Finite Group Databasebeta

← #29835243

Subgroups — #29835243

Order 15616.

564 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456554C211yesmaximal, characteristic, complemented
2#21454727C211yesmaximal, characteristic, complemented
3#21448035C211yesmaximal, characteristic, complemented
4#21448032C211yesmaximal, characteristic, complemented
5#21447989C211yesmaximal, characteristic, complemented
6#21447988C211yesmaximal, characteristic, complemented
7#21447894C211yesmaximal, characteristic, complemented
8#21447893C211yesmaximal, characteristic, complemented
9#21447766C211yesmaximal, characteristic, complemented
10#21447763C211yesmaximal, characteristic, complemented
11#21447719C211yesmaximal, characteristic, complemented
12#21447717C211yesmaximal, characteristic, complemented
13#21447715C211yesmaximal, characteristic, complemented
14#21447629C211yesmaximal, characteristic, complemented
15C61×AGL(1,ℤ/16)C211yesmaximal, characteristic, complemented
16#16840559C2211yescharacteristic
17#16840527—21no—
18#16840309—21no—
19#16840200C2211yescharacteristic
20#16840198C2211yescharacteristic
21#16840197C2211yescharacteristic
22#16840159C2211yescharacteristic, complemented
23#16840137C2211yescharacteristic, complemented
24#16840107—21no—
25#16840016C2211yescharacteristic
26#16840014—21no—
27#16840012C2211yescharacteristic
28#16839885C2211yescharacteristic
29#16839881—21no—
30#16839879C2211yescharacteristic
31#16839878C2211yescharacteristic
32#16839877—21no—
33#16839875C2211yescharacteristic, complemented
34#16839872—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#16839576C2211yescharacteristic, complemented
44#16839575C2211yescharacteristic, complemented
45#16839574C2211yescharacteristic, 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
59C2×#11758222—41no—
60#41×D61—21no—
61#14519809—21no—
62#14519806—21no—
63#14519804—21no—
64#14519801—21no—
65#14519799C2311yescharacteristic
66#14519797—21no—
67C2×D488C2311yescharacteristic
68C2×#11758199—21no—
69#14519785—21no—
70#14519785—41no—
71#14519784—21no—
72#14519783—21no—
73#14519783—41no—
74C2×#11758189—41no—
75C2×#11758189D411yescharacteristic
76C2×#11758189C2311yescharacteristic
77M16×D61—21no—
78M16×D61—41no—
79#14519776—21no—
80#14519773—21no—
81#14519763—41no—
82#14519752—21no—
83#14519750—41no—
84#14519748—21no—
85C4×D244—21no—
86C4×C22⋊D61—41no—
87C42×D61D411yescharacteristic
88#14519722—21no—
89#14519721—21no—
90#14519720—21no—
91D976—21no—
92#14519718—21no—
93#14519718—21no—
94#14519717C2311yescharacteristic
95#14519717C2311yescharacteristic
96#14519713C2311yescharacteristic
97#14519712—21no—
98#14519709C2311yescharacteristic
99#14519708—21no—
100#14519705—21no—
101#14519704—21no—
102#14519703—21no—
103#14519699—21no—
104#14519698—21no—
105#14519697—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—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128#11758226—21no—
129Q8×D61—21no—
130#11758224—21no—
131#11758224—41no—
132D4×D61—21no—
133D4×D61—41no—
134#11758222—21no—
135#11758222—41no—
136#11758222—84no—
137C2×D244—21no—
138C2×D244—41no—
139C2×C4×D61—21no—
140C2×C4×D61—41no—
141C2×C4×D61—41no—
142C2×C4×D61C2×D411yescharacteristic
143C2×Dic122—41no—
144Q16×C61#4111yescharacteristic, complemented
145SD16×C61—21no—
146D8×C61—21no—
147D8×C61—21no—
148D8×C61#4111yescharacteristic, complemented
149M16×C61—21no—
150M16×C61—41no—
151C2×C8×C61—41no—
152C2×C8×C61C2×D411yescharacteristic
153C2×C8×C61C2411yescharacteristic
154C4⋊C4×C61—21no—
155C22⋊C4×C61—41no—
156C42×C61C2×D411yescharacteristic
157C22⋊Dic61—41no—
158#11758202—21no—
159#11758201—21no—
160#11758200—21no—
161#11758199—21no—
162#11758199—41no—
163#11758199—41no—
164#11758198—41no—
165#11758198—41no—
166C4⋊Dic61—21no—
167#11758196—41no—
168C4×Dic61—21no—
169C4×Dic61C2×D411yescharacteristic
170#11758194—21no—
171#11758194—41no—
172C2×#92418—41no—
173C2×#92418C2×D411yescharacteristic
174C2×#92418C2×D411yescharacteristic
175Dic244#4111yescharacteristic, complemented
176D488—21no—
177D488—21no—
178D488#4111yescharacteristic, complemented
179#11758190—21no—
180#11758189—21no—
181#11758189—21no—
182#11758189—21no—
183#11758189—21no—
184#11758189—41no—
185#11758189—41no—
186#11758189—41no—
187C8×D61—41no—
188C8×D61—41no—
189C8×D61—41no—
190C16×C61—21no—
191C16×C61—21no—
192#11758185—21no—
193#11758185—21no—
194C23×C61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61AGL(1,ℤ/8)11yescharacteristic
198Q8×C61—21no—
199D4×C61—21no—
200D4×C61—41no—
201D4×C61—41no—
202D4×C61—41no—
203C2×C4×C61—21no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—82no—
207C2×C4×C61C22×D411yescharacteristic
208C22⋊D61—41no—
209C22⋊D61—41no—
210C22⋊D61—41no—
211C22⋊D61—82no—
212C22⋊D61—82no—
213C22⋊D61—82no—
214C2×Dic61—21no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61AGL(1,ℤ/8)11yescharacteristic
218D244—21no—
219D244—41no—
220D244—41no—
221D244—41no—
222D244—82no—
223C4×D61—21no—
224C4×D61—21no—
225C4×D61—42no—
226C4×D61—41no—
227C4×D61—41no—
228C4×D61—41no—
229C4×D61—82no—
230Dic122—21no—
231Dic122—41no—
232Dic122—82no—
233C8×C61—21no—
234C8×C61—41no—
235C8×C61C2×#4111yescharacteristic
236C8×C61C2×#4111yescharacteristic
237#92418—21no—
238#92418—41no—
239#92418AGL(1,ℤ/8)11yescharacteristic
240#92418AGL(1,ℤ/8)11yescharacteristic
241#33688—611nomaximal, Sylow
242C22×C61—41no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
247C2×D61—21no—
248C2×D61—41no—
249C2×D61—41no—
250C2×D61—81no—
251C2×D61—81no—
252C2×D61—81no—
253C2×D61—161no—
254C4×C61—41no—
255C4×C61—41no—
256C4×C61—81no—
257C4×C61#52511yescharacteristic
258C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
259Dic61—22no—
260Dic61—21no—
261Dic61—41no—
262Dic61—81no—
263#3184—611no—
264#2956—611no—
265#2238—611no—
266#2238—611no—
267#2230—611no—
268#2230—611no—
269#2213—611no—
270#2213—611no—
271#2190—611no—
272#2190—611no—
273AGL(1,ℤ/16)—611no—
274AGL(1,ℤ/16)—611no—
275AGL(1,ℤ/16)—611no—
276AGL(1,ℤ/16)D6111yescharacteristic, complemented
277#2165—611no—
278C2×C61—21no—
279C2×C61—81no—
280C2×C61—161no—
281C2×C61#310911yescharacteristic
282D61—41no—
283D61—81no—
284D61—161no—
285#575—611no—
286#568—1221no—
287#532—1221no—
288#509—611no—
289#509—611no—
290#509—611no—
291#509C2×D6111yescharacteristic
292#498—611no—
293#493—611no—
294#486—1221no—
295#469—611no—
296#469—611no—
297#469—1221no—
298#444—611no—
299#444—1221no—
300#444C2×D6111yescharacteristic
301#443—611no—
302#443—611no—
303#443—1221no—
304#442—611no—
305#442—1221no—
306#442C2×D6111yescharacteristic, complemented
307#430—611no—
308C2×#104—1221no—
309C4×M16—611no—
310#365—611no—
311#365—611no—
312#365—611no—
313#365C2×D6111yescharacteristic, complemented
314#364—611no—
315#364—611no—
316#361—611no—
317#361—611no—
318#361—611no—
319#361C2×D6111yescharacteristic, complemented
320#360—611no—
321#360—611no—
322#360—611no—
323#360C2×D6111yescharacteristic
324#349—611no—
325#349—611no—
326#347—611no—
327#347C2×D6111yescharacteristic, complemented
328C61#3368811yescharacteristic, complemented, Sylow
329C2×#41—1221no—
330C2×#41—2441no—
331#137—1221no—
332#137—1221no—
333AGL(1,ℤ/8)—1221no—
334AGL(1,ℤ/8)—1221no—
335#135—611no—
336#135—1221no—
337#135C22×D6111yescharacteristic
338C2×D8—611no—
339C2×D8—1221no—
340C2×D8C22×D6111yescharacteristic
341#131—21no—
342#131—1221no—
343#131—1221no—
344#131—1221no—
345#131—2441no—
346#131—2441no—
347C2×M16—611no—
348C2×M16—611no—
349C2×M16—1221no—
350C2×M16—2441no—
351C2×M16—2441no—
352C4⋊Q8—1221no—
353#127—1221no—
354#124—2441no—
355#121—1221no—
356#121—1221no—
357#121—2441no—
358C4×D4—21no—
359C4×D4—1221no—
360C4×D4—2441no—
361C2×C42—611no—
362SD32—21no—
363SD32—1221no—
364SD32—1221no—
365SD32—1221no—
366D16—21no—
367D16—1221no—
368D16—1221no—
369D16—1221no—
370M32—611no—
371M32—611no—
372M32C22×D6111yescharacteristic
373M32C22×D6111yescharacteristic
374#108—611no—
375#108—1221no—
376#108C22×D6111yescharacteristic
377#106—611no—
378#106—1221no—
379#106C22×D6111yescharacteristic
380#104—21no—
381#104—1221no—
382#104—1221no—
383#104—1221no—
384#102—21no—
385#102—1221no—
386#102—1221no—
387#102—1221no—
388#97—611no—
389#97C22×D6111yescharacteristic
390C4×C8—611no—
391C4×C8—611no—
392#41—21no—
393#41—1221no—
394#41—1221no—
395#41—1221no—
396#41—4884no—
397#41—2441no—
398#41—2441no—
399C2×Q8—1221no—
400C2×Q8—2441no—
401C2×D4—21no—
402C2×D4—1221no—
403C2×D4—1221no—
404C2×D4—2441no—
405C2×D4—2441no—
406C2×D4—2441no—
407C2×D4—2441no—
408C2×D4—2441no—
409C22×C4—41no—
410C22×C4—611no—
411C22×C4—1221no—
412C22×C4—2441no—
413C22×C4—2441no—
414Q16—611no—
415Q16—1221no—
416Q16#1175822211yescharacteristic, complemented
417SD16—21no—
418SD16—1221no—
419SD16—1221no—
420SD16—1221no—
421D8—21no—
422D8—21no—
423D8—611no—
424D8—1221no—
425D8—1221no—
426D8—1221no—
427D8—2441no—
428D8—2441no—
429D8#1175822211yescharacteristic, complemented
430M16—21no—
431M16—41no—
432M16—1221no—
433M16—1221no—
434M16—1221no—
435M16—1221no—
436M16—1221no—
437M16—2441no—
438M16—2441no—
439M16—2441no—
440M16—2441no—
441C2×C8—41no—
442C2×C8—611no—
443C2×C8—611no—
444C2×C8—2441no—
445C2×C8—2441no—
446C2×C8—2441no—
447C2×C8—2441no—
448C2×C8D4×D6111yescharacteristic
449C2×C8C23×D6111yescharacteristic
450C4⋊C4—21no—
451C4⋊C4—1221no—
452C4⋊C4—2441no—
453C22⋊C4—41no—
454C22⋊C4—2441no—
455C22⋊C4—2441no—
456C22⋊C4—2441no—
457C42—611no—
458C42—1221no—
459C42D4×D6111yescharacteristic
460C16—21no—
461C16—21no—
462C16—1221no—
463C16—1221no—
464C23—41no—
465C23—611no—
466C23—2441no—
467C23—2441no—
468Q8—21no—
469Q8—1221no—
470Q8—4882no—
471Q8—2441no—
472D4—21no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—1221no—
477D4—4882no—
478D4—4882no—
479D4—4882no—
480D4—4882no—
481D4—2441no—
482D4—2441no—
483D4—2441no—
484D4—2441no—
485D4—2441no—
486D4—2441no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—82no—
491C2×C4—611no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—4882no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4C2×D4×D6111yescharacteristic
503C8—21no—
504C8—41no—
505C8—611no—
506C8—611no—
507C8—1221no—
508C8—2441no—
509C8#41×D6111yescharacteristic
510C8C2×#1175822211yescharacteristic
511C22—41no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—1221no—
516C22—9761no—
517C22—4881no—
518C22—4881no—
519C22—4881no—
520C22—2441no—
521C22—2441no—
522C22AGL(1,ℤ/8)×D6111yescharacteristic
523C4—41no—
524C4—41no—
525C4—81no—
526C4—1222no—
527C4—1221no—
528C4—4881no—
529C4—2441no—
530C4#1684027111yescharacteristic
531C4AGL(1,ℤ/8)×D6111yescharacteristic
532C2—21no—
533C2—81no—
534C2—161no—
535C2—9761no—
536C2—4881no—
537C2—2441no—
538C2#2145590811yescharacteristic
subgroupG/H
centerC2#21455908
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×AGL(1,ℤ/16)C2
socleC2×C61#3109
hypercenterAGL(1,ℤ/16)D61
hypocenterC61#33688
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. #21456554C2semidirect
2. #21454727C2semidirect
3. #21448035C2semidirect
4. #21448032C2semidirect
5. #21447989C2semidirect
6. #21447988C2semidirect
7. #21447894C2semidirect
8. #21447893C2semidirect
9. #21447766C2semidirect
10. #21447763C2semidirect
11. #21447719C2semidirect
12. #21447717C2semidirect
13. #21447715C2semidirect
14. #21447629C2semidirect
15. C61×AGL(1,ℤ/16)C2semidirect
22. #16840159C22semidirect
23. #16840137C22semidirect
33. #16839875C22semidirect
38. #16839586C22semidirect
39. #16839585C22semidirect
40. #16839584C22semidirect
41. #16839582C22semidirect
42. #16839582C22semidirect
43. #16839576C22semidirect
44. #16839575C22semidirect
45. #16839574C22semidirect
51. #16839540C22semidirect
54. C61×#442C22semidirect
55. C61×#365C22semidirect
56. C61×#361C22semidirect
58. C61×#347C22semidirect
144. Q16×C61#41semidirect
148. D8×C61#41semidirect
175. Dic244#41semidirect
178. D488#41semidirect
276. AGL(1,ℤ/16)D61semidirect
306. #442C2×D61semidirect
313. #365C2×D61semidirect
319. #361C2×D61semidirect
327. #347C2×D61semidirect
328. C61#33688semidirect
416. Q16#11758222semidirect
429. D8#11758222semidirect

Maximal subgroups

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

subgroupindex
#214565542
#214547272
#214480352
#214480322
#214479892
#214479882
#214478942
#214478932
#214477662
#214477632
#214477192
#214477172
#214477152
#214476292
C61×AGL(1,ℤ/16)2
#3368861