Finite Group Databasebeta

← #29669556

Subgroups — #29669556

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458725C211yesmaximal, characteristic, complemented
2#21458178C211yesmaximal, characteristic, complemented
3#21457549C211yesmaximal, characteristic
4#21445440C211yesmaximal, characteristic, complemented
5#21445289C211yesmaximal, characteristic, complemented
6#21445225C211yesmaximal, characteristic, complemented
7#21445216C211yesmaximal, characteristic, complemented
8#21445201C211yesmaximal, characteristic, complemented
9#21445040C211yesmaximal, characteristic, complemented
10#21444969C211yesmaximal, characteristic, complemented
11#21444946C211yesmaximal, characteristic, complemented
12#21444890C211yesmaximal, characteristic, complemented
13#21444656C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15C61×#3362C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17#16840472C2211yescharacteristic
18#16840417C2211yescharacteristic
19#16840168—21no—
20#16840165C2211yescharacteristic
21#16840160C2211yescharacteristic
22#16840134C2211yescharacteristic
23#16840134C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840091—21no—
26#16840079—21no—
27#16840069—21no—
28#16840057C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840049C2211yescharacteristic
31#16840047C2211yescharacteristic
32#16840039—21no—
33#16839989—21no—
34#16839988—21no—
35#16839972C2211yescharacteristic
36#16839954C2211yescharacteristic
37#16839924—21no—
38#16839905—21no—
39#16839893C2211yescharacteristic
40#16839860—21no—
41C4×#11758201C2211yescharacteristic
42C8×D244C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44#106×D61—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57C61×#550C2211yescharacteristic
58C61×#486—21no—
59C61×#486C2211yescharacteristic
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#460—21no—
63C61×#460C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758222—21no—
68#14519797—44no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—41no—
73C2×#11758201D411yescharacteristic
74C2×#11758199—21no—
75C22×#92418—21no—
76C2×C8×D61—21no—
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519769—21no—
80#14519759—41no—
81#14519756—21no—
82#14519755—41no—
83#14519753—21no—
84#14519748—21no—
85C4×D244—21no—
86C4×D244C2311yescharacteristic
87C4⋊C4×D61—21no—
88C4⋊C4×D61—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—21no—
92#14519709—21no—
93#14519709—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708—21no—
97#14519708D411yescharacteristic
98C4⋊#92418—21no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519701—21no—
103#14519701—21no—
104#14519701—41no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519698—21no—
108#14519698—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123#127×C61—21no—
124#127×C61C2311yescharacteristic
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133#106×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C2×C22⋊D61—41no—
160#11758222—42no—
161#11758222—81no—
162C2×D244C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—41no—
165C2×C4×D61—41no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168SD16×C61—41no—
169SD16×C61—42no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
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
188C22⋊Dic61—41no—
189#11758202—42no—
190#11758201SD1614yescomplemented
191#11758200—42no—
192#11758200—81no—
193#11758200—82no—
194#11758199—42no—
195#11758198—21no—
196#11758198—41no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—41no—
203#11758196—41no—
204#11758196—41no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—42no—
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C8×D61—42no—
213C23×C61—21no—
214C23×C61—41no—
215C22×D61—21no—
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—81no—
220Q8×C61—161no—
221Q8×C61C2×SD1612yes—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—82no—
229D4×C61—81no—
230D4×C61—161no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
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—82no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247D244—21no—
248D244C2×SD1612yes—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—82no—
252C4×D61—161no—
253Dic122—41no—
254Dic122—41no—
255Dic122—81no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—41no—
259#92418—21no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418C2×SD1612yes—
265#19768—611nomaximal, Sylow
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61D4211yescharacteristic
273C2×D61—22no—
274C2×D61—41no—
275C2×D61—41no—
276C4×C61—22no—
277C4×C61—21no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61C22×SD1611yescharacteristic
284C4×C61#57511yescharacteristic
285C4×C61#57711yescharacteristic
286C4×C61#57811yescharacteristic
287Dic61—41no—
288Dic61—81no—
289Dic61—81no—
290#3385—611no—
291#3362D6111yescharacteristic, complemented
292#3335—611no—
293#3284—611no—
294#1714—611no—
295#1686—611no—
296#1677—611no—
297#1675—611no—
298#1672—611no—
299#1645—611no—
300#1635—611no—
301#1631—611no—
302#1622—611no—
303#1582—611no—
304SD16⋊C8—611no—
305C2×C61—81no—
306C2×C61—161no—
307C2×C61#326911yescharacteristic
308C2×C61D4×SD1611yescharacteristic
309C2×C61#328811yescharacteristic
310D61—42no—
311C2×#135—1221no—
312Q82C2×D6111yescharacteristic
313#554—611no—
314#550C2×D6111yescharacteristic
315#547—611no—
316#501—1221no—
317#500—611no—
318#498—611no—
319#492—611no—
320#492—611no—
321#487—611no—
322#486—21no—
323#486C2×D6111yescharacteristic
324#483—1221no—
325#481—1221no—
326#479—1221no—
327#477—611no—
328#476—21no—
329#476—611no—
330#476—611no—
331#476C2×D6111yescharacteristic
332#475—611no—
333#474—1221no—
334#465—1221no—
335#465—1221no—
336#462—611no—
337#460—21no—
338#460C2×D6111yescharacteristic
339#459—611no—
340#452—1221no—
341#449—1221no—
342#446—611no—
343C8×Q8C2×D6111yescharacteristic
344#440—1221no—
345C4×SD16—21no—
346C4×SD16—611no—
347C4×SD16C2×D6111yescharacteristic
348C8×D4—611no—
349C8×D4—611no—
350C2×#106—1221no—
351#335—611no—
352#334—611no—
353#333—611no—
354#333—611no—
355#332—611no—
356#331—611no—
357Q8⋊C8—611no—
358Q8⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361#322—611no—
362C61#1976811yescharacteristic, complemented, Sylow
363C2×#41—21no—
364C2×#41—1221no—
365#135—2444no—
366C2×Q16—1221no—
367C2×SD16—21no—
368C2×SD16—611no—
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×SD16—2441no—
372C2×SD16C22⋊D6111yescharacteristic
373C2×D8—1221no—
374C22×C8—1221no—
375C22×C8—1221no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8—41no—
379C4⋊Q8—611no—
380C4⋊Q8—611no—
381C4⋊Q8C22×D6111yescharacteristic
382#127—21no—
383#127C22×D6111yescharacteristic
384#125—1221no—
385#123—2441no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8—2441no—
389#121—21no—
390#121—41no—
391#121—1221no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8C22×D6111yescharacteristic
395C4×Q8C22×D6111yescharacteristic
396C4×D4—21no—
397C4×D4—611no—
398C4×D4—1221no—
399C4×D4C22×D6111yescharacteristic
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402#107—1221no—
403#107—1221no—
404#107—1221no—
405#106—21no—
406#106—611no—
407#106—1221no—
408#106—1221no—
409#106—1221no—
410#106—1221no—
411#106—1221no—
412#106C22⋊D6111yescharacteristic
413C4⋊C8—21no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—1221no—
418C4⋊C8C22×D6111yescharacteristic
419#103—611no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—2441no—
424#103C22⋊D6111yescharacteristic
425#102—21no—
426#102—21no—
427#102—21no—
428#102—21no—
429#102—611no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102C22⋊D6111yescharacteristic
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436#97—1221no—
437C4×C8—21no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8C22×D6111yescharacteristic
441#41—42no—
442#41—81no—
443#41—4881no—
444#41—2442no—
445C2×Q8—21no—
446C2×Q8—41no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8D4×D6111yescharacteristic
450C2×Q8C2×C22⋊D6111yescharacteristic
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4—611no—
457C2×D4—2441no—
458C2×D4C2×C22⋊D6111yescharacteristic
459C22×C4—21no—
460C22×C4—41no—
461C22×C4—1221no—
462C22×C4—2441no—
463C22×C4—2441no—
464Q16—2442no—
465SD16—41no—
466SD16—42no—
467SD16—614no—
468SD16—4881no—
469SD16—4882no—
470SD16—2442no—
471D8—2442no—
472C2×C8—21no—
473C2×C8—611no—
474C2×C8—611no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—2442no—
480C2×C8—2442no—
481C2×C8C2×C22⋊D6111yescharacteristic
482C2×C8C2×C22⋊D6111yescharacteristic
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—611no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—2441no—
495C4⋊C4—2441no—
496C4⋊C4—2441no—
497C4⋊C4D4×D6111yescharacteristic
498C4⋊C4C2×C22⋊D6111yescharacteristic
499C4⋊C4C2×C22⋊D6111yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—41no—
502C22⋊C4—1221no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C42—21no—
506C42—21no—
507C42—41no—
508C42C23×D6111yescharacteristic
509C23—21no—
510C23—41no—
511C23—1221no—
512Q8—21no—
513Q8—21no—
514Q8—21no—
515Q8—81no—
516Q8—161no—
517Q8—4881no—
518Q8—2441no—
519Q8—2441no—
520Q8SD16×D6112yes—
521D4—21no—
522D4—21no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—82no—
528D4—81no—
529D4—161no—
530D4—612no—
531D4—1221no—
532D4—4882no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—1221no—
543C2×C4—9761no—
544C2×C4—4882no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4#1451982211yescharacteristic
552C8—21no—
553C8—21no—
554C8—41no—
555C8—612no—
556C8—1221no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C8—2441no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—1222no—
568C22—2441no—
569C22—2441no—
570C22D4×C22⋊D6111yescharacteristic
571C4—22no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—4881no—
579C4—4881no—
580C4—2441no—
581C4C2×SD16×D6111yescharacteristic
582C4#1684056011yescharacteristic
583C4#1684056811yescharacteristic
584C4#1684057711yescharacteristic
585C2—81no—
586C2—161no—
587C2—2442no—
588C2#2145737511yescharacteristic
589C2SD16×C22⋊D6111yescharacteristic
590C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3362C2
socleC22×C61D42
prime radical#21457549C2
hypercenter#3362D61
hypocenterC61#19768
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3362

Complements

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

NH ≅ G/Ntype
1. #21458725C2semidirect
2. #21458178C2semidirect
4. #21445440C2semidirect
5. #21445289C2semidirect
6. #21445225C2semidirect
7. #21445216C2semidirect
8. #21445201C2semidirect
9. #21445040C2semidirect
10. #21444969C2semidirect
11. #21444946C2semidirect
12. #21444890C2semidirect
13. #21444656C2semidirect
14. #21444629C2semidirect
15. C61×#3362C2semidirect
190. #11758201SD16semidirect
291. #3362D61semidirect
362. C61#19768semidirect

Maximal subgroups

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

subgroupindex
#214587252
#214581782
#214575492
#214454402
#214452892
#214452252
#214452162
#214452012
#214450402
#214449692
#214449462
#214448902
#214446562
#214446292
C61×#33622
#1976861