Finite Group Databasebeta

← #29670766

Subgroups — #29670766

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458622C211yesmaximal, characteristic, complemented
2#21458515C211yesmaximal, characteristic
3#21458513C211yesmaximal, characteristic
4#21445482C211yesmaximal, characteristic
5#21445389C211yesmaximal, characteristic
6#21445263C211yesmaximal, characteristic
7#21445251C211yesmaximal, characteristic, complemented
8#21445186C211yesmaximal, characteristic, complemented
9#21445096C211yesmaximal, characteristic, complemented
10#21445094C211yesmaximal, characteristic, complemented
11#21445003C211yesmaximal, characteristic
12#21444915C211yesmaximal, characteristic
13#21444670C211yesmaximal, characteristic, complemented
14#21444636C211yesmaximal, characteristic, complemented
15C61×#3395C211yesmaximal, characteristic, complemented
16#16840497C2211yescharacteristic
17#16840491C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840132—21no—
20#16840119—21no—
21#16840118—21no—
22#16840114C2211yescharacteristic
23#16840110C2211yescharacteristic
24#16840110C2211yescharacteristic
25#16840109C2211yescharacteristic
26#16840066—21no—
27#16840065—21no—
28#16840059—21no—
29#16840047C2211yescharacteristic
30#16840044C2211yescharacteristic
31#16840044C2211yescharacteristic
32#16839986—21no—
33#16839984—21no—
34#16839972—21no—
35#16839972C2211yescharacteristic
36#16839962C2211yescharacteristic
37#16839962C2211yescharacteristic
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839867—21no—
42C4×#11758202C2211yescharacteristic
43C4×#11758201—21no—
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×#494—21no—
59C61×#494C2211yescharacteristic
60C61×#462—21no—
61C61×#462—21no—
62C61×#462C2211yescharacteristic
63C61×#462C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×Q16×C61—21no—
66C4×Q16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202D411yescharacteristic
69C2×#11758201—21no—
70C2×#11758201—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519774C2311yescharacteristic
74#14519768—21no—
75#14519768—41no—
76#14519766—21no—
77#14519765—21no—
78#14519762—41no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122C2311yescharacteristic
82C4×D244—21no—
83#14519739—21no—
84#14519710D411yescharacteristic
85#14519708—21no—
86#14519708—21no—
87#14519708—21no—
88#14519708—41no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519706—21no—
95#14519701—21no—
96#14519701D411yescharacteristic
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700D411yescharacteristic
103#14519699—21no—
104#14519699—21no—
105#14519697—21no—
106#14519697—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×C61—21no—
116C2×Q16×C61—21no—
117C2×Q16×C61—21no—
118C2×Q16×C61—21no—
119C2×Q16×C61D411yescharacteristic
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—21no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#125×C61—21no—
125#125×C61—21no—
126#125×C61—41no—
127#125×C61—41no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61—21no—
131C4×Q8×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×Q8×C61C2311yescharacteristic
135#107×C61D411yescharacteristic
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—41no—
156Q16×C61—42no—
157Q16×C61—81no—
158Q16×C61—81no—
159Q16×C61—81no—
160C2×C8×C61—21no—
161C2×C8×C61C2×D411yescharacteristic
162C2×C8×C61C2×D411yescharacteristic
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—41no—
183C42×C61—41no—
184C42×C61C2411yescharacteristic
185#11758202—22no—
186#11758202—42no—
187#11758201—42no—
188#11758201—42no—
189#11758198—41no—
190#11758198—41no—
191#11758198—41no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—41no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—21no—
198#11758196—41no—
199C4×Dic61—21no—
200C4×Dic61—41no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C22×D61—41no—
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61#13512yes—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—41no—
232D244—41no—
233D244—41no—
234D244—41no—
235D244—41no—
236D244—81no—
237C4×D61—161no—
238Dic122—21no—
239Dic122—21no—
240Dic122—41no—
241Dic122—41no—
242C8×C61—21no—
243C8×C61—21no—
244C8×C61—41no—
245#92418—21no—
246#92418—21no—
247#92418—41no—
248#92418—41no—
249#92418—41no—
250#92418—41no—
251#19849—611nomaximal, Sylow
252C22×C61D4211yescharacteristic
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C4×C61—22no—
257C4×C61—21no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—81no—
264C4×C61—81no—
265C4×C61C2×#13511yescharacteristic
266C4×C61#57511yescharacteristic
267C4×C61#57711yescharacteristic
268C4×C61#57811yescharacteristic
269Dic61—41no—
270Dic61—41no—
271Dic61—81no—
272#3395D6111yescharacteristic, complemented
273#3374—611no—
274#3365—611no—
275#3365—611no—
276#1720—611no—
277#1702—611no—
278#1682—611no—
279#1680—611no—
280#1670—611no—
281#1653—611no—
282#1653—611no—
283#1640—611no—
284#1626—611no—
285#1584—611no—
286#1579—611no—
287C2×C61#327111yescharacteristic
288C2×C61#328511yescharacteristic
289C2×C61#328911yescharacteristic
290D61—161no—
291#559—611no—
292#557—611no—
293#557C2×D6111yescharacteristic
294#557C2×D6111yescharacteristic
295#498—611no—
296#494—21no—
297#494C2×D6111yescharacteristic
298#491—1221no—
299#488—1221no—
300#488—1221no—
301#487—611no—
302#486—611no—
303#486—611no—
304#486—611no—
305#479—1221no—
306#478—1221no—
307#478—1221no—
308#475—611no—
309#475—611no—
310#475—611no—
311#464—1221no—
312#464—1221no—
313#462—21no—
314#462—21no—
315#462—611no—
316#462—1221no—
317#462C2×D6111yescharacteristic
318#462C2×D6111yescharacteristic
319#460—611no—
320#460—611no—
321#446—611no—
322C8×Q8—611no—
323C8×Q8—611no—
324C8×Q8C2×D6111yescharacteristic
325#441—1221no—
326C4×Q16—21no—
327C4×Q16—611no—
328C4×Q16C2×D6111yescharacteristic
329C4×SD16—1221no—
330C4×SD16—1221no—
331#335—611no—
332#334—611no—
333#332—611no—
334#332—611no—
335#332—611no—
336#331—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341#322—611no—
342C61#1984911yescharacteristic, complemented, Sylow
343C2×Q16—21no—
344C2×Q16—21no—
345C2×Q16—21no—
346C2×Q16—21no—
347C2×Q16—611no—
348C2×Q16—1221no—
349C2×Q16C22⋊D6111yescharacteristic
350C2×SD16—1221no—
351C2×SD16—1221no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—611no—
355C4⋊Q8—1221no—
356C4⋊Q8C22×D6111yescharacteristic
357C4⋊Q8C22×D6111yescharacteristic
358#127—611no—
359#125—21no—
360#125—21no—
361#125—41no—
362#125—41no—
363#125—1221no—
364#125—2441no—
365#124—1221no—
366#124—1221no—
367#123—2441no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8—611no—
373C4×Q8—1221no—
374C4×Q8—1221no—
375C4×Q8C22×D6111yescharacteristic
376C4×Q8C22×D6111yescharacteristic
377C4×Q8C22×D6111yescharacteristic
378C4×D4—1221no—
379C4×D4—1221no—
380#107—611no—
381#107C22⋊D6111yescharacteristic
382#106—1221no—
383#106—1221no—
384#106—1221no—
385#106—2441no—
386C4⋊C8—21no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8C22×D6111yescharacteristic
394#103—21no—
395#103—21no—
396#103—611no—
397#103—611no—
398#103—1221no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103C22⋊D6111yescharacteristic
405#103C22⋊D6111yescharacteristic
406#102—1221no—
407#102—1221no—
408#102—1221no—
409#102—1221no—
410#102—1221no—
411#102—1221no—
412#102—2441no—
413#97—1221no—
414C4×C8—21no—
415C4×C8—611no—
416C4×C8—611no—
417C4×C8—1221no—
418C4×C8—1221no—
419C4×C8C22×D6111yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—611no—
423C2×Q8—1221no—
424C2×Q8D4×D6111yescharacteristic
425C2×Q8C2×C22⋊D6111yescharacteristic
426C2×Q8C2×C22⋊D6111yescharacteristic
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—1221no—
430C22×C4—2441no—
431Q16—41no—
432Q16—42no—
433Q16—81no—
434Q16—81no—
435Q16—81no—
436Q16—1222no—
437Q16—2442no—
438SD16—2442no—
439SD16—2442no—
440C2×C8—21no—
441C2×C8—611no—
442C2×C8—611no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8C2×C22⋊D6111yescharacteristic
448C2×C8C2×C22⋊D6111yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—611no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C4⋊C4D4×D6111yescharacteristic
470C4⋊C4C2×C22⋊D6111yescharacteristic
471C4⋊C4C2×C22⋊D6111yescharacteristic
472C22⋊C4—2441no—
473C22⋊C4—2441no—
474C22⋊C4—2441no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—41no—
479C42—41no—
480C42—1221no—
481C42—2441no—
482C42C23×D6111yescharacteristic
483C23—2441no—
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—1221no—
494Q8—1221no—
495Q8—2441no—
496Q8—2441no—
497Q8#1451980012yes—
498D4—4881no—
499D4—2441no—
500D4—2441no—
501D4—2441no—
502D4—2441no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—9761no—
514C2×C4—2441no—
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4C22×C22⋊D6111yescharacteristic
517C2×C4#1451982211yescharacteristic
518C8—21no—
519C8—21no—
520C8—41no—
521C8—1221no—
522C8—1221no—
523C8—2441no—
524C8—2441no—
525C8—2441no—
526C8—2441no—
527C22—4881no—
528C22—4881no—
529C22—4881no—
530C22D4×C22⋊D6111yescharacteristic
531C4—22no—
532C4—21no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—81no—
539C4—81no—
540C4—4881no—
541C4—2441no—
542C4—2441no—
543C4C2×#1451980011yescharacteristic
544C4#1684056011yescharacteristic
545C4#1684057311yescharacteristic
546C4#1684057511yescharacteristic
547C2—9761no—
548C2#2145740011yescharacteristic
549C2#2145756011yescharacteristic
550C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3395C2
socleC22×C61D42
prime radical#14519774C23
hypercenter#3395D61
hypocenterC61#19849
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3395

Complements

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

NH ≅ G/Ntype
1. #21458622C2semidirect
7. #21445251C2semidirect
8. #21445186C2semidirect
9. #21445096C2semidirect
10. #21445094C2semidirect
13. #21444670C2semidirect
14. #21444636C2semidirect
15. C61×#3395C2semidirect
272. #3395D61semidirect
342. C61#19849semidirect

Maximal subgroups

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

subgroupindex
#214586222
#214585152
#214585132
#214454822
#214453892
#214452632
#214452512
#214451862
#214450962
#214450942
#214450032
#214449152
#214446702
#214446362
C61×#33952
#1984961