Finite Group Databasebeta

← #29669795

Subgroups — #29669795

Order 15616.

622 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458450C211yesmaximal, characteristic, complemented
2#21458237C211yesmaximal, characteristic, complemented
3#21458115C211yesmaximal, characteristic, complemented
4#21445469C211yesmaximal, characteristic, complemented
5#21445349C211yesmaximal, characteristic, complemented
6#21445270C211yesmaximal, characteristic, complemented
7#21445254C211yesmaximal, characteristic, complemented
8#21445221C211yesmaximal, characteristic, complemented
9#21445047C211yesmaximal, characteristic, complemented
10#21444982C211yesmaximal, characteristic, complemented
11#21444946C211yesmaximal, characteristic, complemented
12#21444901C211yesmaximal, characteristic, complemented
13#21444648C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3305C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17Q8×D244C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840134C2211yescharacteristic
20#16840113C2211yescharacteristic
21#16840112C2211yescharacteristic
22#16840108—21no—
23#16840108C2211yescharacteristic
24#16840106—21no—
25#16840086—21no—
26#16840083—21no—
27#16840057C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840049C2211yescharacteristic
30#16840042—21no—
31#16840042C2211yescharacteristic
32#16840007—21no—
33#16839996—21no—
34#16839967C2211yescharacteristic
35#16839954—21no—
36#16839954C2211yescharacteristic
37#16839899—21no—
38Q8×#92418C2211yescharacteristic
39C4×#11758199—21no—
40C4×#11758199C2211yescharacteristic
41#16839841C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839798—21no—
44#16839769—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#555C2211yescharacteristic
57C61×#550C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#482—21no—
60C61×#477C2211yescharacteristic
61C61×#466—21no—
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C61×#416—21no—
67C2×Q8×D61—21no—
68C2×#11758200—21no—
69C2×#11758200—41no—
70C2×#11758199—21no—
71C2×#11758199D411yescharacteristic
72C2×#11758189—21no—
73C22×#92418—21no—
74#14519777—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519763—41no—
78#14519757—21no—
79#14519757—41no—
80#14519751—21no—
81#14519748—21no—
82C4×D244—21no—
83C4×D244C2311yescharacteristic
84#14519736—21no—
85#14519735—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—21no—
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519702—21no—
98#14519702—21no—
99#14519701—21no—
100#14519701—41no—
101#14519700—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C2×SD16×C61D411yescharacteristic
117C2×D8×C61—21no—
118C22×C8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120#127×C61—21no—
121#127×C61C2311yescharacteristic
122#124×C61—41no—
123#121×C61—21no—
124#121×C61—21no—
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#117×C61—21no—
133#107×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144#41×C61—42no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C2×C22⋊D61—41no—
160Q8×D61—42no—
161Q8×D61—81no—
162C2×D244C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168SD16×C61—22no—
169D8×C61—81no—
170C2×C8×C61—21no—
171C2×C8×C61—81no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61—41no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189#11758200—81no—
190#11758200—82no—
191#11758199—41no—
192#11758199—42no—
193#11758198—21no—
194#11758198—41no—
195#11758198—41no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61C2×D411yescharacteristic
202C4×Dic61—41no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—42no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210#11758189—81no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—41no—
214C22×D61—21no—
215Q8×C61—21no—
216Q8×C61—81no—
217Q8×C61#13512yes—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—82no—
227D4×C61—81no—
228D4×C61—161no—
229D4×C61—161no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—82no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C22⋊D61—161no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247D244—21no—
248D244—21no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—82no—
252Dic122—41no—
253Dic122—41no—
254Dic122—82no—
255Dic122—81no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—41no—
259#92418—21no—
260#92418—21no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#19784—611nomaximal, Sylow
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61D4211yescharacteristic
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61C2×#13511yescharacteristic
286C4×C61#57511yescharacteristic
287C4×C61#57611yescharacteristic
288C4×C61#57811yescharacteristic
289Dic61—41no—
290Dic61—81no—
291Dic61—81no—
292#3358—611no—
293#3341—611no—
294#3329—611no—
295#3305D6111yescharacteristic, complemented
296#1719—611no—
297#1697—611no—
298#1683—611no—
299#1681—611no—
300#1676—611no—
301#1646—611no—
302#1637—611no—
303#1631—611no—
304#1624—611no—
305#1581—611no—
306SD16⋊C8—611no—
307C2×C61—81no—
308C2×C61—81no—
309C2×C61—161no—
310C2×C61#326911yescharacteristic
311C2×C61#328011yescharacteristic
312C2×C61#328811yescharacteristic
313D61—81no—
314#555—611no—
315#555C2×D6111yescharacteristic
316#550C2×D6111yescharacteristic
317D4×Q8—611no—
318#500—611no—
319#492—611no—
320#487—611no—
321#487—611no—
322#486—611no—
323#486—1221no—
324#486C2×D6111yescharacteristic
325#485—1221no—
326#482—21no—
327#482—1221no—
328#482—1221no—
329#477—611no—
330#477C2×D6111yescharacteristic
331#476—611no—
332#476—611no—
333#474—611no—
334#474—1221no—
335#468—1221no—
336#466—21no—
337#466—1221no—
338#461—611no—
339#459—611no—
340#459—1221no—
341#459C2×D6111yescharacteristic
342#449—21no—
343#448—1221no—
344C8×Q8—611no—
345C4×SD16C2×D6111yescharacteristic
346C4×D8—611no—
347C4×D8—1221no—
348#436—611no—
349C8×D4—611no—
350C8×D4C2×D6111yescharacteristic
351#427—1221no—
352#419—1221no—
353#416—21no—
354#335—611no—
355#335—611no—
356#333—611no—
357#333—611no—
358#332—611no—
359#331—611no—
360Q8⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364#322—611no—
365C61#1978411yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C22×Q8—1221no—
368C2×SD16—1221no—
369C2×SD16—2441no—
370C2×SD16C22⋊D6111yescharacteristic
371C2×D8—21no—
372C2×D8—611no—
373C2×D8—1221no—
374C2×M16—1221no—
375C22×C8—21no—
376C22×C8—1221no—
377C4⋊Q8—611no—
378C4⋊Q8—611no—
379C4⋊Q8—1221no—
380C4⋊Q8C22×D6111yescharacteristic
381#127—21no—
382#127C22×D6111yescharacteristic
383#124—41no—
384#124—2441no—
385C22⋊Q8—1221no—
386C22⋊Q8—2441no—
387#121—21no—
388#121—21no—
389#121—21no—
390#121—41no—
391#121—1221no—
392#121—1221no—
393C4×Q8C22×D6111yescharacteristic
394C4×D4—21no—
395C4×D4—611no—
396C4×D4—1221no—
397C4×D4C22×D6111yescharacteristic
398C4×D4C22×D6111yescharacteristic
399#117—21no—
400#117—21no—
401#117—1221no—
402#117—1221no—
403#107—21no—
404#107—611no—
405#107—1221no—
406#107—1221no—
407#107—1221no—
408#107—1221no—
409#106—1221no—
410#106—1221no—
411#106C22⋊D6111yescharacteristic
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—1221no—
416C4⋊C8C22×D6111yescharacteristic
417#103—21no—
418#103—1221no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—2441no—
423#103C22⋊D6111yescharacteristic
424#102—21no—
425#102—21no—
426#102—21no—
427#102—611no—
428#102—611no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102C22⋊D6111yescharacteristic
436C22⋊C8—21no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—1221no—
442C4×C8C22×D6111yescharacteristic
443#41—42no—
444#41—81no—
445C2×Q8—21no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—4881no—
449C2×Q8—2441no—
450C2×Q8—2442no—
451C2×Q8C2×C22⋊D6111yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—611no—
459C2×D4—2441no—
460C2×D4D4×D6111yescharacteristic
461C2×D4C2×C22⋊D6111yescharacteristic
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—41no—
465C22×C4—1221no—
466C22×C4—2441no—
467SD16—22no—
468SD16—4881no—
469SD16—4882no—
470D8—81no—
471D8—2441no—
472D8—2442no—
473M16—4881no—
474C2×C8—21no—
475C2×C8—81no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—2442no—
483C2×C8C2×C22⋊D6111yescharacteristic
484C2×C8C2×C22⋊D6111yescharacteristic
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—611no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—2441no—
494C4⋊C4D4×D6111yescharacteristic
495C4⋊C4C2×C22⋊D6111yescharacteristic
496C4⋊C4C2×C22⋊D6111yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—1221no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C22⋊C4—2441no—
506C42—21no—
507C42—41no—
508C42—2441no—
509C42C23×D6111yescharacteristic
510C23—21no—
511C23—21no—
512C23—41no—
513C23—1221no—
514Q8—21no—
515Q8—81no—
516Q8—4881no—
517Q8—4882no—
518Q8—2441no—
519Q8—2441no—
520Q8#1451979712yes—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—82no—
530D4—81no—
531D4—161no—
532D4—161no—
533D4—1221no—
534D4—1221no—
535D4—9761no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—82no—
546C2×C4—1221no—
547C2×C4—4882no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C22×C22⋊D6111yescharacteristic
554C2×C4#1451982211yescharacteristic
555C8—21no—
556C8—21no—
557C8—41no—
558C8—1221no—
559C8—1221no—
560C8—2441no—
561C8—2441no—
562C8—2441no—
563C8—2441no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—2441no—
574C22—2441no—
575C22—2441no—
576C22D4×C22⋊D6111yescharacteristic
577C4—22no—
578C4—21no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—4881no—
584C4—4881no—
585C4—2441no—
586C4C2×#1451979711yescharacteristic
587C4#1684055911yescharacteristic
588C4#1684056511yescharacteristic
589C4#1684057611yescharacteristic
590C2—81no—
591C2—81no—
592C2—161no—
593C2—4881no—
594C2#2145737411yescharacteristic
595C2#2145750011yescharacteristic
596C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3305C2
socleC22×C61D42
hypercenter#3305D61
hypocenterC61#19784
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3305

Complements

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

NH ≅ G/Ntype
1. #21458450C2semidirect
2. #21458237C2semidirect
3. #21458115C2semidirect
4. #21445469C2semidirect
5. #21445349C2semidirect
6. #21445270C2semidirect
7. #21445254C2semidirect
8. #21445221C2semidirect
9. #21445047C2semidirect
10. #21444982C2semidirect
11. #21444946C2semidirect
12. #21444901C2semidirect
13. #21444648C2semidirect
14. SD16⋊#92418C2semidirect
15. C61×#3305C2semidirect
295. #3305D61semidirect
365. C61#19784semidirect

Maximal subgroups

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

subgroupindex
#214584502
#214582372
#214581152
#214454692
#214453492
#214452702
#214452542
#214452212
#214450472
#214449822
#214449462
#214449012
#214446482
SD16⋊#924182
C61×#33052
#1978461