Finite Group Databasebeta

← #29671131

Subgroups — #29671131

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458868C211yesmaximal, characteristic
2#21458822C211yesmaximal, characteristic, complemented
3#21458513C211yesmaximal, characteristic
4#21445489C211yesmaximal, characteristic
5#21445311C211yesmaximal, characteristic, complemented
6#21445263C211yesmaximal, characteristic
7#21445210C211yesmaximal, characteristic, complemented
8#21445169C211yesmaximal, characteristic
9#21445061C211yesmaximal, characteristic, complemented
10#21445060C211yesmaximal, characteristic, complemented
11#21445004C211yesmaximal, characteristic
12#21444915C211yesmaximal, characteristic
13#21444670C211yesmaximal, characteristic, complemented
14#21444636C211yesmaximal, characteristic, complemented
15C61×#3383C211yesmaximal, characteristic, complemented
16#16840497C2211yescharacteristic
17#16840491C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840154—21no—
20#16840141—21no—
21#16840141C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840118—21no—
24#16840110C2211yescharacteristic
25#16840109C2211yescharacteristic
26#16840065—21no—
27#16840047C2211yescharacteristic
28#16839986—21no—
29#16839984—21no—
30#16839972—21no—
31#16839972C2211yescharacteristic
32#16839969—21no—
33#16839969C2211yescharacteristic
34#16839969C2211yescharacteristic
35#16839964—21no—
36#16839962C2211yescharacteristic
37#16839962C2211yescharacteristic
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839860—21no—
42C4×#11758202—21no—
43C4×#11758202C2211yescharacteristic
44#16839847—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839492C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#557C2211yescharacteristic
57C61×#557C2211yescharacteristic
58C61×#500C2211yescharacteristic
59C61×#499—21no—
60C61×#479—21no—
61C61×#479—21no—
62C61×#475C2211yescharacteristic
63C61×#475C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×Q16×C61—21no—
66C4×Q16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202D411yescharacteristic
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77#14519777—21no—
78#14519777C2311yescharacteristic
79#14519774C2311yescharacteristic
80#14519768—21no—
81#14519768—41no—
82#14519766—21no—
83#14519765—21no—
84#14519762—41no—
85Q8×Dic61—21no—
86C4×Dic122—21no—
87C4×Dic122C2311yescharacteristic
88C4×D244—21no—
89#14519739—21no—
90#14519710—41no—
91#14519710D411yescharacteristic
92#14519708—21no—
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519700D411yescharacteristic
105#14519699—21no—
106#14519699—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519688—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×Q16×C61D411yescharacteristic
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61C2311yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120#125×C61—21no—
121#125×C61—21no—
122#125×C61—41no—
123#125×C61—41no—
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131#107×C61—21no—
132#107×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D411yescharacteristic
148C2×Q8×C61C2×D411yescharacteristic
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×C4×D61—41no—
153C2×Dic122—21no—
154C2×Dic122C2×D411yescharacteristic
155Q16×C61—22no—
156C2×C8×C61—21no—
157C2×C8×C61C2×D411yescharacteristic
158C2×C8×C61C2×D411yescharacteristic
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—41no—
167C4⋊C4×C61—41no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—41no—
179C42×C61—41no—
180C42×C61C2411yescharacteristic
181#11758202—22no—
182#11758202—42no—
183#11758202—81no—
184#11758202—81no—
185#11758202—81no—
186#11758202—81no—
187#11758201—42no—
188#11758201—42no—
189#11758201—81no—
190#11758201—81no—
191#11758198—41no—
192#11758198—41no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—41no—
197C4⋊Dic61—41no—
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—21no—
200#11758196—41no—
201C4×Dic61—21no—
202C4×Dic61—41no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C22×D61—41no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61#13712yes—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—41no—
234D244—41no—
235D244—41no—
236D244—41no—
237D244—41no—
238D244—81no—
239C4×D61—161no—
240Dic122—21no—
241Dic122—21no—
242Dic122—41no—
243Dic122—41no—
244C8×C61—21no—
245C8×C61—21no—
246C8×C61—41no—
247#92418—21no—
248#92418—41no—
249#92418—41no—
250#92418—41no—
251#92418—41no—
252#92418#13512yes—
253#19873—611nomaximal, Sylow
254C22×C61D4211yescharacteristic
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C4×C61—22no—
259C4×C61—21no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—81no—
266C4×C61—81no—
267C4×C61C2×#13511yescharacteristic
268C4×C61C2×#13711yescharacteristic
269C4×C61#57611yescharacteristic
270C4×C61#57811yescharacteristic
271Dic61—41no—
272Dic61—41no—
273Dic61—81no—
274#3400—611no—
275#3395—611no—
276#3383D6111yescharacteristic, complemented
277#3365—611no—
278#1722—611no—
279#1690—611no—
280#1682—611no—
281#1674—611no—
282#1666—611no—
283#1648—611no—
284#1648—611no—
285#1640—611no—
286#1626—611no—
287#1584—611no—
288#1579—611no—
289C2×C61#327011yescharacteristic
290C2×C61#328711yescharacteristic
291C2×C61#328911yescharacteristic
292D61—161no—
293#559—611no—
294#557—611no—
295#557C2×D6111yescharacteristic
296#557C2×D6111yescharacteristic
297#500—611no—
298#500C2×D6111yescharacteristic
299#499—21no—
300#497—1221no—
301#494—611no—
302#494—1221no—
303#492—611no—
304#488—1221no—
305#486—611no—
306#486—611no—
307#479—21no—
308#479—21no—
309#478—1221no—
310#475—611no—
311#475C2×D6111yescharacteristic
312#475C2×D6111yescharacteristic
313#464—1221no—
314#464—1221no—
315#462—611no—
316#462—611no—
317#462—611no—
318#462—1221no—
319#462—1221no—
320#461—1221no—
321#460—611no—
322#460—611no—
323#446—611no—
324C8×Q8—611no—
325C8×Q8—611no—
326C8×Q8C2×D6111yescharacteristic
327#440—1221no—
328C4×Q16—21no—
329C4×Q16—611no—
330C4×Q16—1221no—
331C4×Q16C2×D6111yescharacteristic
332C4×SD16—1221no—
333#335—611no—
334#334—611no—
335#332—611no—
336#332—611no—
337#332—611no—
338#331—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342Q8⋊C8—611no—
343#322—611no—
344C61#1987311yescharacteristic, complemented, Sylow
345C2×Q16—611no—
346C2×Q16—1221no—
347C2×Q16—1221no—
348C2×Q16—1221no—
349C2×Q16—1221no—
350C2×Q16—1221no—
351C2×Q16C22⋊D6111yescharacteristic
352C2×SD16—1221no—
353C2×SD16—1221no—
354C2×SD16—1221no—
355C2×SD16—1221no—
356C4⋊Q8—21no—
357C4⋊Q8—21no—
358C4⋊Q8—611no—
359C4⋊Q8—1221no—
360C4⋊Q8C22×D6111yescharacteristic
361C4⋊Q8C22×D6111yescharacteristic
362#127—611no—
363#125—21no—
364#125—21no—
365#125—41no—
366#125—41no—
367#125—1221no—
368#125—2441no—
369#124—1221no—
370#124—1221no—
371#123—2441no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—611no—
377C4×Q8—1221no—
378C4×Q8—1221no—
379C4×Q8C22×D6111yescharacteristic
380C4×Q8C22×D6111yescharacteristic
381C4×Q8C22×D6111yescharacteristic
382C4×D4—1221no—
383C4×D4—1221no—
384#107—21no—
385#107—21no—
386#107—611no—
387#107—2441no—
388#107C22⋊D6111yescharacteristic
389#106—21no—
390#106—21no—
391#106—1221no—
392C4⋊C8—21no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399C4⋊C8C22×D6111yescharacteristic
400#103—21no—
401#103—21no—
402#103—611no—
403#103—611no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103C22⋊D6111yescharacteristic
409#103C22⋊D6111yescharacteristic
410#102—1221no—
411#102—1221no—
412#102—1221no—
413#102—1221no—
414#102—2441no—
415#97—1221no—
416C4×C8—21no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—1221no—
420C4×C8—1221no—
421C4×C8C22×D6111yescharacteristic
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—611no—
425C2×Q8—1221no—
426C2×Q8D4×D6111yescharacteristic
427C2×Q8C2×C22⋊D6111yescharacteristic
428C2×Q8C2×C22⋊D6111yescharacteristic
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C22×C4—2441no—
433Q16—22no—
434Q16—1222no—
435Q16—4881no—
436Q16—4881no—
437Q16—4881no—
438Q16—4881no—
439Q16—2442no—
440SD16—4881no—
441SD16—4881no—
442SD16—2442no—
443SD16—2442no—
444C2×C8—21no—
445C2×C8—611no—
446C2×C8—611no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8C2×C22⋊D6111yescharacteristic
452C2×C8C2×C22⋊D6111yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—611no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4D4×D6111yescharacteristic
474C4⋊C4C2×C22⋊D6111yescharacteristic
475C4⋊C4C2×C22⋊D6111yescharacteristic
476C22⋊C4—2441no—
477C22⋊C4—2441no—
478C22⋊C4—2441no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—41no—
483C42—41no—
484C42—1221no—
485C42—2441no—
486C42C23×D6111yescharacteristic
487C23—2441no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—41no—
494Q8—41no—
495Q8—41no—
496Q8—41no—
497Q8—1221no—
498Q8—1221no—
499Q8—2441no—
500Q8—2441no—
501Q8#1451980812yes—
502D4—4881no—
503D4—2441no—
504D4—2441no—
505D4—2441no—
506D4—2441no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—9761no—
518C2×C4—2441no—
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4C22×C22⋊D6111yescharacteristic
521C2×C4#1451982211yescharacteristic
522C8—21no—
523C8—21no—
524C8—41no—
525C8—612no—
526C8—1221no—
527C8—2441no—
528C8—2441no—
529C8—2441no—
530C8—2441no—
531C22—4881no—
532C22—4881no—
533C22—4881no—
534C22D4×C22⋊D6111yescharacteristic
535C4—21no—
536C4—22no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—81no—
543C4—81no—
544C4—4881no—
545C4—2441no—
546C4—2441no—
547C4#135×D6111yescharacteristic
548C4C2×#1451980811yescharacteristic
549C4#1684056411yescharacteristic
550C4#1684057711yescharacteristic
551C2—9761no—
552C2#2145739011yescharacteristic
553C2#2145757311yescharacteristic
554C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3383C2
socleC22×C61D42
prime radical#14519774C23
hypercenter#3383D61
hypocenterC61#19873
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3383

Complements

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

NH ≅ G/Ntype
2. #21458822C2semidirect
5. #21445311C2semidirect
7. #21445210C2semidirect
9. #21445061C2semidirect
10. #21445060C2semidirect
13. #21444670C2semidirect
14. #21444636C2semidirect
15. C61×#3383C2semidirect
276. #3383D61semidirect
344. C61#19873semidirect

Maximal subgroups

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

subgroupindex
#214588682
#214588222
#214585132
#214454892
#214453112
#214452632
#214452102
#214451692
#214450612
#214450602
#214450042
#214449152
#214446702
#214446362
C61×#33832
#1987361