Finite Group Databasebeta

← #29834404

Subgroups — #29834404

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458951C212yesmaximal, complemented
2#21456246C211yesmaximal, characteristic, complemented
3#443×D61C211yesmaximal, characteristic, complemented
4#21447901C211yesmaximal, characteristic, complemented
5#21447900C211yesmaximal, characteristic, complemented
6#21447892C211yesmaximal, characteristic, complemented
7#21447887C212yesmaximal, complemented
8#21447886C212yesmaximal, complemented
9#364×D61C211yesmaximal, characteristic, complemented
10#21447565C212yesmaximal, complemented
11C61×#2213C211yesmaximal, characteristic, complemented
12C2×#14519796—21no—
13C2×#14519796C2212yes—
14#135×D61C2211yescharacteristic
15#131×D61—21no—
16#16840310—21no—
17#16840205C2212yes—
18#16840202C2212yes—
19#16840201—22no—
20#16840198C2212yes—
21#16840196—22no—
22#16840193—22no—
23#16840191—22no—
24M32×D61C2212yes—
25#16840145—21no—
26#16840145C2212yescomplemented
27#16840142C2211yescharacteristic, complemented
28#16840134—21no—
29#16839880C2211yescharacteristic
30#16839879C2211yescharacteristic
31#16839878C2211yescharacteristic
32#16839877—21no—
33#16839876—21no—
34#108×D61C2211yescharacteristic
35#104×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839583C2211yescharacteristic, complemented
38#16839582C2212yescomplemented
39#16839579C2212yescomplemented
40#16839577C2212yescomplemented
41#16839575C2212yescomplemented
42C61×#509C2212yes—
43C61×#493C2211yescharacteristic, complemented
44C61×#443C2211yescharacteristic
45C61×#364C2211yescharacteristic, complemented
46C61×#361C2212yescomplemented
47C2×#11758224—22no—
48#41×D61—21no—
49#14519800D411yescharacteristic, complemented
50#14519799C2311yescharacteristic
51#14519798—21no—
52#14519797—21no—
53#14519796—22no—
54#14519796—22no—
55#14519796—22no—
56#14519796—22no—
57#14519796—41no—
58#14519796—41no—
59#14519796—41no—
60#14519796—41no—
61#14519796D411yescharacteristic, complemented
62C2×Dic244—21no—
63C2×Dic244C2312yes—
64Q16×D61D411yescharacteristic, complemented
65C2×#11758200—22no—
66C2×#11758200—22no—
67SD16×D61—21no—
68D8×D61D411yescharacteristic, complemented
69#14519785—21no—
70#14519785—41no—
71#14519784—21no—
72#14519783—21no—
73#14519783—41no—
74M16×D61—21no—
75M16×D61—41no—
76C2×C8×D61—41no—
77C2×C8×D61D411yescharacteristic
78C2×C8×D61C2311yescharacteristic
79#14519777—21no—
80#14519764—22no—
81#14519748—42no—
82D4×Dic61—42no—
83C42×D61D411yescharacteristic
84Dic488—22no—
85#14519723—22no—
86#14519722—22no—
87#14519721—22no—
88#14519721—22no—
89#14519718—22no—
90#14519717C2312yes—
91#14519716—22no—
92C16×D61—22no—
93#14519713C2311yescharacteristic
94#14519712—21no—
95#14519705—21no—
96#14519704—21no—
97#14519703—21no—
98C4×#92418D411yescharacteristic
99C8×Dic61D411yescharacteristic
100C8×Dic61C2311yescharacteristic
101#135×C61C2311yescharacteristic
102C2×D8×C61—21no—
103C2×D8×C61C2312yes—
104#131×C61—21no—
105#127×C61—21no—
106SD32×C61—22no—
107D16×C61—22no—
108M32×C61C2312yes—
109#108×C61C2311yescharacteristic
110#104×C61—21no—
111C4×C8×C61C2311yescharacteristic
112#41×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—42no—
115C2×C22⋊D61—42no—
116C22×Dic61—42no—
117#11758226—21no—
118Q8×D61—21no—
119#11758224—21no—
120#11758224—41no—
121#11758224—42no—
122#11758224—42no—
123#11758224—42no—
124#11758224—42no—
125D4×D61—21no—
126D4×D61—41no—
127#11758222—21no—
128#11758222—41no—
129C2×C4×D61—21no—
130C2×C4×D61—41no—
131C2×C4×D61C2×D411yescharacteristic
132C2×Dic122—22no—
133Q16×C61C2×D411yescharacteristic
134SD16×C61—21no—
135D8×C61—22no—
136D8×C61—22no—
137D8×C61—41no—
138D8×C61—41no—
139D8×C61C2×D411yescharacteristic
140M16×C61—21no—
141M16×C61—41no—
142C2×C8×C61—41no—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C22⋊Dic61—44no—
147#11758202—21no—
148#11758201—21no—
149#11758200—21no—
150#11758200—42no—
151#11758200—42no—
152#11758200—42no—
153#11758200—42no—
154#11758199—21no—
155C4⋊Dic61—42no—
156C4×Dic61—21no—
157C4×Dic61C2×D411yescharacteristic
158#11758194—21no—
159#11758194—41no—
160C2×#92418—41no—
161C2×#92418C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163Dic244—22no—
164Dic244—22no—
165Dic244—41no—
166Dic244—41no—
167Dic244C2×D411yescharacteristic
168D488C2×D411yescharacteristic
169#11758190—21no—
170#11758189—41no—
171#11758189—41no—
172#11758189—41no—
173C8×D61—41no—
174C8×D61—41no—
175C8×D61—41no—
176C8×D61D811yescharacteristic
177C8×D61D811yescharacteristic
178C8×D61D811yescharacteristic
179C8×D61D811yescharacteristic
180C8×D61C2×D411yescharacteristic
181C8×D61C2×D411yescharacteristic
182C8×D61#4111yescharacteristic
183C8×D61#4111yescharacteristic
184C16×C61—22no—
185#11758185—22no—
186C23×C61—42no—
187C22×D61—41no—
188C22×D61AGL(1,ℤ/8)11yescharacteristic
189Q8×C61—21no—
190D4×C61—21no—
191D4×C61—41no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—84no—
195C2×C4×C61—21no—
196C2×C4×C61—41no—
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—41no—
199C22⋊D61—41no—
200C22⋊D61—41no—
201C22⋊D61—82no—
202C22⋊D61—82no—
203C2×Dic61—21no—
204C2×Dic61—41no—
205C2×Dic61—42no—
206C2×Dic61—82no—
207C2×Dic61—82no—
208C2×Dic61AGL(1,ℤ/8)11yescharacteristic
209D244—21no—
210D244—41no—
211C4×D61—42no—
212C4×D61—41no—
213C4×D61—41no—
214C4×D61—41no—
215C4×D61#12111yescharacteristic
216C4×D61#12111yescharacteristic
217C4×D61C2×D811yescharacteristic
218C4×D61C2×D811yescharacteristic
219Dic122—21no—
220Dic122—41no—
221Dic122—42no—
222Dic122—42no—
223C8×C61—41no—
224C8×C61C2×D811yescharacteristic
225C8×C61C2×D811yescharacteristic
226C8×C61C22×D411yescharacteristic
227C8×C61C2×#4111yescharacteristic
228#92418—41no—
229#92418#12111yescharacteristic
230#92418#12111yescharacteristic
231#92418C2×D811yescharacteristic
232#92418C2×D811yescharacteristic
233#33594—611nomaximal, Sylow
234C22×C61—41no—
235C22×C61—84no—
236C22×C61—82no—
237C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C2×D61—41no—
239C2×D61—41no—
240C2×D61—82no—
241C2×D61#45911yescharacteristic
242C2×D61#45911yescharacteristic
243C4×C61—22no—
244C4×C61—41no—
245C4×C61C2×#12111yescharacteristic
246C4×C61C22×D811yescharacteristic
247Dic61—41no—
248Dic61—41no—
249Dic61—82no—
250Dic61#45911yescharacteristic
251Dic61#45911yescharacteristic
252#3414—612no—
253#3147—611no—
254C2×#443—611no—
255#2214—611no—
256#2214—611no—
257#2213—611no—
258#2213D6111yescharacteristic, complemented
259#2212—612no—
260#2212—612no—
261C2×#364—611no—
262#2148—612no—
263C2×C61—21no—
264C2×C61—81no—
265C2×C61—162no—
266C2×C61C2×#45911yescharacteristic
267D61—21no—
268D61—21no—
269D61—81no—
270C2×#135—611no—
271C2×#135—612no—
272C2×#135—1221no—
273C2×#131—1221no—
274#532—1221no—
275#510—612no—
276#510—612no—
277#510—1222no—
278#509—612no—
279#509—1222no—
280#509C2×D6112yes—
281#508—1222no—
282#508—1222no—
283C2×M32—612no—
284#495—612no—
285#495—1221no—
286#494—611no—
287#493C2×D6111yescharacteristic, complemented
288#492—1221no—
289#443—611no—
290#443—611no—
291#443—611no—
292#443—1221no—
293#443—1221no—
294#443C2×D6111yescharacteristic
295C2×#108—611no—
296C2×#104—1221no—
297C2×C4×C8—611no—
298#364—612no—
299#364—611no—
300#364C2×D6111yescharacteristic, complemented
301#362—612no—
302#362—612no—
303#361—612no—
304#361C2×D6112yescomplemented
305C61#3359411yescharacteristic, complemented, Sylow
306C2×#41—1222no—
307C2×#41—1221no—
308#135—611no—
309#135—611no—
310#135—611no—
311#135—1222no—
312#135—1222no—
313#135—1222no—
314#135—1222no—
315#135—1221no—
316#135—1221no—
317#135—2441no—
318#135—2441no—
319#135—2441no—
320#135—2441no—
321#135C22×D6111yescharacteristic
322C2×Q16—611no—
323C2×Q16—612no—
324C2×Q16—1221no—
325C2×SD16—1221no—
326C2×SD16—1222no—
327C2×SD16—1222no—
328C2×D8—21no—
329C2×D8—611no—
330C2×D8C22×D6112yes—
331#131—21no—
332#131—1221no—
333#131—1221no—
334#131—1221no—
335#131—2441no—
336#131—2441no—
337C2×M16—1221no—
338C2×M16—2441no—
339C22×C8—611no—
340C22×C8—611no—
341C22×C8—2441no—
342C4⋊Q8—1221no—
343#127—21no—
344#124—1222no—
345#121—2442no—
346C4×D4—2442no—
347C2×C42—611no—
348Q32—1222no—
349Q32—1222no—
350SD32—22no—
351SD32—1222no—
352SD32—1222no—
353SD32—1222no—
354D16—22no—
355D16—1222no—
356M32—612no—
357M32—1222no—
358M32C22×D6112yes—
359C2×C16—1222no—
360#108—611no—
361#108—1221no—
362#108C22×D6111yescharacteristic
363#104—21no—
364#104—1221no—
365#104—1221no—
366#104—1221no—
367C4×C8—611no—
368C4×C8—611no—
369C4×C8—611no—
370C4×C8C22×D6111yescharacteristic
371#41—21no—
372#41—1221no—
373#41—1221no—
374#41—1221no—
375#41—2441no—
376#41—2442no—
377#41—2442no—
378#41—2442no—
379#41—2442no—
380#41—2441no—
381C2×Q8—1221no—
382C2×Q8—1222no—
383C2×D4—22no—
384C2×D4—42no—
385C2×D4—1221no—
386C2×D4—2441no—
387C2×D4—2442no—
388C22×C4—611no—
389C22×C4—1221no—
390C22×C4—2441no—
391C22×C4—2442no—
392Q16—611no—
393Q16—1221no—
394Q16—1222no—
395Q16—1222no—
396Q16—2441no—
397Q16—2441no—
398Q16D4×D6111yescharacteristic
399SD16—21no—
400SD16—1221no—
401SD16—1221no—
402SD16—1221no—
403SD16—2442no—
404SD16—2442no—
405SD16—2442no—
406SD16—2442no—
407D8—22no—
408D8—22no—
409D8—41no—
410D8—41no—
411D8—611no—
412D8—1221no—
413D8D4×D6111yescharacteristic
414M16—21no—
415M16—41no—
416M16—1221no—
417M16—2441no—
418M16—2441no—
419M16—2441no—
420M16—2441no—
421C2×C8—41no—
422C2×C8—611no—
423C2×C8—611no—
424C2×C8—611no—
425C2×C8—611no—
426C2×C8—611no—
427C2×C8—611no—
428C2×C8—611no—
429C2×C8—611no—
430C2×C8—611no—
431C2×C8—611no—
432C2×C8—2441no—
433C2×C8—2441no—
434C2×C8—2441no—
435C2×C8—2441no—
436C2×C8D4×D6111yescharacteristic
437C2×C8C23×D6111yescharacteristic
438C4⋊C4—2442no—
439C22⋊C4—2444no—
440C42—611no—
441C42—1221no—
442C42D4×D6111yescharacteristic
443C16—22no—
444C16—1222no—
445C23—42no—
446C23—611no—
447C23—2441no—
448Q8—21no—
449Q8—1221no—
450Q8—2441no—
451Q8—2442no—
452Q8—2442no—
453D4—21no—
454D4—41no—
455D4—42no—
456D4—42no—
457D4—84no—
458D4—1221no—
459D4—4882no—
460D4—4882no—
461D4—2441no—
462D4—2441no—
463D4—2441no—
464D4—2441no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—611no—
468C2×C4—611no—
469C2×C4—611no—
470C2×C4—611no—
471C2×C4—611no—
472C2×C4—1221no—
473C2×C4—4882no—
474C2×C4—4882no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2442no—
480C2×C4—2441no—
481C2×C4C2×D4×D6111yescharacteristic
482C8—41no—
483C8—611no—
484C8—611no—
485C8—611no—
486C8—611no—
487C8—2441no—
488C8D8×D6111yescharacteristic
489C8D8×D6111yescharacteristic
490C8C2×D4×D6111yescharacteristic
491C8#41×D6111yescharacteristic
492C22—41no—
493C22—84no—
494C22—82no—
495C22—611no—
496C22—611no—
497C22—4882no—
498C22—2441no—
499C22—2441no—
500C22AGL(1,ℤ/8)×D6111yescharacteristic
501C4—22no—
502C4—41no—
503C4—611no—
504C4—611no—
505C4—4882no—
506C4—2441no—
507C4—2441no—
508C4#121×D6111yescharacteristic
509C4C2×D8×D6111yescharacteristic
510C2—21no—
511C2—81no—
512C2—162no—
513C2—1221no—
514C2—1221no—
515C2—4881no—
516C2#459×D6111yescharacteristic
subgroupG/H
centerC2#459×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2213C2
socleC2×C61C2×#459
hypercenter#2213D61
hypocenterC61#33594
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2213

Complements

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

NH ≅ G/Ntype
1. #21458951C2semidirect
2. #21456246C2semidirect
3. #443×D61C2semidirect
4. #21447901C2semidirect
5. #21447900C2semidirect
6. #21447892C2semidirect
7. #21447887C2semidirect
8. #21447886C2semidirect
9. #364×D61C2semidirect
10. #21447565C2semidirect
11. C61×#2213C2semidirect
26. #16840145C22semidirect
27. #16840142C22semidirect
37. #16839583C22semidirect
38. #16839582C22semidirect
39. #16839579C22semidirect
40. #16839577C22semidirect
41. #16839575C22semidirect
43. C61×#493C22semidirect
45. C61×#364C22semidirect
46. C61×#361C22semidirect
49. #14519800D4semidirect
61. #14519796D4semidirect
64. Q16×D61D4semidirect
68. D8×D61D4semidirect
258. #2213D61semidirect
287. #493C2×D61semidirect
300. #364C2×D61semidirect
304. #361C2×D61semidirect
305. C61#33594semidirect

Maximal subgroups

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

subgroupindex
#214589512
#214562462
#443×D612
#214479012
#214479002
#214478922
#214478872
#214478862
#364×D612
#214475652
C61×#22132
#3359461