Finite Group Databasebeta

← #29835277

Subgroups — #29835277

Order 15616.

524 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#21456438C211yesmaximal, characteristic, complemented
2#21454717C211yesmaximal, characteristic, complemented
3#21448040C211yesmaximal, characteristic, complemented
4#21448035C211yesmaximal, characteristic, complemented
5#21447987C211yesmaximal, characteristic, complemented
6#21447889C211yesmaximal, characteristic, complemented
7#21447888C211yesmaximal, characteristic, complemented
8#21447771C211yesmaximal, characteristic, complemented
9#21447766C211yesmaximal, characteristic
10#21447716C211yesmaximal, characteristic, complemented
11#21447714C211yesmaximal, characteristic, complemented
12#21447709C211yesmaximal, characteristic, complemented
13#21447707C211yesmaximal, characteristic, complemented
14#21447634C211yesmaximal, characteristic, complemented
15C61×#2230C211yesmaximal, characteristic, complemented
16#16840559C2211yescharacteristic
17#131×D61—21no—
18#16840386—21no—
19#16840205C2211yescharacteristic
20#16840202C2211yescharacteristic
21#16840200C2211yescharacteristic
22#16840197C2211yescharacteristic
23#16840146C2211yescharacteristic, complemented
24#16840124—21no—
25#16840021C2211yescharacteristic
26#16840016C2211yescharacteristic
27#16839992—21no—
28#16839884C2211yescharacteristic
29#16839883C2211yescharacteristic
30#16839881—21no—
31#16839880C2211yescharacteristic
32#16839877—21no—
33#16839873C2211yescharacteristic, complemented
34#16839868C2211yescharacteristic, complemented
35#16839849—21no—
36#16839809C2211yescharacteristic
37#16839777—21no—
38#16839716C2211yescharacteristic
39#16839584C2211yescharacteristic, complemented
40#16839584C2211yescharacteristic, complemented
41#16839583C2211yescharacteristic, complemented
42#16839579C2211yescharacteristic, complemented
43#16839577C2211yescharacteristic, complemented
44#16839576C2211yescharacteristic, complemented
45#16839574C2211yescharacteristic, complemented
46#16839571C2211yescharacteristic
47#16839571C2211yescharacteristic
48#16839543C2211yescharacteristic
49#16839543C2211yescharacteristic
50#16839539C2211yescharacteristic, complemented
51#16839539C2211yescharacteristic, complemented
52C61×#498C2211yescharacteristic, complemented
53C61×#443C2211yescharacteristic
54C61×#365C2211yescharacteristic, complemented
55C61×#365C2211yescharacteristic, complemented
56C61×#364C2211yescharacteristic, complemented
57C61×#360C2211yescharacteristic
58C61×#360C2211yescharacteristic
59#41×D61—21no—
60#14519809—21no—
61#14519806—21no—
62#14519804—21no—
63#14519801—21no—
64#14519799C2311yescharacteristic
65#14519797—21no—
66C2×Dic244C2311yescharacteristic
67C2×#11758190—21no—
68C2×D488C2311yescharacteristic
69#14519785—21no—
70#14519785—41no—
71#14519784—21no—
72#14519783—21no—
73#14519783—41no—
74C2×#11758189C2311yescharacteristic
75M16×D61—21no—
76M16×D61—41no—
77C2×C8×D61—41no—
78C2×C8×D61D411yescharacteristic
79#14519772—41no—
80#14519766—41no—
81#14519765—21no—
82#14519762—41no—
83#14519757—21no—
84#14519753—41no—
85#14519752—21no—
86#14519742—21no—
87#14519739—21no—
88#14519734D411yescharacteristic
89Dic488—21no—
90#14519723—21no—
91#14519722—21no—
92#14519722—21no—
93#14519721—21no—
94#14519720—21no—
95D976—21no—
96#14519718—21no—
97#14519717C2311yescharacteristic
98#14519717C2311yescharacteristic
99#14519713C2311yescharacteristic
100#14519712—21no—
101#14519705—21no—
102#14519704—21no—
103#14519703—21no—
104#14519700—21no—
105#14519700—21no—
106#14519697—21no—
107#14519697—21no—
108#14519688D411yescharacteristic
109#14519687C2311yescharacteristic
110C8×Dic61D411yescharacteristic
111#135×C61C2311yescharacteristic
112#131×C61—21no—
113C4⋊Q8×C61—21no—
114M32×C61C2311yescharacteristic
115M32×C61C2311yescharacteristic
116#108×C61C2311yescharacteristic
117#106×C61—21no—
118#106×C61C2311yescharacteristic
119#106×C61C2311yescharacteristic
120#104×C61—21no—
121C4×C8×C61C2311yescharacteristic
122#41×C61—21no—
123C2×Q8×C61—41no—
124#11758226—21no—
125Q8×D61—21no—
126#11758224—21no—
127#11758224—41no—
128D4×D61—21no—
129D4×D61—41no—
130#11758222—21no—
131#11758222—41no—
132C2×D244—21no—
133C2×D244—41no—
134C2×C4×D61—41no—
135C2×C4×D61—41no—
136C2×C4×D61C2×D411yescharacteristic
137C2×Dic122—21no—
138Q16×C61#4111yescharacteristic
139SD16×C61—21no—
140D8×C61#4111yescharacteristic
141M16×C61—21no—
142M16×C61—41no—
143C2×C8×C61—41no—
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61C2411yescharacteristic
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—41no—
149C42×C61C2×D411yescharacteristic
150#11758202—21no—
151#11758201—21no—
152#11758200—21no—
153#11758199—21no—
154#11758198—21no—
155#11758198—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198—41no—
160C4⋊Dic61—41no—
161#11758196—21no—
162#11758196—41no—
163C4×Dic61—41no—
164C4×Dic61C2×D411yescharacteristic
165#11758194—21no—
166#11758194—41no—
167C2×#92418—41no—
168C2×#92418C2×D411yescharacteristic
169C2×#92418C2×D411yescharacteristic
170Dic244—21no—
171Dic244—21no—
172Dic244#4111yescharacteristic
173D488—21no—
174D488—21no—
175D488#4111yescharacteristic
176#11758190—21no—
177#11758190—41no—
178#11758190—41no—
179#11758189—21no—
180#11758189—21no—
181#11758189—41no—
182#11758189—41no—
183#11758189—41no—
184C8×D61—21no—
185C8×D61—21no—
186C8×D61—41no—
187C8×D61—41no—
188C8×D61—41no—
189C16×C61—21no—
190C16×C61—21no—
191#11758185—21no—
192#11758185—21no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61AGL(1,ℤ/8)11yescharacteristic
196Q8×C61—21no—
197Q8×C61—161no—
198D4×C61—21no—
199D4×C61—41no—
200C2×C4×C61—21no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61C22×D411yescharacteristic
205C22⋊D61—41no—
206C22⋊D61—41no—
207C22⋊D61—41no—
208C2×Dic61—21no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61AGL(1,ℤ/8)11yescharacteristic
212D244—21no—
213D244—41no—
214D244—41no—
215D244—41no—
216D244—161no—
217C4×D61—21no—
218C4×D61—21no—
219C4×D61—41no—
220C4×D61—41no—
221C4×D61—41no—
222C4×D61—161no—
223Dic122—21no—
224Dic122—41no—
225Dic122—41no—
226Dic122—41no—
227C8×C61—41no—
228C8×C61AGL(1,ℤ/8)11yescharacteristic
229C8×C61AGL(1,ℤ/8)11yescharacteristic
230C8×C61C2×#4111yescharacteristic
231C8×C61C2×#4111yescharacteristic
232#92418—21no—
233#92418—21no—
234#92418—41no—
235#33691—611nomaximal, Sylow
236C22×C61—41no—
237C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C2×D61—21no—
239C2×D61—41no—
240C2×D61—41no—
241C2×D61—82no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—81no—
245C4×C61—41no—
246C4×C61—41no—
247C4×C61—81no—
248C4×C61—81no—
249C4×C61#52511yescharacteristic
250C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
251Dic61—21no—
252Dic61—41no—
253Dic61—41no—
254Dic61—81no—
255#3173—611no—
256#2955—611no—
257#2239—611no—
258#2238—611no—
259#2230—611no—
260#2230D6111yescharacteristic, complemented
261#2212—611no—
262#2212—611no—
263#2191—611no—
264#2190—611no—
265AGL(1,ℤ/16)—611no—
266AGL(1,ℤ/16)—611no—
267#2180—611no—
268#2180—611no—
269#2166—611no—
270C2×C61—21no—
271C2×C61—81no—
272C2×C61#310911yescharacteristic
273D61—41no—
274D61—81no—
275D61—161no—
276#575—611no—
277C2×#131—1221no—
278#542—1221no—
279#510—611no—
280#510—611no—
281#509—611no—
282#509—611no—
283#498C2×D6111yescharacteristic, complemented
284#495—611no—
285#489—1221no—
286#470—611no—
287#469—611no—
288#465—1221no—
289#444—611no—
290#444—611no—
291#444—1221no—
292#443—611no—
293#443—1221no—
294#443C2×D6111yescharacteristic
295#442—611no—
296#441—611no—
297C4×SD16—1221no—
298#430—611no—
299#421—1221no—
300#405—611no—
301#365—611no—
302#365—611no—
303#365C2×D6111yescharacteristic, complemented
304#365C2×D6111yescharacteristic, complemented
305#364—611no—
306#364C2×D6111yescharacteristic, complemented
307#362—611no—
308#362—611no—
309#361—611no—
310#361—611no—
311#360—611no—
312#360—611no—
313#360C2×D6111yescharacteristic
314#360C2×D6111yescharacteristic
315#349—611no—
316#349—611no—
317#347—611no—
318#347—611no—
319C61#3369111yescharacteristic, 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×D8—611no—
331#131—21no—
332#131—1221no—
333#131—1221no—
334#131—1221no—
335#131—2441no—
336#131—2441no—
337C2×M16—611no—
338C2×M16—1221no—
339C2×M16—2441no—
340C22×C8—611no—
341C22×C8—2441no—
342C4⋊Q8—21no—
343#126—2441no—
344#124—1221no—
345#124—2441no—
346#123—2441no—
347C22⋊Q8—1221no—
348C22⋊Q8—2441no—
349#121—1221no—
350C4×Q8—1221no—
351C4×D4—1221no—
352#117—611no—
353Q32—1221no—
354Q32—1221no—
355SD32—1221no—
356SD32—1221no—
357SD32—1221no—
358SD32—1221no—
359D16—1221no—
360D16—1221no—
361M32—611no—
362M32—611no—
363M32C22×D6111yescharacteristic
364M32C22×D6111yescharacteristic
365#108—611no—
366#108—1221no—
367#108C22×D6111yescharacteristic
368#106—21no—
369#106C22×D6111yescharacteristic
370#106C22×D6111yescharacteristic
371#104—21no—
372#104—1221no—
373#104—1221no—
374#104—1221no—
375#103—1221no—
376#103—1221no—
377#102—1221no—
378#102—1221no—
379#97—611no—
380#97—611no—
381C4×C8—611no—
382C4×C8C22×D6111yescharacteristic
383#41—21no—
384#41—1221no—
385#41—1221no—
386#41—1221no—
387#41—2441no—
388#41—2441no—
389C2×Q8—41no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×D4—1221no—
393C2×D4—1221no—
394C2×D4—2441no—
395C2×D4—2441no—
396C22×C4—611no—
397C22×C4—2441no—
398C22×C4—2441no—
399Q16—611no—
400Q16—1221no—
401Q16—1221no—
402Q16—1221no—
403Q16#1175822611yescharacteristic
404SD16—21no—
405SD16—1221no—
406SD16—1221no—
407SD16—1221no—
408SD16—2441no—
409SD16—2441no—
410D8—611no—
411D8—1221no—
412D8—1221no—
413D8—1221no—
414D8#1175822611yescharacteristic
415M16—21no—
416M16—41no—
417M16—1221no—
418M16—1221no—
419M16—1221no—
420M16—2441no—
421M16—2441no—
422M16—2441no—
423M16—2441no—
424C2×C8—41no—
425C2×C8—611no—
426C2×C8—611no—
427C2×C8—1221no—
428C2×C8—1221no—
429C2×C8—2441no—
430C2×C8—2441no—
431C2×C8—2441no—
432C2×C8—2441no—
433C2×C8D4×D6111yescharacteristic
434C2×C8C23×D6111yescharacteristic
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—41no—
438C4⋊C4—1221no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C22⋊C4—1221no—
442C22⋊C4—2441no—
443C22⋊C4—2441no—
444C22⋊C4—2441no—
445C22⋊C4—2441no—
446C22⋊C4—2441no—
447C42—611no—
448C42—2441no—
449C42D4×D6111yescharacteristic
450C16—21no—
451C16—21no—
452C16—1221no—
453C16—1221no—
454C23—611no—
455C23—2441no—
456C23—2441no—
457Q8—21no—
458Q8—161no—
459Q8—1221no—
460Q8—2441no—
461Q8—2441no—
462Q8—2441no—
463D4—21no—
464D4—41no—
465D4—1221no—
466D4—9761no—
467D4—2441no—
468D4—2441no—
469D4—2441no—
470D4—2441no—
471D4—2441no—
472D4—2441no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—611no—
478C2×C4—1221no—
479C2×C4—1221no—
480C2×C4—1221no—
481C2×C4—9761no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4C2×D4×D6111yescharacteristic
488C8—41no—
489C8—1221no—
490C8—1221no—
491C8—2441no—
492C8#1451980411yescharacteristic
493C8#1451980411yescharacteristic
494C8#41×D6111yescharacteristic
495C8C2×#1175822611yescharacteristic
496C22—41no—
497C22—1221no—
498C22—4881no—
499C22—4881no—
500C22—4882no—
501C22—4881no—
502C22—2441no—
503C22—2441no—
504C22AGL(1,ℤ/8)×D6111yescharacteristic
505C4—41no—
506C4—41no—
507C4—81no—
508C4—81no—
509C4—1221no—
510C4—4881no—
511C4—2441no—
512C4—2441no—
513C4#1684027511yescharacteristic
514C4C2×#1451980411yescharacteristic
515C2—21no—
516C2—81no—
517C2—9761no—
518C2—4881no—
519C2—2441no—
520C2#2145591211yescharacteristic
subgroupG/H
centerC2#21455912
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2230C2
socleC2×C61#3109
prime radical#21447766C2
hypercenter#2230D61
hypocenterC61#33691
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2230

Complements

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

NH ≅ G/Ntype
1. #21456438C2semidirect
2. #21454717C2semidirect
3. #21448040C2semidirect
4. #21448035C2semidirect
5. #21447987C2semidirect
6. #21447889C2semidirect
7. #21447888C2semidirect
8. #21447771C2semidirect
10. #21447716C2semidirect
11. #21447714C2semidirect
12. #21447709C2semidirect
13. #21447707C2semidirect
14. #21447634C2semidirect
15. C61×#2230C2semidirect
23. #16840146C22semidirect
33. #16839873C22semidirect
34. #16839868C22semidirect
39. #16839584C22semidirect
40. #16839584C22semidirect
41. #16839583C22semidirect
42. #16839579C22semidirect
43. #16839577C22semidirect
44. #16839576C22semidirect
45. #16839574C22semidirect
50. #16839539C22semidirect
51. #16839539C22semidirect
52. C61×#498C22semidirect
54. C61×#365C22semidirect
55. C61×#365C22semidirect
56. C61×#364C22semidirect
260. #2230D61semidirect
283. #498C2×D61semidirect
303. #365C2×D61semidirect
304. #365C2×D61semidirect
306. #364C2×D61semidirect
319. C61#33691semidirect

Maximal subgroups

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

subgroupindex
#214564382
#214547172
#214480402
#214480352
#214479872
#214478892
#214478882
#214477712
#214477662
#214477162
#214477142
#214477092
#214477072
#214476342
C61×#22302
#3369161