
465D4—2441no—
466D4—2441no—
467D4—2442no—
468D4—2442no—
469D4—2441no—
470C2×C4—41no—
471C2×C4—611no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—611no—
476C2×C4—1221no—
477C2×C4—1221no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4D4×#692311yescharacteristic
487C8—611no—
488C8—611no—
489C8—611no—
490C8—611no—
491C8—611no—
492C8—611no—
493C8—2441no—
494C8—2441no—
495C8—2442no—
496C8D4×#692311yescharacteristic
497C8C23×#692311yescharacteristic
498C22—41no—
499C22—42no—
500C22—82no—
501C22—611no—
502C22—611no—
503C22—4882no—
504C22—2441no—
505C22—2441no—
506C22D8×#692311yescharacteristic
507C4—41no—
508C4—611no—
509C4—611no—
510C4—1222no—
511C4—2441no—
512C4—2441no—
513C4—2442no—
514C4D8×#692311yescharacteristic
515C4C2×D4×#692311yescharacteristic
516C2—21no—
517C2—82no—
518C2—81no—
519C2—1221no—
520C2—1221no—
521C2—4881no—
522C2C2×D8×#692311yescharacteristic
subgroupG/H
centerC2C2×D8×#6923
commutatorC8×C61C23×C4
FrattiniC8C23×#6923
FittingC61×#509C4
socleC2×C61C2×C4×D8
prime radical#21458951C2
cosocleC8×D61C24
hypercenter#509#6923
hypocenterC61#33463
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. D61⋊#443C2semidirect
2. C2×#16840804C2semidirect
3. #21462223C2semidirect
4. #21462221C2semidirect
5. #21462219C2semidirect
6. #21462218C2semidirect
7. #21462175C2semidirect
8. #21462134C2semidirect
9. #21462133C2semidirect
10. M32×#6923C2semidirect
15. #16840806C22semidirect
16. #16840806C22semidirect
21. #16840804C22semidirect
22. #16840804C22semidirect
23. #16840804C22semidirect
24. #16840804C22semidirect
30. #16840691C22semidirect
31. #16840690C22semidirect
32. #16840677C22semidirect
33. #16840676C22semidirect
34. #16840676C22semidirect
35. #16840676C22semidirect
36. #16840649C22semidirect
37. C16×#6923C22semidirect
40. #16840205C4semidirect
41. #16840202C4semidirect
42. #16840201C4semidirect
43. #16840198C4semidirect
44. #16840196C4semidirect
48. C61×#509C4semidirect
102. Dic488C2×C4semidirect
103. #14519723C2×C4semidirect
104. #14519722C2×C4semidirect
105. #14519721C2×C4semidirect
106. #14519721C2×C4semidirect
107. #14519718C2×C4semidirect
113. SD32×C61C2×C4semidirect
114. D16×C61C2×C4semidirect
282. #509#6923semidirect
310. C61#33463semidirect
353. SD32C2×#6923semidirect
355. D16C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4432
C2×#168408042
#214622232
#214622212
#214622192
#214622182
#214621752
#214621342
#214621332
M32×#69232
#214589512
#3346361