Finite Group Databasebeta

← #29835259

Subgroups — #29835259

Order 15616.

526 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456546C211yesmaximal, characteristic, complemented
2#21454727C211yesmaximal, characteristic, complemented
3#21448040C211yesmaximal, characteristic, complemented
4#21448032C211yesmaximal, characteristic, complemented
5#21447994C211yesmaximal, characteristic, complemented
6#21447993C211yesmaximal, characteristic, complemented
7#21447887C211yesmaximal, characteristic, complemented
8#21447886C211yesmaximal, characteristic, complemented
9#21447771C211yesmaximal, characteristic, complemented
10#21447763C211yesmaximal, characteristic
11#21447715C211yesmaximal, characteristic, complemented
12#21447712C211yesmaximal, characteristic, complemented
13#21447710C211yesmaximal, characteristic, complemented
14#21447629C211yesmaximal, characteristic, complemented
15C61×AGL(1,ℤ/16)C211yesmaximal, characteristic, complemented
16#16840559C2211yescharacteristic
17#16840527—21no—
18#16840294—21no—
19#16840205C2211yescharacteristic
20#16840202C2211yescharacteristic
21#16840198C2211yescharacteristic
22#16840162C2211yescharacteristic, complemented
23#16840145C2211yescharacteristic, complemented
24#16840115—21no—
25#16840021C2211yescharacteristic
26#16840012C2211yescharacteristic
27#16840007—21no—
28#16839885C2211yescharacteristic
29#16839881—21no—
30#16839879C2211yescharacteristic
31#16839878C2211yescharacteristic
32#16839877—21no—
33#16839870C2211yescharacteristic, complemented
34#16839862—21no—
35#16839809C2211yescharacteristic
36#104×D61—21no—
37#16839714C2211yescharacteristic
38#16839586C2211yescharacteristic, complemented
39#16839585C2211yescharacteristic, complemented
40#16839584C2211yescharacteristic, complemented
41#16839582C2211yescharacteristic, complemented
42#16839582C2211yescharacteristic, complemented
43#16839579C2211yescharacteristic, complemented
44#16839577C2211yescharacteristic, complemented
45#16839575C2211yescharacteristic, complemented
46#16839573C2211yescharacteristic
47#16839572C2211yescharacteristic
48#16839571C2211yescharacteristic
49#16839543C2211yescharacteristic
50#16839543C2211yescharacteristic
51#16839540C2211yescharacteristic, complemented
52C61×#509C2211yescharacteristic
53C61×#444C2211yescharacteristic
54C61×#442C2211yescharacteristic, complemented
55C61×#365C2211yescharacteristic, complemented
56C61×#361C2211yescharacteristic, complemented
57C61×#360C2211yescharacteristic
58C61×#347C2211yescharacteristic, complemented
59#41×D61—21no—
60#14519809—21no—
61#14519806—21no—
62#14519804—21no—
63#14519801—21no—
64#14519799C2311yescharacteristic
65#14519797—21no—
66C2×Dic244C2311yescharacteristic
67C2×#11758200—21no—
68#14519785—21no—
69#14519785—41no—
70#14519784—21no—
71#14519783—21no—
72#14519783—41no—
73C2×#11758189—41no—
74C2×#11758189D411yescharacteristic
75C2×#11758189C2311yescharacteristic
76M16×D61—21no—
77M16×D61—41no—
78#14519770—41no—
79#14519768—21no—
80#14519764—21no—
81#14519759—41no—
82#14519757—21no—
83#14519748—21no—
84C4×Dic122—21no—
85C4×C22⋊D61—41no—
86#14519736—41no—
87C42×D61D411yescharacteristic
88Dic488—21no—
89#14519723—21no—
90#14519722—21no—
91#14519721—21no—
92#14519721—21no—
93#14519718—21no—
94#14519717C2311yescharacteristic
95#14519717C2311yescharacteristic
96#14519713C2311yescharacteristic
97#14519712—21no—
98#14519710—21no—
99#14519709C2311yescharacteristic
100#14519705—21no—
101#14519704—21no—
102#14519703—21no—
103#14519702—21no—
104#14519700—21no—
105#14519698—21no—
106#14519688D411yescharacteristic
107C8×Dic61D411yescharacteristic
108C8×Dic61C2311yescharacteristic
109#135×C61C2311yescharacteristic
110C2×D8×C61C2311yescharacteristic
111#131×C61—21no—
112C4×D4×C61—21no—
113SD32×C61—21no—
114D16×C61—21no—
115M32×C61C2311yescharacteristic
116M32×C61C2311yescharacteristic
117#108×C61C2311yescharacteristic
118#106×C61C2311yescharacteristic
119#104×C61—21no—
120#102×C61—21no—
121#97×C61C2311yescharacteristic
122#41×C61—21no—
123C2×D4×C61—21no—
124C22×C4×C61—41no—
125C2×C22⋊D61—41no—
126#11758226—21no—
127Q8×D61—21no—
128#11758224—21no—
129#11758224—41no—
130D4×D61—21no—
131D4×D61—41no—
132#11758222—21no—
133#11758222—41no—
134C2×C4×D61—21no—
135C2×C4×D61—41no—
136C2×C4×D61C2×D411yescharacteristic
137C2×Dic122—21no—
138Q16×C61#4111yescharacteristic
139SD16×C61—21no—
140D8×C61—21no—
141D8×C61—21no—
142D8×C61#4111yescharacteristic
143M16×C61—21no—
144M16×C61—41no—
145C2×C8×C61—41no—
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61C2411yescharacteristic
148C4⋊C4×C61—21no—
149C22⋊C4×C61—41no—
150C42×C61C2×D411yescharacteristic
151C22⋊Dic61—41no—
152C22⋊Dic61—41no—
153C22⋊Dic61—41no—
154#11758202—21no—
155#11758201—21no—
156#11758200—21no—
157#11758200—41no—
158#11758200—41no—
159#11758199—21no—
160#11758198—41no—
161C4⋊Dic61—21no—
162C4⋊Dic61—41no—
163C4⋊Dic61—41no—
164#11758196—41no—
165#11758196—41no—
166C4×Dic61—21no—
167C4×Dic61—41no—
168C4×Dic61C2×D411yescharacteristic
169#11758194—21no—
170#11758194—41no—
171C2×#92418—41no—
172C2×#92418C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174Dic244—21no—
175Dic244—21no—
176Dic244#4111yescharacteristic
177D488#4111yescharacteristic
178#11758190—21no—
179#11758189—21no—
180#11758189—21no—
181#11758189—21no—
182#11758189—21no—
183#11758189—41no—
184#11758189—41no—
185#11758189—41no—
186C8×D61—41no—
187C8×D61—41no—
188C8×D61—41no—
189C16×C61—21no—
190C16×C61—21no—
191#11758185—21no—
192#11758185—21no—
193C23×C61—41no—
194C22×D61—41no—
195C22×D61AGL(1,ℤ/8)11yescharacteristic
196Q8×C61—21no—
197D4×C61—21no—
198D4×C61—41no—
199D4×C61—41no—
200D4×C61—41no—
201C2×C4×C61—21no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—82no—
205C2×C4×C61C22×D411yescharacteristic
206C22⋊D61—41no—
207C22⋊D61—41no—
208C22⋊D61—41no—
209C22⋊D61—82no—
210C2×Dic61—21no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61AGL(1,ℤ/8)11yescharacteristic
215D244—21no—
216D244—41no—
217C4×D61—21no—
218C4×D61—21no—
219C4×D61—41no—
220C4×D61—41no—
221C4×D61—41no—
222C4×D61—81no—
223Dic122—21no—
224Dic122—41no—
225Dic122—41no—
226Dic122—41no—
227C8×C61—21no—
228C8×C61—41no—
229C8×C61C2×#4111yescharacteristic
230C8×C61C2×#4111yescharacteristic
231#92418—21no—
232#92418—41no—
233#92418AGL(1,ℤ/8)11yescharacteristic
234#92418AGL(1,ℤ/8)11yescharacteristic
235#33690—611nomaximal, Sylow
236C22×C61—41no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
241C2×D61—21no—
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—161no—
245C4×C61—41no—
246C4×C61—41no—
247C4×C61—81no—
248C4×C61#52511yescharacteristic
249C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
250Dic61—21no—
251Dic61—41no—
252Dic61—41no—
253Dic61—81no—
254Dic61—81no—
255#3183—611no—
256#2956—611no—
257#2239—611no—
258#2238—611no—
259#2231—611no—
260#2231—611no—
261#2212—611no—
262#2212—611no—
263#2191—611no—
264#2190—611no—
265AGL(1,ℤ/16)—611no—
266AGL(1,ℤ/16)D6111yescharacteristic, complemented
267#2180—611no—
268#2180—611no—
269#2165—611no—
270C2×C61—21no—
271C2×C61—81no—
272C2×C61—161no—
273C2×C61#310911yescharacteristic
274D61—41no—
275D61—81no—
276#575—611no—
277#568—1221no—
278#529—1221no—
279#510—611no—
280#510—611no—
281#509—611no—
282#509C2×D6111yescharacteristic
283#499—611no—
284#495—611no—
285#488—1221no—
286#470—611no—
287#469—611no—
288#468—1221no—
289#444—611no—
290#444—1221no—
291#444C2×D6111yescharacteristic
292#443—611no—
293#443—611no—
294#443—1221no—
295#442C2×D6111yescharacteristic, complemented
296#441—611no—
297#440—1221no—
298#430—611no—
299C2×#104—1221no—
300C4×M16—611no—
301#365—611no—
302#365—611no—
303#365—611no—
304#365C2×D6111yescharacteristic, complemented
305#364—611no—
306#364—611no—
307#362—611no—
308#362—611no—
309#361—611no—
310#361C2×D6111yescharacteristic, complemented
311#360—611no—
312#360—611no—
313#360—611no—
314#360C2×D6111yescharacteristic
315#349—611no—
316#349—611no—
317#347—611no—
318#347C2×D6111yescharacteristic, complemented
319C61#3369011yescharacteristic, complemented, Sylow
320C2×#41—1221no—
321#137—1221no—
322#137—1221no—
323AGL(1,ℤ/8)—1221no—
324AGL(1,ℤ/8)—1221no—
325#135—611no—
326#135—1221no—
327#135C22×D6111yescharacteristic
328C2×Q16—611no—
329C2×SD16—1221no—
330C2×D8C22×D6111yescharacteristic
331#131—21no—
332#131—1221no—
333#131—1221no—
334#131—1221no—
335#131—2441no—
336#131—2441no—
337C2×M16—611no—
338C2×M16—611no—
339C2×M16—1221no—
340C2×M16—2441no—
341C2×M16—2441no—
342#126—2441no—
343#125—1221no—
344#124—1221no—
345#123—2441no—
346C22⋊Q8—1221no—
347#121—1221no—
348C4×Q8—1221no—
349C4×D4—21no—
350C4×D4—2441no—
351#117—2441no—
352C2×C42—611no—
353Q32—1221no—
354Q32—1221no—
355SD32—21no—
356SD32—1221no—
357SD32—1221no—
358SD32—1221no—
359D16—21no—
360D16—1221no—
361M32—611no—
362M32—611no—
363M32C22×D6111yescharacteristic
364M32C22×D6111yescharacteristic
365#108—611no—
366#108—1221no—
367#108C22×D6111yescharacteristic
368#107—1221no—
369#106—611no—
370#106C22×D6111yescharacteristic
371#104—21no—
372#104—1221no—
373#104—1221no—
374#104—1221no—
375#103—1221no—
376#103—1221no—
377#102—21no—
378#102—1221no—
379#97—611no—
380#97C22×D6111yescharacteristic
381C4×C8—611no—
382C4×C8—611no—
383#41—21no—
384#41—1221no—
385#41—1221no—
386#41—1221no—
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—41no—
396C22×C4—611no—
397C22×C4—1221no—
398C22×C4—2441no—
399Q16—611no—
400Q16—1221no—
401Q16—1221no—
402Q16—1221no—
403Q16#1175822211yescharacteristic
404SD16—21no—
405SD16—1221no—
406SD16—1221no—
407SD16—1221no—
408SD16—2441no—
409SD16—2441no—
410D8—21no—
411D8—21no—
412D8—611no—
413D8—1221no—
414D8#1175822211yescharacteristic
415M16—21no—
416M16—41no—
417M16—1221no—
418M16—1221no—
419M16—1221no—
420M16—1221no—
421M16—1221no—
422M16—2441no—
423M16—2441no—
424M16—2441no—
425M16—2441no—
426C2×C8—41no—
427C2×C8—611no—
428C2×C8—611no—
429C2×C8—2441no—
430C2×C8—2441no—
431C2×C8—2441no—
432C2×C8—2441no—
433C2×C8D4×D6111yescharacteristic
434C2×C8C23×D6111yescharacteristic
435C4⋊C4—21no—
436C4⋊C4—1221no—
437C4⋊C4—2441no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C22⋊C4—41no—
442C22⋊C4—2441no—
443C22⋊C4—2441no—
444C22⋊C4—2441no—
445C22⋊C4—2441no—
446C42—611no—
447C42—1221no—
448C42—2441no—
449C42D4×D6111yescharacteristic
450C16—21no—
451C16—21no—
452C16—1221no—
453C16—1221no—
454C23—41no—
455C23—611no—
456C23—2441no—
457Q8—21no—
458Q8—1221no—
459Q8—2441no—
460Q8—2441no—
461Q8—2441no—
462D4—21no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—1221no—
467D4—4882no—
468D4—2441no—
469D4—2441no—
470D4—2441no—
471D4—2441no—
472C2×C4—21no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—82no—
476C2×C4—611no—
477C2×C4—1221no—
478C2×C4—1221no—
479C2×C4—1221no—
480C2×C4—4881no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4C2×D4×D6111yescharacteristic
488C8—21no—
489C8—41no—
490C8—611no—
491C8—611no—
492C8—1221no—
493C8—2441no—
494C8#41×D6111yescharacteristic
495C8C2×#1175822211yescharacteristic
496C22—41no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—1221no—
501C22—9761no—
502C22—2441no—
503C22—2441no—
504C22AGL(1,ℤ/8)×D6111yescharacteristic
505C4—41no—
506C4—41no—
507C4—81no—
508C4—1221no—
509C4—4881no—
510C4—4881no—
511C4—2441no—
512C4—2441no—
513C4#1684027111yescharacteristic
514C4AGL(1,ℤ/8)×D6111yescharacteristic
515C2—21no—
516C2—81no—
517C2—161no—
518C2—4881no—
519C2—2441no—
520C2#2145590811yescharacteristic
subgroupG/H
centerC2#21455908
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×AGL(1,ℤ/16)C2
socleC2×C61#3109
prime radical#21447763C2
hypercenterAGL(1,ℤ/16)D61
hypocenterC61#33690
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ AGL(1,ℤ/16)

Complements

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

NH ≅ G/Ntype
1. #21456546C2semidirect
2. #21454727C2semidirect
3. #21448040C2semidirect
4. #21448032C2semidirect
5. #21447994C2semidirect
6. #21447993C2semidirect
7. #21447887C2semidirect
8. #21447886C2semidirect
9. #21447771C2semidirect
11. #21447715C2semidirect
12. #21447712C2semidirect
13. #21447710C2semidirect
14. #21447629C2semidirect
15. C61×AGL(1,ℤ/16)C2semidirect
22. #16840162C22semidirect
23. #16840145C22semidirect
33. #16839870C22semidirect
38. #16839586C22semidirect
39. #16839585C22semidirect
40. #16839584C22semidirect
41. #16839582C22semidirect
42. #16839582C22semidirect
43. #16839579C22semidirect
44. #16839577C22semidirect
45. #16839575C22semidirect
51. #16839540C22semidirect
54. C61×#442C22semidirect
55. C61×#365C22semidirect
56. C61×#361C22semidirect
58. C61×#347C22semidirect
266. AGL(1,ℤ/16)D61semidirect
295. #442C2×D61semidirect
304. #365C2×D61semidirect
310. #361C2×D61semidirect
318. #347C2×D61semidirect
319. C61#33690semidirect

Maximal subgroups

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

subgroupindex
#214565462
#214547272
#214480402
#214480322
#214479942
#214479932
#214478872
#214478862
#214477712
#214477632
#214477152
#214477122
#214477102
#214476292
C61×AGL(1,ℤ/16)2
#3369061