Finite Group Databasebeta

← #30624327

Subgroups — #30624327

Order 15616.

652 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840809C211yesmaximal, characteristic, complemented
2#21462573C211yesmaximal, characteristic, complemented
3#21462229C212yesmaximal, complemented
4#21462228C212yesmaximal, complemented
5#21462225C212yesmaximal, complemented
6#21462224C212yesmaximal, complemented
7#21462167C211yesmaximal, characteristic, complemented
8#21462147C211yesmaximal, characteristic, complemented
9#21462140C211yesmaximal, characteristic, complemented
10#21462034C211yesmaximal, characteristic, complemented
11#21458951C211yesmaximal, characteristic
12C2×#14519864—21no—
13#41×#6923—21no—
14D61⋊M32C2211yescharacteristic
15#16840809—21no—
16#16840809—21no—
17#16840809C2212yes—
18#16840809C2212yes—
19#16840803C2211yescharacteristic, complemented
20#16840803C2211yescharacteristic, complemented
21#16840802C2211yescharacteristic, complemented
22#16840802C2211yescharacteristic, complemented
23#16840801—21no—
24#16840801—21no—
25C2×#14519847C2211yescharacteristic
26#16840772C2211yescharacteristic
27C2×#14519841—21no—
28#16840752—21no—
29C2×#14519831C2211yescharacteristic
30#16840687C2212yescomplemented
31#16840686C2212yescomplemented
32#16840681C2212yescomplemented
33#16840680C2212yescomplemented
34#16840678C2212yes—
35#16840678C2212yes—
36#16840653C2212yescomplemented
37#16840652C2212yescomplemented
38C2×#14519796C2211yescharacteristic
39#135×D61C2211yescharacteristic
40#16840205C411yescharacteristic, complemented
41#16840202C411yescharacteristic, complemented
42#16840201C2212yes—
43#16840198C411yescharacteristic, complemented
44#16840196C2212yes—
45#16840193C412yescomplemented
46#16840191C412yescomplemented
47M32×D61C2211yescharacteristic
48C61×#509C411yescharacteristic, complemented
49#14519864—22no—
50#14519864—22no—
51#14519864—41no—
52#14519864—41no—
53C2×#11758213—21no—
54C2×#11758217—41no—
55C2×D61⋊C8D411yescharacteristic
56C22×#92420—41no—
57Q8×#6923—21no—
58Q8×#6923—21no—
59D4×#6923—21no—
60D4×#6923—21no—
61D4×#6923—41no—
62D4×#6923—41no—
63#14519856—21no—
64#14519856—41no—
65C2×C4×#6923—41no—
66C2×C4×#6923D411yescharacteristic
67#14519852C2312yes—
68#14519851—21no—
69D61⋊C16—21no—
70#14519847—21no—
71#14519847C2312yes—
72#14519846—21no—
73#14519845C2311yescharacteristic
74#14519845C2311yescharacteristic
75#14519841—22no—
76#14519841—22no—
77#14519840—21no—
78#14519840—21no—
79#14519839—21no—
80#14519839—21no—
81#14519831D411yescharacteristic
82#14519831D411yescharacteristic
83#14519831C2311yescharacteristic
84#14519831C2311yescharacteristic
85C2×#11758224—21no—
86#41×D61—21no—
87#14519800C2×C411yescharacteristic
88#14519799C2×C411yescharacteristic
89#14519798—21no—
90#14519797—21no—
91#14519796—21no—
92#14519796—21no—
93#14519796C2×C411yescharacteristic
94#14519796C2×C412yescomplemented
95#14519796C2312yes—
96C2×Dic244C2×C411yescharacteristic, complemented
97Q16×D61C2311yescharacteristic
98C2×#11758200—21no—
99SD16×D61—21no—
100D8×D61C2311yescharacteristic
101C2×C8×D61C2311yescharacteristic
102Dic488C2×C412yescomplemented
103#14519723C2×C412yescomplemented
104#14519722C2×C412yescomplemented
105#14519721C2×C412yescomplemented
106#14519721C2×C412yescomplemented
107#14519718C2×C412yescomplemented
108#14519717C2×C411yescharacteristic, complemented
109#14519716C2312yes—
110C16×D61C2×C412yescomplemented
111#135×C61C2×C411yescharacteristic
112C2×D8×C61C2×C411yescharacteristic, complemented
113SD32×C61C2×C412yescomplemented
114D16×C61C2×C412yescomplemented
115M32×C61C2×C411yescharacteristic, complemented
116C22×#6923—21no—
117C22×#6923—41no—
118C22×#6923—41no—
119#41×C61—21no—
120C2×D4×C61—21no—
121C2×C22⋊D61—41no—
122C22×Dic61—41no—
123#11758226—21no—
124Q8×D61—21no—
125#11758224—21no—
126#11758224—22no—
127#11758224—22no—
128#11758224—41no—
129#11758224—41no—
130#11758224—41no—
131D4×D61—21no—
132D4×D61—41no—
133#11758222—21no—
134#11758222—41no—
135C2×C4×D61—41no—
136C2×C4×D61C2×D411yescharacteristic
137C2×Dic122—21no—
138C22⋊#6923—41no—
139C22⋊#6923—41no—
140C22⋊#6923—41no—
141#11758217—42no—
142#11758217—42no—
143C2×#92420—21no—
144C2×#92420—42no—
145C2×#92420—42no—
146C2×#92420—41no—
147C4⋊#6923—21no—
148C4⋊#6923—21no—
149C4⋊#6923—41no—
150C4⋊#6923—41no—
151C4×#6923—21no—
152C4×#6923—41no—
153C4×#6923—41no—
154C4×#6923—41no—
155C4×#6923C2×D411yescharacteristic
156C4×#6923C2×D411yescharacteristic
157#11758213—22no—
158#11758213—41no—
159D61⋊C8—21no—
160D61⋊C8C2×D411yescharacteristic
161D61⋊C8C2×D411yescharacteristic
162Q16×C61C22×C411yescharacteristic
163SD16×C61—21no—
164D8×C61—21no—
165D8×C61C22×C411yescharacteristic
166D8×C61C22×C412yes—
167C2×C8×C61C22×C411yescharacteristic
168#11758202—21no—
169#11758201—21no—
170#11758200—21no—
171#11758200—22no—
172#11758200—22no—
173#11758199—21no—
174C2×#92418#4111yescharacteristic
175Dic244—21no—
176Dic244C22×C411yescharacteristic
177Dic244C22×C412yes—
178D488C22×C411yescharacteristic
179#11758190—21no—
180C8×D61C22×C411yescharacteristic
181C8×D61C2×D411yescharacteristic
182C8×D61#4111yescharacteristic
183C8×D61C2411yescharacteristic
184#11758187—22no—
185C16×C61C22×C412yes—
186#11758185C22×C412yes—
187C23×C61—41no—
188C22×D61—41no—
189C22×D61AGL(1,ℤ/8)11yescharacteristic
190C2×#6923—21no—
191C2×#6923—21no—
192C2×#6923—41no—
193C2×#6923—41no—
194C2×#6923—81no—
195C2×#6923—82no—
196C2×#6923—82no—
197Q8×C61—21no—
198D4×C61—21no—
199D4×C61—22no—
200D4×C61—41no—
201D4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61C4×D411yescharacteristic
204C22⋊D61—41no—
205C22⋊D61—42no—
206C22⋊D61—41no—
207C22⋊D61—42no—
208C22⋊D61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61AGL(1,ℤ/8)11yescharacteristic
214D244—21no—
215D244—41no—
216C4×D61—41no—
217C4×D61—41no—
218C4×D61—41no—
219C4×D61C4×D411yescharacteristic
220C4×D61AGL(1,ℤ/8)11yescharacteristic
221C4×D61AGL(1,ℤ/8)11yescharacteristic
222C4×D61C22×D411yescharacteristic
223Dic122—21no—
224Dic122—22no—
225Dic122—41no—
226Dic122—41no—
227#92420—21no—
228#92420—21no—
229#92420—42no—
230C8×C61C4×D411yescharacteristic
231C8×C61C23×C411yescharacteristic
232#92418C4×D411yescharacteristic
233#92418C2×#4111yescharacteristic
234#33468—611nomaximal, Sylow
235C22×C61—42no—
236C22×C61—41no—
237C22×C61—82no—
238C22×C61#44211yescharacteristic
239C2×D61—41no—
240C2×D61—41no—
241C2×D61—81no—
242C2×D61—81no—
243C2×D61#44211yescharacteristic
244C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
245#6923—41no—
246#6923—41no—
247#6923—81no—
248#6923—81no—
249C4×C61—41no—
250C4×C61#44211yescharacteristic
251C4×C61C2×C4×D411yescharacteristic
252Dic61—41no—
253Dic61—42no—
254Dic61#44211yescharacteristic
255Dic61C2×AGL(1,ℤ/8)11yescharacteristic
256#3414—611no—
257C2×#444—611no—
258#2943—611no—
259AGL(1,ℤ/16)—612no—
260AGL(1,ℤ/16)—612no—
261#2180—612no—
262#2180—612no—
263#2155—611no—
264#2148—611no—
265#2145—611no—
266C2×#347—611no—
267C2×C61—21no—
268C2×C61—81no—
269C2×C61—82no—
270C2×C61C2×#44211yescharacteristic
271D61—21no—
272D61—21no—
273D61—81no—
274C2×#135—611no—
275C2×#135—611no—
276C2×#131—1221no—
277C4×#41—1221no—
278#510—611no—
279#510—612no—
280#510—611no—
281#509—611no—
282#509—612no—
283#509#692311yescharacteristic, complemented
284#508—612no—
285#508—612no—
286C2×M32—611no—
287C2×M32—611no—
288#444—612no—
289#444—612no—
290#444—1221no—
291#444—1221no—
292#442—611no—
293#442—611no—
294#441—611no—
295#441—611no—
296#440—1221no—
297#440—1221no—
298C2×#108—611no—
299#427—611no—
300C2×#104—1221no—
301#416—1221no—
302C2×#97—611no—
303#365—612no—
304#365—612no—
305#362—612no—
306#361—612no—
307#360—612no—
308#360—612no—
309#347—612no—
310#347—612no—
311C61#3346811yescharacteristic, complemented, Sylow
312C2×#41—1221no—
313C2×#41—1221no—
314#135—611no—
315#135—612no—
316#135—611no—
317#135—612no—
318#135—611no—
319#135—1221no—
320#135—1221no—
321#135—1221no—
322#135—1221no—
323#135C2×#692311yescharacteristic
324C2×Q16—611no—
325C2×Q16—611no—
326C2×SD16—1221no—
327C2×SD16—1221no—
328C2×D8—611no—
329C2×D8C2×#692311yescharacteristic, complemented
330#131—1222no—
331#131—1222no—
332#131—2441no—
333#131—2441no—
334C2×M16—1221no—
335C2×M16—2441no—
336C22×C8—611no—
337C22×C8—611no—
338C22×C8—2441no—
339C4×Q8—1221no—
340C4×Q8—1221no—
341C4×D4—1221no—
342C4×D4—1221no—
343C4×D4—2441no—
344C4×D4—2441no—
345#117—1221no—
346#117—2441no—
347C2×C42—611no—
348C2×C42—2441no—
349Q32—612no—
350Q32—612no—
351SD32—612no—
352SD32—612no—
353SD32—612no—
354SD32C2×#692312yescomplemented
355D16—612no—
356D16C2×#692312yescomplemented
357M32—611no—
358M32—612no—
359M32—612no—
360M32—1221no—
361M32C2×#692311yescharacteristic, complemented
362C2×C16—612no—
363C2×C16—1221no—
364#108—612no—
365#108—1221no—
366#107—1221no—
367#106—611no—
368#106—611no—
369#104—1222no—
370#104—1222no—
371#103—1221no—
372#103—1221no—
373#102—1221no—
374#102—1221no—
375#97—611no—
376#97—611no—
377#97—611no—
378#97—611no—
379#41—21no—
380#41—1222no—
381#41—1222no—
382#41—1221no—
383#41—1221no—
384#41—1221no—
385#41—2441no—
386#41—2441no—
387#41—2441no—
388#41—2441no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×D4—21no—
392C2×D4—1221no—
393C2×D4—2441no—
394C2×D4—2441no—
395C22×C4—611no—
396C22×C4—1221no—
397C22×C4—2441no—
398C22×C4—2441no—
399C22×C4—2441no—
400C22×C4—2441no—
401Q16—611no—
402Q16—612no—
403Q16—1221no—
404Q16—1221no—
405Q16C22×#692311yescharacteristic
406SD16—21no—
407SD16—1222no—
408SD16—1221no—
409SD16—1222no—
410SD16—1221no—
411SD16—1221no—
412D8—21no—
413D8—611no—
414D8—1221no—
415D8C22×#692312yes—
416D8C22×#692311yescharacteristic
417M16—1222no—
418M16—2442no—
419M16—2442no—
420M16—2441no—
421C2×C8—611no—
422C2×C8—611no—
423C2×C8—611no—
424C2×C8—611no—
425C2×C8—611no—
426C2×C8—611no—
427C2×C8—611no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—2442no—
431C2×C8—2442no—
432C2×C8—2441no—
433C2×C8C22×#692311yescharacteristic
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—2441no—
437C4⋊C4—2441no—
438C22⋊C4—2441no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C42—611no—
442C42—611no—
443C42—1221no—
444C42—2441no—
445C42—2441no—
446C42—2441no—
447C16—612no—
448C16—1222no—
449C16C22×#692312yes—
450C23—41no—
451C23—611no—
452C23—2441no—
453Q8—21no—
454Q8—1221no—
455Q8—1222no—
456Q8—2441no—
457Q8—2441no—
458D4—22no—
459D4—21no—
460D4—41no—
461D4—41no—
462D4—1221no—
463D4—2441no—
464D4—2441no—
465D4—2441no—
466D4—2442no—
467D4—2442no—
468D4—2441no—
469C2×C4—41no—
470C2×C4—611no—
471C2×C4—611no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—1221no—
476C2×C4—1221no—
477C2×C4—4882no—
478C2×C4—4882no—
479C2×C4—4881no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2442no—
483C2×C4—2441no—
484C2×C4—2442no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4D4×#692311yescharacteristic
490C8—611no—
491C8—611no—
492C8—1221no—
493C8—1221no—
494C8—2442no—
495C8D4×#692311yescharacteristic
496C8C23×#692311yescharacteristic
497C22—41no—
498C22—42no—
499C22—82no—
500C22—611no—
501C22—611no—
502C22—4881no—
503C22—4881no—
504C22—2441no—
505C22—2441no—
506C22#1684080311yescharacteristic
507C4—41no—
508C4—611no—
509C4—611no—
510C4—4881no—
511C4—4881no—
512C4—2441no—
513C4—2441no—
514C4—2441no—
515C4—2442no—
516C4#1684080311yescharacteristic
517C4C2×D4×#692311yescharacteristic
518C2—21no—
519C2—82no—
520C2—81no—
521C2—1221no—
522C2—1221no—
523C2—4881no—
524C2C2×#1684080311yescharacteristic
subgroupG/H
centerC2C2×#16840803
commutatorC8×C61C23×C4
FrattiniC8C23×#6923
FittingC61×#509C4
socleC2×C61C2×#442
prime radical#21458951C2
cosocleC8×D61C24
hypercenter#509#6923
hypocenterC61#33468
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. C2×#16840809C2semidirect
2. #21462573C2semidirect
3. #21462229C2semidirect
4. #21462228C2semidirect
5. #21462225C2semidirect
6. #21462224C2semidirect
7. #21462167C2semidirect
8. #21462147C2semidirect
9. #21462140C2semidirect
10. #21462034C2semidirect
19. #16840803C22semidirect
20. #16840803C22semidirect
21. #16840802C22semidirect
22. #16840802C22semidirect
30. #16840687C22semidirect
31. #16840686C22semidirect
32. #16840681C22semidirect
33. #16840680C22semidirect
36. #16840653C22semidirect
37. #16840652C22semidirect
40. #16840205C4semidirect
41. #16840202C4semidirect
43. #16840198C4semidirect
45. #16840193C4semidirect
46. #16840191C4semidirect
48. C61×#509C4semidirect
94. #14519796C2×C4semidirect
96. C2×Dic244C2×C4semidirect
102. Dic488C2×C4semidirect
103. #14519723C2×C4semidirect
104. #14519722C2×C4semidirect
105. #14519721C2×C4semidirect
106. #14519721C2×C4semidirect
107. #14519718C2×C4semidirect
108. #14519717C2×C4semidirect
110. C16×D61C2×C4semidirect
112. C2×D8×C61C2×C4semidirect
113. SD32×C61C2×C4semidirect
114. D16×C61C2×C4semidirect
115. M32×C61C2×C4semidirect
283. #509#6923semidirect
311. C61#33468semidirect
329. C2×D8C2×#6923semidirect
354. SD32C2×#6923semidirect
356. D16C2×#6923semidirect
361. M32C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408092
#214625732
#214622292
#214622282
#214622252
#214622242
#214621672
#214621472
#214621402
#214620342
#214589512
#3346861