Finite Group Databasebeta

← #30613157

Subgroups — #30613157

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462665C211yesmaximal, characteristic, complemented
2#21462652C211yesmaximal, characteristic, complemented
3#21461455C211yesmaximal, characteristic, complemented
4#21461450C211yesmaximal, characteristic, complemented
5#21461432C211yesmaximal, characteristic, complemented
6#21461428C211yesmaximal, characteristic, complemented
7#21461426C211yesmaximal, characteristic, complemented
8#21461421C211yesmaximal, characteristic, complemented
9Q16×#92420C211yesmaximal, characteristic, complemented
10#21461403C211yesmaximal, characteristic, complemented
11#21461390C211yesmaximal, characteristic, complemented
12#21461172C211yesmaximal, characteristic, complemented
13#21461110C211yesmaximal, characteristic, complemented
14#21461020C211yesmaximal, characteristic, complemented
15#21455838C211yesmaximal, characteristic
16#16840813C2211yescharacteristic
17Q8×#92420C2211yescharacteristic
18Q8×#92420C2211yescharacteristic
19Q8×#92420C2211yescharacteristic
20#16840797—21no—
21#16840797—21no—
22#16840797C2211yescharacteristic
23#16840797C2211yescharacteristic
24#16840796C2211yescharacteristic
25#16840795—21no—
26#16840794—21no—
27#16840790C2211yescharacteristic
28#16840787—21no—
29#16840783—21no—
30#16840783C2211yescharacteristic
31#16840778C2211yescharacteristic
32#16840727—21no—
33C4×#11758217—21no—
34C4×#11758217C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40Q8⋊#92420C2211yescharacteristic
41Q8⋊#92420C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44#16840610C2211yescharacteristic
45#16840610C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50C8×#92420C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840298C2211yescharacteristic
53#41×Dic61C2211yescharacteristic
54#16840100C411yescharacteristic
55#16840087C411yescharacteristic
56#16840067C411yescharacteristic
57#16840060C411yescharacteristic
58#16839986C411yescharacteristic
59#16839979C411yescharacteristic
60#16839907C411yescharacteristic
61Q16×Dic61C2211yescharacteristic
62SD16×Dic61C2211yescharacteristic
63#16839847C2211yescharacteristic
64#16839842C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#452C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758217—21no—
69C2×#11758217—21no—
70C2×#11758217D411yescharacteristic
71C2×#11758217D411yescharacteristic
72C2×D61⋊C8—21no—
73C22×#92420—21no—
74#14519844—21no—
75C4⋊#92420—21no—
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420C2311yescharacteristic
79C4⋊#92420C2311yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82#14519834—21no—
83#14519834—21no—
84C22⋊#92420—21no—
85C22⋊#92420—21no—
86C22⋊#92420—21no—
87C22⋊#92420—21no—
88#14519832—21no—
89#14519832—21no—
90#14519832D411yescharacteristic
91#14519832D411yescharacteristic
92C4×#92420—21no—
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758201C2×C411yescharacteristic
102C2×#11758199C2×C411yescharacteristic
103#14519772—21no—
104#14519768C2×C411yescharacteristic
105#14519766C2×C411yescharacteristic
106#14519761—21no—
107#14519751C2×C411yescharacteristic
108Q8×Dic61C2×C411yescharacteristic
109Q8×Dic61C2311yescharacteristic
110Q8×Dic61C2311yescharacteristic
111D4×Dic61—21no—
112D4×Dic61—21no—
113D4×Dic61C2×C411yescharacteristic
114D4×Dic61C2311yescharacteristic
115#14519739C2311yescharacteristic
116#14519737—21no—
117#14519736—21no—
118#14519736C2×C411yescharacteristic
119#14519735—21no—
120C2×C4×Dic61—21no—
121C2×C4×Dic61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519710C2×C411yescharacteristic
124#14519709C2×C411yescharacteristic
125#14519708C2×C411yescharacteristic
126#14519706C2×C411yescharacteristic
127#14519706C2×C411yescharacteristic
128#14519701C2×C411yescharacteristic
129#14519701C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519699C2×C411yescharacteristic
132#14519699C2×C411yescharacteristic
133#14519698C2×C411yescharacteristic
134C22⋊#92418C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×#41×C61C2×C411yescharacteristic
138C2×Q16×C61C2×C411yescharacteristic
139C2×SD16×C61C2×C411yescharacteristic
140C22⋊Q8×C61C2×C411yescharacteristic
141#103×C61C2×C411yescharacteristic
142#102×C61C2×C411yescharacteristic
143C22⋊C8×C61C2×C411yescharacteristic
144#41×C61—22no—
145#41×C61—41no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×Q8×C61C22×C411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61C22×C411yescharacteristic
152C2×C22⋊D61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61#4111yescharacteristic
157C2×D244C22×C411yescharacteristic
158C2×C4×D61—21no—
159#11758217—22no—
160#11758217—22no—
161#11758217—42no—
162#11758217—42no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—42no—
170C2×#92420C2×D411yescharacteristic
171C2×#92420C2×D411yescharacteristic
172C2×#92420C2×D411yescharacteristic
173C2×#92420C2×D411yescharacteristic
174#11758213—42no—
175D61⋊C8—42no—
176Q16×C61—22no—
177SD16×C61—41no—
178C2×C8×C61C22×C411yescharacteristic
179C2×C8×C61C22×C411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61C22×C411yescharacteristic
182C22⋊C4×C61—21no—
183C22⋊Dic61—21no—
184C22⋊Dic61—21no—
185C22⋊Dic61—21no—
186#11758201—22no—
187#11758199—41no—
188#11758198—21no—
189#11758198—21no—
190#11758198—21no—
191#11758198—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61C22×C411yescharacteristic
197C4⋊Dic61C22×C411yescharacteristic
198C4⋊Dic61C22×C411yescharacteristic
199#11758196—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61C22×C411yescharacteristic
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61#4111yescharacteristic
207C4×Dic61C2411yescharacteristic
208C2×#92418C22×C411yescharacteristic
209C2×#92418C22×C411yescharacteristic
210C23×C61—21no—
211C23×C61C4×D411yescharacteristic
212C22×D61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61#13112yes—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61—42no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—42no—
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C23×C411yescharacteristic
228C22⋊D61—81no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—81no—
238C2×Dic61C4×D411yescharacteristic
239C2×Dic61C4×D411yescharacteristic
240C2×Dic61C22×D411yescharacteristic
241C2×Dic61C2×#4111yescharacteristic
242D244—21no—
243D244—21no—
244C4×D61—41no—
245C4×D61—41no—
246#92420—22no—
247#92420—21no—
248#92420—21no—
249#92420—41no—
250#92420—41no—
251#92420—41no—
252#92420—41no—
253#92420—41no—
254#92420#13512yes—
255#92420#13512yes—
256C8×C61—21no—
257C8×C61—21no—
258#92418—21no—
259#92418—21no—
260#17999—611nomaximal, Sylow
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61C2×C4×D411yescharacteristic
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—41no—
271C4×C61—21no—
272C4×C61—22no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61C2×#13111yescharacteristic
276C4×C61#56811yescharacteristic
277Dic61—21no—
278Dic61—21no—
279Dic61—21no—
280Dic61—41no—
281Dic61—41no—
282Dic61—41no—
283Dic61C2×#13511yescharacteristic
284Dic61C2×#13511yescharacteristic
285#3102—611no—
286#2971—611no—
287#2969—611no—
288#1590—611no—
289#1589—611no—
290#1584—611no—
291#1583—611no—
292#1583—611no—
293#1581—611no—
294C8×Q16—611no—
295C8×SD16—611no—
296#1572—611no—
297#1494—611no—
298#1476—611no—
299C8×M16—611no—
300C2×C61—41no—
301C2×C61—81no—
302C2×C61#293311yescharacteristic
303C2×C61C4×#13511yescharacteristic
304C2×C61#295611yescharacteristic
305D61—81no—
306#529—611no—
307C4×#41—611no—
308#485—611no—
309#482—611no—
310#479—611no—
311#478—611no—
312#464—611no—
313#463—611no—
314#452#692311yescharacteristic
315#449—611no—
316#446—611no—
317C8×Q8—611no—
318C8×Q8—611no—
319C8×Q8—611no—
320C4×Q16—611no—
321C4×SD16—611no—
322C4×SD16—611no—
323C4×D8—611no—
324#436—611no—
325#436—611no—
326#436—611no—
327#436—1221no—
328#436—1221no—
329#436—1221no—
330#435—1221no—
331C8×D4—611no—
332#433—1221no—
333#432—611no—
334#432—1221no—
335#431—611no—
336#431—611no—
337#405—1221no—
338C4×M16—611no—
339C4×M16—1221no—
340C2×C4×C8—611no—
341#335—611no—
342#335—611no—
343#334—611no—
344#334—611no—
345Q8⋊C8—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348Q8⋊C8—611no—
349D4⋊C8—611no—
350D4⋊C8—611no—
351D4⋊C8—611no—
352D4⋊C8—611no—
353#322—611no—
354#322—611no—
355C82—611no—
356C82—611no—
357C61#1799911yescharacteristic, complemented, Sylow
358C2×#41C2×#692311yescharacteristic
359C2×Q16C2×#692311yescharacteristic
360C2×SD16—611no—
361C2×SD16C2×#692311yescharacteristic
362C2×D8—611no—
363C2×M16—611no—
364C2×M16—611no—
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16—1221no—
368C22×C8—1221no—
369C22×C8—1221no—
370#126—1221no—
371#125—611no—
372#124—611no—
373#123—1221no—
374C22⋊Q8C2×#692311yescharacteristic
375#121—611no—
376C4×Q8—611no—
377C4×Q8—611no—
378C4×Q8—611no—
379C4×D4—611no—
380C4×D4—611no—
381C4×D4—611no—
382C4×D4—1221no—
383C4×D4—1221no—
384C4×D4—1221no—
385#117—611no—
386#117—1221no—
387#117—1221no—
388C2×C42—611no—
389C2×C42—1221no—
390#107—611no—
391#107—611no—
392#106—611no—
393#106—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404#103—611no—
405#103—611no—
406#103—611no—
407#103C2×#692311yescharacteristic
408#102—611no—
409#102—611no—
410#102—611no—
411#102C2×#692311yescharacteristic
412C22⋊C8—611no—
413C22⋊C8—1221no—
414C22⋊C8—1221no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8—1221no—
419C22⋊C8C2×#692311yescharacteristic
420#97—611no—
421#97—611no—
422#97—1221no—
423#97—1221no—
424C4×C8—611no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—1221no—
431C4×C8—1221no—
432C4×C8—1221no—
433C4×C8—1221no—
434C4×C8—1221no—
435#41—22no—
436#41—41no—
437C2×Q8C22×#692311yescharacteristic
438C2×Q8C22×#692311yescharacteristic
439C2×D4—21no—
440C2×D4—611no—
441C2×D4—1221no—
442C2×D4C22×#692311yescharacteristic
443C22×C4—21no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4C22×#692311yescharacteristic
450Q16—22no—
451SD16—41no—
452SD16—1222no—
453D8—2441no—
454M16—1222no—
455M16—1222no—
456M16—2442no—
457M16—2442no—
458M16—2442no—
459C2×C8—611no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—611no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—2442no—
472C2×C8—2442no—
473C2×C8C22×#692311yescharacteristic
474C2×C8C22×#692311yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—611no—
477C4⋊C4—611no—
478C4⋊C4—611no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4C22×#692311yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—1221no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C22⋊C4—1221no—
492C22⋊C4—1221no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—1221no—
498C42—1221no—
499C42—1221no—
500C42—1221no—
501C23—21no—
502C23—1221no—
503C23D4×#692311yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8#1451986412yes—
508D4—21no—
509D4—21no—
510D4—42no—
511D4—1221no—
512D4—1221no—
513D4—4881no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—42no—
520C2×C4—611no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—4881no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4D4×#692311yescharacteristic
536C2×C4C23×#692311yescharacteristic
537C8—21no—
538C8—21no—
539C8—612no—
540C8—612no—
541C8—1221no—
542C8—1221no—
543C8—1222no—
544C8—1221no—
545C8—1221no—
546C8—2441no—
547C8—2441no—
548C8—2441no—
549C8—2441no—
550C8—2441no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—2441no—
558C22—2441no—
559C22—2441no—
560C22C2×D4×#692311yescharacteristic
561C4—21no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—611no—
566C4—611no—
567C4—1221no—
568C4—1221no—
569C4—1221no—
570C4—2441no—
571C4—2441no—
572C4—2441no—
573C4C2×#1451986411yescharacteristic
574C4#1684084011yescharacteristic
575C2—41no—
576C2—81no—
577C2—4881no—
578C2#2146255111yescharacteristic
579C2#135×#692311yescharacteristic
580C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#452C4
socleC22×C61C2×C4×D4
prime radical#16839907C4
cosocleC4×Dic61C24
hypercenter#452#6923
hypocenterC61#17999
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #21462665C2semidirect
2. #21462652C2semidirect
3. #21461455C2semidirect
4. #21461450C2semidirect
5. #21461432C2semidirect
6. #21461428C2semidirect
7. #21461426C2semidirect
8. #21461421C2semidirect
9. Q16×#92420C2semidirect
10. #21461403C2semidirect
11. #21461390C2semidirect
12. #21461172C2semidirect
13. #21461110C2semidirect
14. #21461020C2semidirect
357. C61#17999semidirect

Maximal subgroups

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

subgroupindex
#214626652
#214626522
#214614552
#214614502
#214614322
#214614282
#214614262
#214614212
Q16×#924202
#214614032
#214613902
#214611722
#214611102
#214610202
#214558382
#1799961