Finite Group Databasebeta

← #30624307

Subgroups — #30624307

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
1C2×#16840806C211yesmaximal, characteristic, complemented
2D61⋊#443C211yesmaximal, characteristic, complemented
3#21462222C212yesmaximal, complemented
4#21462221C212yesmaximal, complemented
5#21462220C212yesmaximal, complemented
6#21462219C212yesmaximal, complemented
7#21462175C211yesmaximal, characteristic, complemented
8#21462135C211yesmaximal, characteristic, complemented
9#21462133C211yesmaximal, characteristic, complemented
10M32×#6923C211yesmaximal, characteristic, complemented
11#21458954C211yesmaximal, characteristic
12C2×#14519865—21no—
13D61⋊#131—21no—
14D61⋊M32C2211yescharacteristic
15#16840806—21no—
16#16840806—21no—
17#16840806C2211yescharacteristic, complemented
18#16840806C2211yescharacteristic, complemented
19#16840806C2212yescomplemented
20#16840806C2212yescomplemented
21#16840805—21no—
22#16840805—21no—
23#16840804C2211yescharacteristic, complemented
24#16840804C2211yescharacteristic, complemented
25C2×#14519848C2211yescharacteristic
26D61⋊#108C2211yescharacteristic
27C2×#14519842—21no—
28D61⋊#104—21no—
29C2×C8×#6923C2211yescharacteristic
30#16840691C2212yescomplemented
31#16840690C2212yescomplemented
32#16840677C2212yescomplemented
33#16840677C2212yescomplemented
34#16840677C2212yescomplemented
35#16840676C2212yescomplemented
36#16840649C2212yescomplemented
37C16×#6923C2212yescomplemented
38C2×#14519800C2211yescharacteristic
39#135×D61C2211yescharacteristic
40#16840204C412yescomplemented
41#16840203C411yescharacteristic, complemented
42#16840200C411yescharacteristic, complemented
43#16840199C412yescomplemented
44#16840197C411yescharacteristic, complemented
45#16840195C2212yes—
46#16840193C2212yes—
47M32×D61C2211yescharacteristic
48C61×#510C411yescharacteristic, complemented
49#14519865—21no—
50#14519865—21no—
51#14519865—22no—
52#14519865—22no—
53#14519865—41no—
54#14519865—41no—
55#14519864—21no—
56#14519864—21no—
57#14519864—41no—
58#14519864—41no—
59C2×#11758213—21no—
60C2×#11758213—41no—
61D61⋊M16—21no—
62D61⋊M16—41no—
63C2×D61⋊C8—41no—
64C2×D61⋊C8—41no—
65C2×D61⋊C8D411yescharacteristic
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#14519842—22no—
78#14519842—22no—
79#14519841—21no—
80#14519841—21no—
81C8×#6923D411yescharacteristic
82C8×#6923D411yescharacteristic
83C8×#6923C2311yescharacteristic
84C8×#6923C2311yescharacteristic
85C2×#11758226—21no—
86#41×D61—21no—
87#14519800—21no—
88#14519800—21no—
89#14519800C2×C412yes—
90#14519800C2312yes—
91#14519800C2311yescharacteristic
92#14519799C2×C411yescharacteristic
93#14519798—21no—
94#14519797—21no—
95#14519796C2311yescharacteristic
96Q16×D61C2×C411yescharacteristic
97C2×#11758201—21no—
98SD16×D61—21no—
99C2×D488C2×C411yescharacteristic
100D8×D61C2×C411yescharacteristic
101C2×C8×D61C2311yescharacteristic
102#14519723C2×C412yescomplemented
103#14519722C2×C412yescomplemented
104#14519720C2×C412yescomplemented
105#14519720C2×C412yescomplemented
106D976C2×C412yescomplemented
107#14519718C2×C412yescomplemented
108#14519717C2×C411yescharacteristic
109#14519716C2×C412yes—
110C16×D61C2312yes—
111#135×C61C2×C411yescharacteristic
112C2×Q16×C61C2×C411yescharacteristic
113Q32×C61C2×C412yescomplemented
114SD32×C61C2×C412yescomplemented
115M32×C61C2×C411yescharacteristic
116C22×#6923—21no—
117#41×C61—21no—
118C2×Q8×C61—21no—
119#11758226—22no—
120#11758226—21no—
121#11758226—22no—
122#11758226—41no—
123#11758226—41no—
124Q8×D61—21no—
125#11758224—21no—
126#11758224—41no—
127D4×D61—21no—
128D4×D61—41no—
129#11758222—21no—
130#11758222—41no—
131C2×D244—21no—
132C2×D244—41no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61C2×D411yescharacteristic
136#11758217—41no—
137#11758217—41no—
138#11758217—41no—
139C2×#92420—41no—
140C2×#92420—41no—
141C2×#92420—41no—
142C2×#92420D811yescharacteristic
143C2×#92420D811yescharacteristic
144C4×#6923—21no—
145C4×#6923C2×D411yescharacteristic
146C4×#6923C2×D411yescharacteristic
147#11758213—21no—
148#11758213—21no—
149#11758213—22no—
150#11758213—41no—
151#11758213—41no—
152#11758213—41no—
153#11758213—42no—
154#11758213—42no—
155D61⋊C8—41no—
156D61⋊C8—41no—
157D61⋊C8—41no—
158D61⋊C8—42no—
159D61⋊C8—42no—
160D61⋊C8D811yescharacteristic
161D61⋊C8D811yescharacteristic
162D61⋊C8C2×D411yescharacteristic
163D61⋊C8C2×D411yescharacteristic
164Q16×C61—21no—
165Q16×C61C22×C412yes—
166Q16×C61C22×C411yescharacteristic
167SD16×C61—21no—
168D8×C61C22×C411yescharacteristic
169C2×C8×C61C22×C411yescharacteristic
170#11758202—21no—
171#11758201—22no—
172#11758201—21no—
173#11758201—22no—
174#11758200—21no—
175#11758199—21no—
176C2×#92418#4111yescharacteristic
177Dic244C22×C411yescharacteristic
178D488—21no—
179D488C22×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—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61#13511yescharacteristic
192C2×#6923—21no—
193C2×#6923—21no—
194C2×#6923—42no—
195Q8×C61—22no—
196Q8×C61—21no—
197Q8×C61—41no—
198D4×C61—21no—
199D4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61C4×D411yescharacteristic
203C22⋊D61—41no—
204C22⋊D61—41no—
205C22⋊D61—41no—
206C2×Dic61—41no—
207C2×Dic61C2×D811yescharacteristic
208D244—21no—
209D244—22no—
210D244—42no—
211D244—41no—
212D244—42no—
213D244—41no—
214C4×D61—41no—
215C4×D61—41no—
216C4×D61—41no—
217C4×D61—42no—
218C4×D61—42no—
219C4×D61C4×D411yescharacteristic
220C4×D61C2×D811yescharacteristic
221C4×D61#13511yescharacteristic
222C4×D61C22×D411yescharacteristic
223Dic122—21no—
224Dic122—41no—
225#92420—42no—
226#92420—41no—
227#92420—41no—
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—41no—
238C22×C61C4×D811yescharacteristic
239C2×D61—41no—
240C2×D61—42no—
241C2×D61—41no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61C4×D811yescharacteristic
245C2×D61C2×#13511yescharacteristic
246#6923—22no—
247#6923—41no—
248C4×C61—41no—
249C4×C61—42no—
250C4×C61C4×D811yescharacteristic
251C4×C61C2×C4×D411yescharacteristic
252Dic61—41no—
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×C61C2×C4×D811yescharacteristic
269D61—21no—
270D61—21no—
271D61—82no—
272D61—81no—
273C2×#135—611no—
274C2×#135—611no—
275C2×#131—1221no—
276C2×#131—1221no—
277#510—611no—
278#510—612no—
279#510#692311yescharacteristic, complemented
280#509—611no—
281#509—612no—
282#509—611no—
283#508—612no—
284#508—612no—
285C2×M32—611no—
286C2×M32—611no—
287#443—611no—
288#443—611no—
289#443—611no—
290#443—612no—
291#443—611no—
292#443—612no—
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—612no—
316#135—611no—
317#135—611no—
318#135—1221no—
319#135—1221no—
320#135—1221no—
321#135—1221no—
322#135C2×#692311yescharacteristic
323C2×Q16—611no—
324C2×Q16C2×#692311yescharacteristic
325C2×SD16—1221no—
326C2×SD16—1221no—
327C2×D8—611no—
328C2×D8—611no—
329#131—1221no—
330#131—1221no—
331#131—1221no—
332#131—1221no—
333#131—1222no—
334#131—1222no—
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—
349Q32C2×#692312yescomplemented
350SD32—612no—
351SD32—612no—
352SD32—612no—
353SD32C2×#692312yescomplemented
354D16—612no—
355D16—612no—
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—1221no—
369#104—1221no—
370#104—1221no—
371#104—1221no—
372#104—1222no—
373#104—1222no—
374C4×C8—611no—
375C4×C8—611no—
376C4×C8—611no—
377C4×C8—611no—
378#41—21no—
379#41—1221no—
380#41—1221no—
381#41—1222no—
382#41—1222no—
383#41—1221no—
384#41—2441no—
385#41—2441no—
386#41—2441no—
387#41—2441no—
388C2×Q8—21no—
389C2×Q8—1221no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—2441no—
393C2×D4—2441no—
394C22×C4—611no—
395C22×C4—1221no—
396C22×C4—2441no—
397C22×C4—2441no—
398Q16—21no—
399Q16—611no—
400Q16—1221no—
401Q16C22×#692311yescharacteristic
402Q16C22×#692312yes—
403SD16—21no—
404SD16—1221no—
405SD16—1222no—
406SD16—1221no—
407SD16—1222no—
408SD16—1221no—
409D8—612no—
410D8—611no—
411D8—1221no—
412D8—1221no—
413D8C22×#692311yescharacteristic
414M16—1221no—
415M16—1221no—
416M16—1222no—
417M16—2441no—
418M16—2442no—
419M16—2441no—
420M16—2442no—
421M16—2441no—
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—2442no—
438C2×C8—2441no—
439C2×C8—2442no—
440C2×C8—2441no—
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—611no—
452C23—2441no—
453C23—2441no—
454Q8—21no—
455Q8—22no—
456Q8—41no—
457Q8—1221no—
458Q8—2441no—
459D4—21no—
460D4—41no—
461D4—1221no—
462D4—1222no—
463D4—2441no—
464D4—2442no—
465D4—2442no—
466D4—2441no—
467D4—2441no—
468D4—2441no—
469D4—2441no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—611no—
476C2×C4—611no—
477C2×C4—1221no—
478C2×C4—1221no—
479C2×C4—2441no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2442no—
485C2×C4—2441no—
486C2×C4D4×#692311yescharacteristic
487C8—611no—
488C8—611no—
489C8—611no—
490C8—611no—
491C8—611no—
492C8—611no—
493C8—2442no—
494C8—2441no—
495C8—2441no—
496C8D4×#692311yescharacteristic
497C8C23×#692311yescharacteristic
498C22—41no—
499C22—611no—
500C22—611no—
501C22—4882no—
502C22—4882no—
503C22—2441no—
504C22—2442no—
505C22—2441no—
506C22D8×#692311yescharacteristic
507C4—41no—
508C4—42no—
509C4—611no—
510C4—611no—
511C4—1222no—
512C4—2441no—
513C4—2441no—
514C4D8×#692311yescharacteristic
515C4C2×D4×#692311yescharacteristic
516C2—21no—
517C2—81no—
518C2—1221no—
519C2—1221no—
520C2—4882no—
521C2—4881no—
522C2C2×D8×#692311yescharacteristic
subgroupG/H
centerC2C2×D8×#6923
commutatorC8×C61C23×C4
FrattiniC8C23×#6923
FittingC61×#510C4
socleC2×C61C2×C4×D8
prime radical#21458954C2
cosocleC8×D61C24
hypercenter#510#6923
hypocenterC61#33463
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#16840806C2semidirect
2. D61⋊#443C2semidirect
3. #21462222C2semidirect
4. #21462221C2semidirect
5. #21462220C2semidirect
6. #21462219C2semidirect
7. #21462175C2semidirect
8. #21462135C2semidirect
9. #21462133C2semidirect
10. M32×#6923C2semidirect
17. #16840806C22semidirect
18. #16840806C22semidirect
19. #16840806C22semidirect
20. #16840806C22semidirect
23. #16840804C22semidirect
24. #16840804C22semidirect
30. #16840691C22semidirect
31. #16840690C22semidirect
32. #16840677C22semidirect
33. #16840677C22semidirect
34. #16840677C22semidirect
35. #16840676C22semidirect
36. #16840649C22semidirect
37. C16×#6923C22semidirect
40. #16840204C4semidirect
41. #16840203C4semidirect
42. #16840200C4semidirect
43. #16840199C4semidirect
44. #16840197C4semidirect
48. C61×#510C4semidirect
102. #14519723C2×C4semidirect
103. #14519722C2×C4semidirect
104. #14519720C2×C4semidirect
105. #14519720C2×C4semidirect
106. D976C2×C4semidirect
107. #14519718C2×C4semidirect
113. Q32×C61C2×C4semidirect
114. SD32×C61C2×C4semidirect
279. #510#6923semidirect
310. C61#33463semidirect
349. Q32C2×#6923semidirect
353. SD32C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408062
D61⋊#4432
#214622222
#214622212
#214622202
#214622192
#214621752
#214621352
#214621332
M32×#69232
#214589542
#3346361