Finite Group Databasebeta

← #29671458

Subgroups — #29671458

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458798C211yesmaximal, characteristic
2#21458747C211yesmaximal, characteristic, complemented
3#21457481C211yesmaximal, characteristic
4#21445407C211yesmaximal, characteristic, complemented
5#21445320C211yesmaximal, characteristic, complemented
6#21445293C211yesmaximal, characteristic, complemented
7#21445198C211yesmaximal, characteristic, complemented
8#21445167C211yesmaximal, characteristic, complemented
9#21445095C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21444945C211yesmaximal, characteristic
12#21444897C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14C8×#11758202C211yesmaximal, characteristic, complemented
15Q8×SD16×C61C211yesmaximal, characteristic, complemented
16C2×#14519808—21no—
17#16840490C2211yescharacteristic
18#16840433C2211yescharacteristic
19#16840168—21no—
20#16840158C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840136C2211yescharacteristic
24#16840135C2211yescharacteristic
25#16840133—21no—
26#16840069—21no—
27#16840054C2211yescharacteristic
28#16840047C2211yescharacteristic
29#16840044—21no—
30#16840010—21no—
31#16839969C2211yescharacteristic
32#16839967C2211yescharacteristic
33#16839966—21no—
34#16839962C2211yescharacteristic
35#16839962C2211yescharacteristic
36#16839961—21no—
37#16839922—21no—
38#16839901—21no—
39#16839893C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839867—21no—
42C4×#11758202C2211yescharacteristic
43#16839849—21no—
44#16839840C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#498—21no—
59C61×#498C2211yescharacteristic
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#475—21no—
63C61×#475C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758222—21no—
68#14519808—44no—
69C2×#11758202—21no—
70C2×#11758202D411yescharacteristic
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78C2×#11758201—41no—
79C2×#11758194—21no—
80#14519777C2311yescharacteristic
81#14519775—21no—
82#14519774C2311yescharacteristic
83#14519773—21no—
84#14519769—21no—
85#14519767—41no—
86#14519751—21no—
87#14519750—41no—
88C4×Dic122—21no—
89C4×Dic122C2311yescharacteristic
90#14519742—21no—
91#14519739—21no—
92#14519710—21no—
93#14519710D411yescharacteristic
94#14519708—21no—
95#14519708—21no—
96#14519708—21no—
97#14519708—41no—
98C4⋊#92418—21no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519706—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519700—21no—
108#14519700D411yescharacteristic
109#14519697—21no—
110#14519697—21no—
111C22⋊#92418—21no—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C8×Dic61—21no—
116C22×Q8×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#106×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C4×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—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158#11758222—42no—
159#11758222—81no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×C4×D61—41no—
164C2×Dic122—21no—
165C2×Dic122C2×D411yescharacteristic
166SD16×C61—22no—
167C2×C8×C61—21no—
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C22⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—41no—
187C42×C61C2411yescharacteristic
188#11758202—42no—
189#11758202SD1614yescomplemented
190#11758201—42no—
191#11758201—42no—
192#11758201—81no—
193#11758201—82no—
194#11758201—81no—
195#11758200—81no—
196#11758198—41no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—21no—
201#11758196—41no—
202#11758196—41no—
203#11758196—41no—
204C4×Dic61—21no—
205C4×Dic61—41no—
206#11758194—42no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213#11758190—42no—
214#11758190—81no—
215C23×C61—21no—
216C22×D61—41no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—21no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—82no—
223Q8×C61—81no—
224Q8×C61—161no—
225Q8×C61C2×SD1612yes—
226D4×C61—21no—
227D4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—82no—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C22⋊D61—82no—
242C22⋊D61—161no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246D244—41no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—81no—
251C4×D61—82no—
252Dic122—21no—
253Dic122—41no—
254Dic122—41no—
255Dic122C2×SD1612yes—
256C8×C61—21no—
257C8×C61—41no—
258C8×C61#13712yescomplemented
259#92418—22no—
260#92418—21no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418C2×SD1612yes—
265#19897—611nomaximal, Sylow
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61D4211yescharacteristic
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61C22×SD1611yescharacteristic
282C4×C61C2×#13711yescharacteristic
283C4×C61#57711yescharacteristic
284C4×C61#57711yescharacteristic
285Dic61—22no—
286Dic61—41no—
287Dic61—81no—
288#3392—611no—
289#3387—611no—
290Q8×SD16D6111yescharacteristic, complemented
291#3278—611no—
292#1705—611no—
293#1691—611no—
294#1686—611no—
295#1672—611no—
296#1666—611no—
297#1653—611no—
298#1651—611no—
299#1631—611no—
300#1623—611no—
301#1590—611no—
302C8×Q16—611no—
303C2×C61—81no—
304C2×C61#327811yescharacteristic
305C2×C61D4×SD1611yescharacteristic
306C2×C61#329111yescharacteristic
307D61—161no—
308C2×#137—1221no—
309Q82C2×D6111yescharacteristic
310#557—611no—
311#550—611no—
312D4×Q8C2×D6111yescharacteristic
313#501—1221no—
314#498—21no—
315#498—611no—
316#498—611no—
317#498C2×D6111yescharacteristic
318#492—611no—
319#492—611no—
320#492—611no—
321#492—1221no—
322#479—1221no—
323#477—611no—
324#476—21no—
325#476C2×D6111yescharacteristic
326#475—21no—
327#475—611no—
328#475—1221no—
329#475C2×D6111yescharacteristic
330#468—1221no—
331#462—611no—
332#461—611no—
333#461—1221no—
334#460—611no—
335#460—611no—
336#460—1221no—
337#452—1221no—
338#448—1221no—
339#446—611no—
340C8×Q8—611no—
341C8×Q8C2×D6111yescharacteristic
342#441—1221no—
343C4×Q16—611no—
344C4×SD16—21no—
345C4×SD16—1221no—
346C4×SD16C2×D6111yescharacteristic
347#436—611no—
348#335—611no—
349#334—611no—
350#333—611no—
351#332—611no—
352#332—611no—
353#331—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357D4⋊C8—611no—
358C82—611no—
359C61#1989711yescharacteristic, complemented, Sylow
360C2×#41—1221no—
361C22×Q8—21no—
362#137—2444no—
363C2×Q16—611no—
364C2×Q16—1221no—
365C2×SD16—1221no—
366C2×SD16—1221no—
367C2×SD16—1221no—
368C2×SD16—1221no—
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×SD16—2441no—
373C2×SD16C22⋊D6111yescharacteristic
374C2×M16—1221no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8—41no—
379C4⋊Q8—611no—
380C4⋊Q8—1221no—
381C4⋊Q8C22×D6111yescharacteristic
382C4⋊Q8C22×D6111yescharacteristic
383#127—611no—
384#127—1221no—
385#125—1221no—
386#125—2441no—
387C22⋊Q8—21no—
388C22⋊Q8—41no—
389#121—1221no—
390#121—2441no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—611no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×Q8C22×D6111yescharacteristic
397C4×Q8C22×D6111yescharacteristic
398C4×D4—21no—
399C4×D4—1221no—
400C4×D4C22×D6111yescharacteristic
401#107—611no—
402#107—1221no—
403#106—21no—
404#106—21no—
405#106—21no—
406#106—21no—
407#106—1221no—
408#106—1221no—
409#106—1221no—
410#106—2441no—
411#106C22⋊D6111yescharacteristic
412C4⋊C8—21no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8C22×D6111yescharacteristic
419#103—21no—
420#103—611no—
421#103—611no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103C22⋊D6111yescharacteristic
427#102—21no—
428#102—1221no—
429#102—1221no—
430#102C22⋊D6111yescharacteristic
431C22⋊C8—1221no—
432#97—1221no—
433C4×C8—21no—
434C4×C8—611no—
435C4×C8—611no—
436C4×C8—1221no—
437C4×C8C22×D6111yescharacteristic
438#41—4881no—
439#41—2442no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—41no—
443C2×Q8—41no—
444C2×Q8—42no—
445C2×Q8—81no—
446C2×Q8—611no—
447C2×Q8—1221no—
448C2×Q8D4×D6111yescharacteristic
449C2×Q8C2×C22⋊D6111yescharacteristic
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4C2×C22⋊D6111yescharacteristic
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—2441no—
459Q16—614no—
460Q16—2442no—
461SD16—22no—
462SD16—4881no—
463SD16—4882no—
464SD16—4881no—
465SD16—4881no—
466SD16—4881no—
467SD16—2442no—
468SD16—2442no—
469SD16—2442no—
470M16—2442no—
471C2×C8—21no—
472C2×C8—611no—
473C2×C8—611no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8C2×C22⋊D6111yescharacteristic
479C2×C8C2×C22⋊D6111yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—611no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—2441no—
494C4⋊C4—2441no—
495C4⋊C4—2441no—
496C4⋊C4D4×D6111yescharacteristic
497C4⋊C4C2×C22⋊D6111yescharacteristic
498C4⋊C4C2×C22⋊D6111yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—41no—
501C22⋊C4—2441no—
502C42—21no—
503C42—21no—
504C42—41no—
505C42—1221no—
506C42—2441no—
507C42C23×D6111yescharacteristic
508C23—21no—
509C23—2441no—
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—41no—
514Q8—41no—
515Q8—82no—
516Q8—81no—
517Q8—161no—
518Q8—612no—
519Q8—1221no—
520Q8—2441no—
521Q8—2441no—
522Q8SD16×D6112yes—
523D4—21no—
524D4—21no—
525D4—9761no—
526D4—4881no—
527D4—4882no—
528D4—2441no—
529D4—2441no—
530D4—2441no—
531D4—2441no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—4882no—
545C2×C4—2441no—
546C2×C4C2×D4×D6111yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4#1451982211yescharacteristic
549C8—21no—
550C8—41no—
551C8—612no—
552C8—1222no—
553C8—1221no—
554C8—2441no—
555C8—2441no—
556C8—2441no—
557C8#1451980812yescomplemented
558C22—41no—
559C22—41no—
560C22—41no—
561C22—4881no—
562C22—4881no—
563C22—4881no—
564C22D4×C22⋊D6111yescharacteristic
565C4—22no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—1222no—
574C4—4881no—
575C4—2441no—
576C4C2×SD16×D6111yescharacteristic
577C4C2×#1451980811yescharacteristic
578C4#1684056811yescharacteristic
579C4#1684057011yescharacteristic
580C2—81no—
581C2—9761no—
582C2#2145748111yescharacteristic
583C2SD16×C22⋊D6111yescharacteristic
584C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×SD16×C61C2
socleC22×C61D42
prime radical#16840433C22
hypercenterQ8×SD16D61
hypocenterC61#19897
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
2. #21458747C2semidirect
4. #21445407C2semidirect
5. #21445320C2semidirect
6. #21445293C2semidirect
7. #21445198C2semidirect
8. #21445167C2semidirect
9. #21445095C2semidirect
10. #21445082C2semidirect
12. #21444897C2semidirect
13. #21444710C2semidirect
14. C8×#11758202C2semidirect
15. Q8×SD16×C61C2semidirect
189. #11758202SD16semidirect
258. C8×C61#137semidirect
290. Q8×SD16D61semidirect
359. C61#19897semidirect
557. C8#14519808semidirect

Maximal subgroups

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

subgroupindex
#214587982
#214587472
#214574812
#214454072
#214453202
#214452932
#214451982
#214451672
#214450952
#214450822
#214449452
#214448972
#214447102
C8×#117582022
Q8×SD16×C612
#1989761