Finite Group Databasebeta

← #29835251

Subgroups — #29835251

Order 15616.

520 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456561C211yesmaximal, characteristic, complemented
2#21454727C211yesmaximal, characteristic
3#21448040C211yesmaximal, characteristic, complemented
4#21448037C211yesmaximal, characteristic, complemented
5#21447989C211yesmaximal, characteristic, complemented
6#21447988C211yesmaximal, characteristic, complemented
7#21447899C211yesmaximal, characteristic, complemented
8#21447898C211yesmaximal, characteristic, complemented
9#21447771C211yesmaximal, characteristic
10#21447768C211yesmaximal, characteristic
11#21447712C211yesmaximal, characteristic, complemented
12#21447710C211yesmaximal, characteristic, complemented
13#21447708C211yesmaximal, characteristic, complemented
14#21447629C211yesmaximal, characteristic, complemented
15C61×#2180C211yesmaximal, characteristic, complemented
16#16840559C2211yescharacteristic
17#16840527—21no—
18#16840316—21no—
19#16840205C2211yescharacteristic
20#16840203C2211yescharacteristic
21#16840202C2211yescharacteristic
22#16840159C2211yescharacteristic, complemented
23#16840140C2211yescharacteristic, complemented
24#16840111—21no—
25#16840021C2211yescharacteristic
26#16840019—21no—
27#16840017C2211yescharacteristic
28#16839885C2211yescharacteristic
29#16839881—21no—
30#16839879C2211yescharacteristic
31#16839878C2211yescharacteristic
32#16839877—21no—
33#16839870C2211yescharacteristic, complemented
34#16839867—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#16839578C2211yescharacteristic, complemented
45#16839577C2211yescharacteristic, complemented
46#16839573C2211yescharacteristic
47#16839572C2211yescharacteristic
48#16839571C2211yescharacteristic
49#16839543C2211yescharacteristic
50#16839543C2211yescharacteristic
51#16839540C2211yescharacteristic, complemented
52C61×#510C2211yescharacteristic
53C61×#444C2211yescharacteristic
54C61×#441C2211yescharacteristic, complemented
55C61×#365C2211yescharacteristic, complemented
56C61×#362C2211yescharacteristic, 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×#11758202—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#14519776—21no—
79#14519775—21no—
80#14519767—41no—
81#14519757—21no—
82#14519755—41no—
83#14519753—21no—
84C4×Dic122—21no—
85C4×Dic122—41no—
86#14519734—41no—
87C42×D61D411yescharacteristic
88Dic488—21no—
89#14519723—21no—
90#14519723—21no—
91#14519722—21no—
92#14519721—21no—
93#14519720—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#14519702—21no—
104#14519701—21no—
105#14519700—21no—
106#14519688D411yescharacteristic
107C8×Dic61D411yescharacteristic
108C8×Dic61C2311yescharacteristic
109#135×C61C2311yescharacteristic
110C2×Q16×C61C2311yescharacteristic
111#131×C61—21no—
112C4×Q8×C61—21no—
113Q32×C61—21no—
114SD32×C61—21no—
115M32×C61C2311yescharacteristic
116M32×C61C2311yescharacteristic
117#108×C61C2311yescharacteristic
118#106×C61C2311yescharacteristic
119#104×C61—21no—
120#103×C61—21no—
121#97×C61C2311yescharacteristic
122#41×C61—21no—
123C2×Q8×C61—21no—
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×C4×D61—21no—
133C2×C4×D61—41no—
134C2×C4×D61C2×D411yescharacteristic
135C2×Dic122—21no—
136C2×Dic122—41no—
137Q16×C61—21no—
138Q16×C61—21no—
139Q16×C61#4111yescharacteristic
140SD16×C61—21no—
141D8×C61#4111yescharacteristic
142M16×C61—21no—
143M16×C61—41no—
144C2×C8×C61—41no—
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2411yescharacteristic
147C4⋊C4×C61—21no—
148C4⋊C4×C61—41no—
149C42×C61—41no—
150C42×C61C2×D411yescharacteristic
151#11758202—21no—
152#11758202—41no—
153#11758202—41no—
154#11758201—21no—
155#11758200—21no—
156#11758199—21no—
157#11758198—41no—
158#11758198—41no—
159C4⋊Dic61—21no—
160C4⋊Dic61—41no—
161#11758196—41no—
162#11758196—41no—
163#11758196—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—
193C22×D61—41no—
194C22×D61AGL(1,ℤ/8)11yescharacteristic
195Q8×C61—21no—
196Q8×C61—41no—
197Q8×C61—41no—
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×Dic61—41no—
212C2×Dic61AGL(1,ℤ/8)11yescharacteristic
213D244—21no—
214D244—41no—
215C4×D61—21no—
216C4×D61—21no—
217C4×D61—42no—
218C4×D61—41no—
219C4×D61—41no—
220C4×D61—41no—
221Dic122—21no—
222Dic122—41no—
223Dic122—41no—
224Dic122—41no—
225Dic122—82no—
226C8×C61—21no—
227C8×C61—41no—
228C8×C61C2×#4111yescharacteristic
229C8×C61C2×#4111yescharacteristic
230#92418—21no—
231#92418—41no—
232#92418AGL(1,ℤ/8)11yescharacteristic
233#92418AGL(1,ℤ/8)11yescharacteristic
234#33689—611nomaximal, Sylow
235C22×C61—41no—
236C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C2×D61—21no—
238C2×D61—41no—
239C2×D61—41no—
240C2×D61—161no—
241C4×C61—41no—
242C4×C61—41no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61#52511yescharacteristic
246C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
247Dic61—22no—
248Dic61—21no—
249Dic61—41no—
250Dic61—81no—
251Dic61—81no—
252#3185—611no—
253#2956—611no—
254#2239—611no—
255#2239—611no—
256#2230—611no—
257#2230—611no—
258#2214—611no—
259#2214—611no—
260#2191—611no—
261#2191—611no—
262#2180—611no—
263#2180—611no—
264#2180—611no—
265#2180D6111yescharacteristic, complemented
266#2165—611no—
267C2×C61—21no—
268C2×C61—81no—
269C2×C61#310911yescharacteristic
270D61—41no—
271D61—81no—
272#575—611no—
273#568—1221no—
274#533—1221no—
275#510—611no—
276#510—611no—
277#510—611no—
278#510C2×D6111yescharacteristic
279#498—611no—
280#494—611no—
281#487—1221no—
282#470—611no—
283#470—611no—
284#470—1221no—
285#444—611no—
286#444—1221no—
287#444C2×D6111yescharacteristic
288#443—611no—
289#443—611no—
290#443—1221no—
291#441—611no—
292#441—1221no—
293#441C2×D6111yescharacteristic, complemented
294#430—611no—
295C2×#104—1221no—
296C4×M16—611no—
297#365—611no—
298#365—611no—
299#365—611no—
300#365C2×D6111yescharacteristic, complemented
301#364—611no—
302#364—611no—
303#362—611no—
304#362—611no—
305#362—611no—
306#362C2×D6111yescharacteristic, complemented
307#360—611no—
308#360—611no—
309#360—611no—
310#360C2×D6111yescharacteristic
311#349—611no—
312#349—611no—
313#347—611no—
314#347C2×D6111yescharacteristic, complemented
315C61#3368911yescharacteristic, complemented, Sylow
316C2×#41—1221no—
317#137—1221no—
318#137—1221no—
319AGL(1,ℤ/8)—1221no—
320AGL(1,ℤ/8)—1221no—
321#135—611no—
322#135—1221no—
323#135C22×D6111yescharacteristic
324C2×Q16—611no—
325C2×Q16—1221no—
326C2×Q16C22×D6111yescharacteristic
327#131—21no—
328#131—1221no—
329#131—1221no—
330#131—1221no—
331#131—2441no—
332#131—2441no—
333C2×M16—611no—
334C2×M16—611no—
335C2×M16—1221no—
336C2×M16—2441no—
337C2×M16—2441no—
338C4⋊Q8—1221no—
339C4⋊Q8—1221no—
340#125—2441no—
341C22⋊Q8—1221no—
342C22⋊Q8—1221no—
343C22⋊Q8—2441no—
344C4×Q8—21no—
345C4×Q8—1221no—
346C4×Q8—2441no—
347#117—2441no—
348C2×C42—611no—
349Q32—21no—
350Q32—1221no—
351Q32—1221no—
352Q32—1221no—
353SD32—21no—
354SD32—1221no—
355SD32—1221no—
356SD32—1221no—
357M32—611no—
358M32—611no—
359M32C22×D6111yescharacteristic
360M32C22×D6111yescharacteristic
361#108—611no—
362#108—1221no—
363#108C22×D6111yescharacteristic
364#106—611no—
365#106—1221no—
366#106C22×D6111yescharacteristic
367#104—21no—
368#104—1221no—
369#104—1221no—
370#104—1221no—
371#103—21no—
372#103—1221no—
373#103—1221no—
374#103—1221no—
375#97—611no—
376#97C22×D6111yescharacteristic
377C4×C8—611no—
378C4×C8—611no—
379#41—21no—
380#41—1221no—
381#41—1221no—
382#41—1221no—
383#41—2441no—
384#41—2441no—
385C2×Q8—21no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×Q8—2441no—
389C2×D4—1221no—
390C2×D4—2441no—
391C22×C4—611no—
392C22×C4—1221no—
393C22×C4—2441no—
394Q16—21no—
395Q16—21no—
396Q16—611no—
397Q16—1221no—
398Q16—1221no—
399Q16—1221no—
400Q16—2441no—
401Q16—2441no—
402Q16#1175822211yescharacteristic
403SD16—21no—
404SD16—1221no—
405SD16—1221no—
406SD16—1221no—
407D8—611no—
408D8—1221no—
409D8#1175822211yescharacteristic
410M16—21no—
411M16—41no—
412M16—1221no—
413M16—1221no—
414M16—1221no—
415M16—1221no—
416M16—1221no—
417M16—2441no—
418M16—2441no—
419M16—2441no—
420M16—2441no—
421C2×C8—41no—
422C2×C8—611no—
423C2×C8—611no—
424C2×C8—2441no—
425C2×C8—2441no—
426C2×C8—2441no—
427C2×C8—2441no—
428C2×C8D4×D6111yescharacteristic
429C2×C8C23×D6111yescharacteristic
430C4⋊C4—21no—
431C4⋊C4—41no—
432C4⋊C4—1221no—
433C4⋊C4—2441no—
434C4⋊C4—2441no—
435C4⋊C4—2441no—
436C4⋊C4—2441no—
437C4⋊C4—2441no—
438C4⋊C4—2441no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C42—41no—
442C42—611no—
443C42—1221no—
444C42—2441no—
445C42D4×D6111yescharacteristic
446C16—21no—
447C16—21no—
448C16—1221no—
449C16—1221no—
450C23—611no—
451C23—2441no—
452Q8—21no—
453Q8—41no—
454Q8—41no—
455Q8—1221no—
456Q8—4882no—
457Q8—2441no—
458Q8—2441no—
459Q8—2441no—
460D4—21no—
461D4—41no—
462D4—1221no—
463D4—2441no—
464D4—2441no—
465D4—2441no—
466D4—2441no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—611no—
472C2×C4—1221no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4C2×D4×D6111yescharacteristic
483C8—21no—
484C8—41no—
485C8—611no—
486C8—611no—
487C8—1221no—
488C8—2441no—
489C8#41×D6111yescharacteristic
490C8C2×#1175822211yescharacteristic
491C22—41no—
492C22—1221no—
493C22—9761no—
494C22—2441no—
495C22—2441no—
496C22AGL(1,ℤ/8)×D6111yescharacteristic
497C4—41no—
498C4—41no—
499C4—81no—
500C4—81no—
501C4—1222no—
502C4—1221no—
503C4—4881no—
504C4—4881no—
505C4—2441no—
506C4#1684027111yescharacteristic
507C4AGL(1,ℤ/8)×D6111yescharacteristic
508C2—21no—
509C2—81no—
510C2—4881no—
511C2—2441no—
512C2#2145590811yescharacteristic
subgroupG/H
centerC2#21455908
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2180C2
socleC2×C61#3109
prime radical#16840559C22
hypercenter#2180D61
hypocenterC61#33689
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #2180

Complements

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

NH ≅ G/Ntype
1. #21456561C2semidirect
3. #21448040C2semidirect
4. #21448037C2semidirect
5. #21447989C2semidirect
6. #21447988C2semidirect
7. #21447899C2semidirect
8. #21447898C2semidirect
11. #21447712C2semidirect
12. #21447710C2semidirect
13. #21447708C2semidirect
14. #21447629C2semidirect
15. C61×#2180C2semidirect
22. #16840159C22semidirect
23. #16840140C22semidirect
33. #16839870C22semidirect
38. #16839586C22semidirect
39. #16839585C22semidirect
40. #16839584C22semidirect
41. #16839582C22semidirect
42. #16839582C22semidirect
43. #16839579C22semidirect
44. #16839578C22semidirect
45. #16839577C22semidirect
51. #16839540C22semidirect
54. C61×#441C22semidirect
55. C61×#365C22semidirect
56. C61×#362C22semidirect
58. C61×#347C22semidirect
265. #2180D61semidirect
293. #441C2×D61semidirect
300. #365C2×D61semidirect
306. #362C2×D61semidirect
314. #347C2×D61semidirect
315. C61#33689semidirect

Maximal subgroups

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

subgroupindex
#214565612
#214547272
#214480402
#214480372
#214479892
#214479882
#214478992
#214478982
#214477712
#214477682
#214477122
#214477102
#214477082
#214476292
C61×#21802
#3368961