Finite Group Databasebeta

← #29670586

Subgroups — #29670586

Order 15616.

628 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#21458611C211yesmaximal, characteristic
2#21458513C211yesmaximal, characteristic, complemented
3#21458461C211yesmaximal, characteristic, complemented
4#21445482C211yesmaximal, characteristic, complemented
5#21445375C211yesmaximal, characteristic, complemented
6#21445265C211yesmaximal, characteristic, complemented
7#21445256C211yesmaximal, characteristic
8#21445186C211yesmaximal, characteristic, complemented
9#21445094C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21445003C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic
13#21444670C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15D4×Q16×C61C211yesmaximal, characteristic, complemented
16#16840497C2211yescharacteristic
17#16840491C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840124—21no—
20#16840118—21no—
21#16840114—21no—
22#16840114C2211yescharacteristic
23#16840110C2211yescharacteristic
24#16840110C2211yescharacteristic
25#16840109C2211yescharacteristic
26#16840065—21no—
27#16840049—21no—
28#16840047C2211yescharacteristic
29#16840044—21no—
30#16840044C2211yescharacteristic
31#16840044C2211yescharacteristic
32#16839986—21no—
33#16839977—21no—
34#16839967—21no—
35#16839967C2211yescharacteristic
36#16839962C2211yescharacteristic
37#16839957C2211yescharacteristic
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839872—21no—
42C4×#11758200—21no—
43C4×#11758200C2211yescharacteristic
44#16839847—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C22×Q16×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58D4×Q8×C61C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#467—21no—
61C61×#462C2211yescharacteristic
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C61×#451—21no—
65C4×Q16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758200D411yescharacteristic
69C2×#11758201—21no—
70C2×#11758199—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#14519708—21no—
85#14519708—21no—
86#14519708—21no—
87#14519708—41no—
88#14519708D411yescharacteristic
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519706—21no—
95#14519701—21no—
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700D411yescharacteristic
102#14519699—21no—
103#14519699—21no—
104#14519698D411yescharacteristic
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—
115C22×Q8×C61—21no—
116C22×Q8×C61—21no—
117C2×Q16×C61—21no—
118C2×Q16×C61—21no—
119C2×Q16×C61—21no—
120C2×Q16×C61—21no—
121C2×Q16×C61—44no—
122C2×Q16×C61D411yescharacteristic
123C22×C8×C61—21no—
124C4⋊Q8×C61—21no—
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127C4⋊Q8×C61C2311yescharacteristic
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131C22⋊Q8×C61—41no—
132C4×Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#103×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61—81no—
153C2×Q8×C61C2×D411yescharacteristic
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—41no—
158C22×C4×C61—41no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164C2×Dic122C2×D411yescharacteristic
165Q16×C61—22no—
166Q16×C61—42no—
167Q16×C61—42no—
168Q16×C61—42no—
169Q16×C61—81no—
170C2×C8×C61—21no—
171C2×C8×C61—42no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61C2411yescharacteristic
191#11758201—42no—
192#11758200—22no—
193#11758200—42no—
194#11758199—42no—
195#11758198—41no—
196#11758198—41no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205C4×Dic61—21no—
206C4×Dic61—41no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C23×C61—21no—
214C22×D61—41no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—82no—
223Q8×C61—82no—
224Q8×C61AGL(1,ℤ/8)12yes—
225D4×C61—21no—
226D4×C61#13512yes—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—82no—
237C2×C4×C61—82no—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—41no—
248D244—81no—
249C4×D61—161no—
250Dic122—21no—
251Dic122—21no—
252Dic122—41no—
253Dic122—41no—
254C8×C61—21no—
255C8×C61—21no—
256C8×C61—41no—
257#92418—21no—
258#92418—21no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#19837—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61D4211yescharacteristic
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C4×C61—22no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61C2×#13511yescharacteristic
280C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
281C4×C61#57711yescharacteristic
282C4×C61#57811yescharacteristic
283Dic61—41no—
284Dic61—41no—
285Dic61—81no—
286#3373—611no—
287#3365—611no—
288#3359—611no—
289D4×Q16D6111yescharacteristic, complemented
290#1720—611no—
291#1700—611no—
292#1682—611no—
293#1681—611no—
294#1670—611no—
295#1653—611no—
296#1651—611no—
297#1640—611no—
298#1627—611no—
299#1584—611no—
300SD16⋊C8—611no—
301C2×C61—42no—
302C2×C61#326511yescharacteristic
303C2×C61#328511yescharacteristic
304C2×C61#328911yescharacteristic
305D61—161no—
306C22×Q16—21no—
307#559—611no—
308#557—611no—
309D4×Q8C2×D6111yescharacteristic
310D4×Q8C2×D6111yescharacteristic
311#498—611no—
312#494C2×D6111yescharacteristic
313#489—1221no—
314#488—1221no—
315#487—611no—
316#487—1221no—
317#486—611no—
318#486—611no—
319#486—611no—
320#478—1221no—
321#476—1221no—
322#475—611no—
323#475—611no—
324#475—611no—
325#475—1221no—
326#467—21no—
327#464—1221no—
328#463—1221no—
329#462C2×D6111yescharacteristic
330#462C2×D6111yescharacteristic
331#461—611no—
332#461—1221no—
333#460—611no—
334#459—611no—
335#451—21no—
336#451—21no—
337#446—611no—
338C8×Q8—611no—
339C8×Q8—611no—
340#442—1221no—
341C4×Q16C2×D6111yescharacteristic
342C4×SD16—611no—
343C4×SD16—1221no—
344C4×SD16—1221no—
345C8×D4C2×D6111yescharacteristic
346#335—611no—
347#335—611no—
348#332—611no—
349#332—611no—
350#332—611no—
351#331—611no—
352Q8⋊C8—611no—
353Q8⋊C8—611no—
354Q8⋊C8—611no—
355D4⋊C8—611no—
356#322—611no—
357C61#1983711yescharacteristic, complemented, Sylow
358C22×Q8—21no—
359C22×Q8—21no—
360C2×Q16—21no—
361C2×Q16—21no—
362C2×Q16—21no—
363C2×Q16—21no—
364C2×Q16—44no—
365C2×Q16C22⋊D6111yescharacteristic
366C2×SD16—611no—
367C2×SD16—1221no—
368C2×SD16—1221no—
369C2×D8—1221no—
370C22×C8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—611no—
374C4⋊Q8—1221no—
375C4⋊Q8C22×D6111yescharacteristic
376C4⋊Q8C22×D6111yescharacteristic
377#127—611no—
378#125—1221no—
379#125—2441no—
380#124—1221no—
381#124—1221no—
382#123—2441no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—41no—
386C22⋊Q8—41no—
387C4×Q8—611no—
388C4×Q8—1221no—
389C4×Q8—1221no—
390C4×Q8C22×D6111yescharacteristic
391C4×Q8C22×D6111yescharacteristic
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—1221no—
395C4×D4—1221no—
396C4×D4C22×D6111yescharacteristic
397#107C22⋊D6111yescharacteristic
398#106—611no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402#106—2441no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—1221no—
407C4⋊C8—1221no—
408C4⋊C8—1221no—
409C4⋊C8C22×D6111yescharacteristic
410#103—21no—
411#103—21no—
412#103—611no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103C22⋊D6111yescharacteristic
420#103C22⋊D6111yescharacteristic
421#102—611no—
422#102—1221no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—2441no—
429C22⋊C8—21no—
430#97—1221no—
431C4×C8—611no—
432C4×C8—611no—
433C4×C8—1221no—
434C4×C8—1221no—
435C4×C8C22×D6111yescharacteristic
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—41no—
439C2×Q8—41no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—81no—
443C2×Q8—81no—
444C2×Q8—611no—
445C2×Q8—1221no—
446C2×Q8C2×C22⋊D6111yescharacteristic
447C2×Q8C2×C22⋊D6111yescharacteristic
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4D4×D6111yescharacteristic
452C22×C4—21no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—2441no—
456Q16—22no—
457Q16—42no—
458Q16—42no—
459Q16—42no—
460Q16—81no—
461SD16—1222no—
462SD16—2442no—
463SD16—2442no—
464D8—2442no—
465C2×C8—21no—
466C2×C8—42no—
467C2×C8—611no—
468C2×C8—611no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8C2×C22⋊D6111yescharacteristic
474C2×C8C2×C22⋊D6111yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—611no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—2441no—
488C4⋊C4—2441no—
489C4⋊C4—2441no—
490C4⋊C4D4×D6111yescharacteristic
491C4⋊C4C2×C22⋊D6111yescharacteristic
492C4⋊C4C2×C22⋊D6111yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C42—21no—
500C42—21no—
501C42—1221no—
502C42—2441no—
503C42C23×D6111yescharacteristic
504C23—21no—
505C23—2441no—
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—82no—
514Q8—82no—
515Q8—1221no—
516Q8—1221no—
517Q8—2441no—
518Q8—2441no—
519Q8#1451980312yes—
520D4—21no—
521D4—4881no—
522D4—2441no—
523D4—2441no—
524D4—2441no—
525D4—2441no—
526D4#1451980012yes—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—9761no—
541C2×C4—2441no—
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4#1451982211yescharacteristic
545C8—21no—
546C8—21no—
547C8—41no—
548C8—1221no—
549C8—1221no—
550C8—2441no—
551C8—2441no—
552C8—2441no—
553C8—2441no—
554C22—22no—
555C22—41no—
556C22—41no—
557C22—4881no—
558C22—4881no—
559C22—4881no—
560C22D4×C22⋊D6111yescharacteristic
561C4—22no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—4881no—
570C4—2441no—
571C4—2441no—
572C4C2×#1451980011yescharacteristic
573C4C2×#1451980311yescharacteristic
574C4#1684057311yescharacteristic
575C4#1684057511yescharacteristic
576C2—42no—
577C2—9761no—
578C2#2145731011yescharacteristic
579C2#2145756011yescharacteristic
580C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×Q16×C61C2
socleC22×C61D42
prime radical#16839957C22
hypercenterD4×Q16D61
hypocenterC61#19837
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
2. #21458513C2semidirect
3. #21458461C2semidirect
4. #21445482C2semidirect
5. #21445375C2semidirect
6. #21445265C2semidirect
8. #21445186C2semidirect
9. #21445094C2semidirect
10. #21445082C2semidirect
11. #21445003C2semidirect
13. #21444670C2semidirect
14. #21444631C2semidirect
15. D4×Q16×C61C2semidirect
289. D4×Q16D61semidirect
357. C61#19837semidirect

Maximal subgroups

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

subgroupindex
#214586112
#214585132
#214584612
#214454822
#214453752
#214452652
#214452562
#214451862
#214450942
#214450822
#214450032
#214449222
#214446702
#214446312
D4×Q16×C612
#1983761