Finite Group Databasebeta

← #29650211

Subgroups — #29650211

Order 15616.

658 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#21455848C211yesmaximal, characteristic
2#446×D61C211yesmaximal, characteristic, complemented
3#21454528C211yesmaximal, characteristic
4#21444711C211yesmaximal, characteristic, complemented
5#21444709C211yesmaximal, characteristic, complemented
6#21444707C211yesmaximal, characteristic, complemented
7#21444658C211yesmaximal, characteristic, complemented
8#21444656C211yesmaximal, characteristic, complemented
9#21444654C211yesmaximal, characteristic, complemented
10#21444652C211yesmaximal, characteristic, complemented
11#21444588C211yesmaximal, characteristic, complemented
12#21444246C211yesmaximal, characteristic, complemented
13#21444216C211yesmaximal, characteristic
14#21444101C211yesmaximal, characteristic, complemented
15C61×#1590C211yesmaximal, characteristic, complemented
16C2×#14519785—21no—
17#16840309C2211yescharacteristic
18C4×Q8×D61C2211yescharacteristic
19#16840082C411yescharacteristic
20#16840079C411yescharacteristic
21#16840046C411yescharacteristic
22#16840045C411yescharacteristic
23#16839961C411yescharacteristic
24#16839960C411yescharacteristic
25#16839902C411yescharacteristic
26#16839899C411yescharacteristic
27#16839893C2211yescharacteristic
28#16839892C2211yescharacteristic
29#16839891—21no—
30#16839890—21no—
31#16839889C2211yescharacteristic
32C4×#11758190C2211yescharacteristic
33C4×#11758201C2211yescharacteristic
34C4×#11758200C2211yescharacteristic
35#16839841C2211yescharacteristic
36#16839838—21no—
37D4×#92418C2211yescharacteristic
38#16839827C2211yescharacteristic
39#16839825—21no—
40#16839813C2211yescharacteristic
41#16839813C2211yescharacteristic
42#16839784C2211yescharacteristic
43C4⋊C8×D61—21no—
44C4⋊C8×D61C2211yescharacteristic
45#16839727—21no—
46#97×D61—21no—
47C4×C8×D61C2211yescharacteristic
48#16839513C2211yescharacteristic
49#16839512C2211yescharacteristic
50#16839512C2211yescharacteristic
51#16839494C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54#16839491C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic
58#16839484C2211yescharacteristic
59C8×#92418C2211yescharacteristic
60C61×#446C2211yescharacteristic
61C4×SD16×C61C2211yescharacteristic
62C61×#436C2211yescharacteristic
63C61×#334C2211yescharacteristic
64C61×Q8⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic
66C82×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×Q8×D61C2×C411yescharacteristic
69C2×#11758190C2×C411yescharacteristic
70C2×#11758200C2×C411yescharacteristic
71C2×#11758201C2×C411yescharacteristic
72#14519785—44no—
73C2×#11758189—21no—
74C2×#11758189—21no—
75C22×#92418—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78C2×C8×D61D411yescharacteristic
79C2×C8×D61D411yescharacteristic
80#14519776C2×C411yescharacteristic
81#14519773C2×C411yescharacteristic
82#14519763—21no—
83#14519752C2×C411yescharacteristic
84#14519750—21no—
85#14519748C2×C411yescharacteristic
86Q8×Dic61C2×C411yescharacteristic
87C4×Dic122—21no—
88C4×Dic122C2311yescharacteristic
89#14519742—21no—
90#14519742C2×C411yescharacteristic
91C4×D244C2311yescharacteristic
92C4×C22⋊D61—21no—
93C4⋊C4×D61—21no—
94C4⋊C4×D61C2×C411yescharacteristic
95C42×D61—21no—
96C42×D61C2311yescharacteristic
97#14519709C2×C411yescharacteristic
98#14519708C2×C411yescharacteristic
99#14519708C2×C411yescharacteristic
100C4⋊#92418—21no—
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519706—21no—
106#14519706—21no—
107#14519706—21no—
108#14519706C2×C411yescharacteristic
109#14519706C2×C411yescharacteristic
110#14519702C2×C411yescharacteristic
111#14519701C2×C411yescharacteristic
112#14519700C2×C411yescharacteristic
113#14519699C2×C411yescharacteristic
114#14519698C2×C411yescharacteristic
115#14519697C2×C411yescharacteristic
116C22⋊#92418—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689C2×C411yescharacteristic
121#14519689C2×C411yescharacteristic
122#14519688—21no—
123#14519687—21no—
124#14519687—21no—
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×SD16×C61C2×C411yescharacteristic
131C2×M16×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#106×C61C2×C411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61—21no—
141#97×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C2×C22⋊D61—21no—
149C2×C22⋊D61—21no—
150C2×C22⋊D61—21no—
151Q8×D61—41no—
152Q8×D61—41no—
153#11758222—42no—
154#11758222—42no—
155C2×D244—21no—
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61C22×C411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61#4111yescharacteristic
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122C22×C411yescharacteristic
166SD16×C61—41no—
167M16×C61—42no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C22×C411yescharacteristic
172C2×C8×C61C22×C411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C411yescharacteristic
177C4⋊C4×C61C22×C411yescharacteristic
178C22⋊C4×C61—21no—
179C42×C61—21no—
180C42×C61C2411yescharacteristic
181C22⋊Dic61—21no—
182#11758201—41no—
183#11758200—41no—
184#11758198—21no—
185#11758198—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61C22×C411yescharacteristic
188C4⋊Dic61C22×C411yescharacteristic
189#11758196—21no—
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61C22×C411yescharacteristic
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61#4111yescharacteristic
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—42no—
203C2×#92418C22×C411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207#11758190—41no—
208#11758189—42no—
209#11758189—42no—
210C8×D61—22no—
211C8×D61—42no—
212C8×D61—42no—
213C8×D61SD1614yescomplemented
214C23×C61—21no—
215C22×D61—21no—
216C22×D61C4×D411yescharacteristic
217Q8×C61—21no—
218Q8×C61—21no—
219D4×C61—21no—
220D4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61C23×C411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C22⋊D61—42no—
230C22⋊D61—42no—
231C22⋊D61—42no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61C4×D411yescharacteristic
236C2×Dic61C4×D411yescharacteristic
237C2×Dic61C4×D411yescharacteristic
238D244—21no—
239D244—21no—
240D244—42no—
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—41no—
244C4×D61—42no—
245C4×D61—42no—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61C2×SD1612yes—
249C4×D61#13512yes—
250Dic122—21no—
251Dic122—21no—
252Dic122—42no—
253Dic122—42no—
254C8×C61—21no—
255C8×C61—21no—
256C8×C61—21no—
257C8×C61—41no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61C2×SD1612yes—
261#92418—21no—
262#92418—21no—
263#92418—21no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#92418C2×SD1612yes—
268#17929—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61C2×C4×D411yescharacteristic
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61C4×SD1612yes—
279C4×C61—21no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61#56811yescharacteristic
284C4×C61#56811yescharacteristic
285C4×C61C22×SD1611yescharacteristic
286C4×C61C2×#13511yescharacteristic
287Dic61—21no—
288Dic61—21no—
289Dic61—41no—
290Dic61—41no—
291Dic61—41no—
292Dic61C4×SD1612yes—
293#3103—611no—
294C2×#446—611no—
295#2931—611no—
296#1590—611no—
297#1590—611no—
298#1590—611no—
299#1590D6111yescharacteristic, complemented
300#1582—611no—
301#1582—611no—
302#1582—611no—
303#1582—611no—
304#1570—611no—
305#1487—611no—
306#1480—611no—
307#1450—611no—
308C2×C61—81no—
309C2×C61#293011yescharacteristic
310C2×C61C2×C4×SD1611yescharacteristic
311C2×C61#295511yescharacteristic
312D61—22no—
313D61—81no—
314C2×#131—1221no—
315#532—611no—
316C2×C4×Q8—611no—
317#481—611no—
318#481—611no—
319#475—611no—
320#475—611no—
321#460—611no—
322#460—611no—
323#448—611no—
324#448—611no—
325#446—611no—
326#446—611no—
327#446—611no—
328#446—1221no—
329#446—1221no—
330#446C2×D6111yescharacteristic
331C4×SD16—611no—
332C4×SD16—611no—
333C4×SD16—611no—
334C4×SD16C2×D6111yescharacteristic
335#436—611no—
336#436—1221no—
337#436C2×D6111yescharacteristic
338C8×D4—611no—
339C8×D4—611no—
340C8×D4—1221no—
341#431—611no—
342#431—611no—
343#423—611no—
344C2×C4⋊C8—611no—
345C2×C4⋊C8—1221no—
346#408—1221no—
347C2×#97—1221no—
348C2×C4×C8—611no—
349#334—611no—
350#334—611no—
351#334—611no—
352#334C2×D6111yescharacteristic
353Q8⋊C8—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356Q8⋊C8C2×D6111yescharacteristic
357D4⋊C8—611no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8C2×D6111yescharacteristic
361#322—611no—
362#322—611no—
363C82—611no—
364C82C2×D6111yescharacteristic
365C61#1792911yescharacteristic, complemented, Sylow
366C2×#41—1221no—
367C22×Q8—611no—
368C2×SD16—611no—
369C2×SD16—611no—
370C2×SD16—611no—
371C2×SD16C4×D6111yescharacteristic
372#131—2444no—
373C2×M16—21no—
374C2×M16—1221no—
375C2×M16—1221no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8—1221no—
379C22×C8—1221no—
380C22×C8—1221no—
381C4⋊Q8—611no—
382#127—611no—
383#124—1221no—
384#121—611no—
385#121—611no—
386#121—1221no—
387C4×Q8—611no—
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—1221no—
391C4×Q8—1221no—
392C4×Q8C22×D6111yescharacteristic
393C4×D4—611no—
394C4×D4—1221no—
395C4×D4C22×D6111yescharacteristic
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1221no—
398C2×C42—611no—
399C2×C42—1221no—
400#106—611no—
401#106—611no—
402#106—611no—
403#106C4×D6111yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—1221no—
410C4⋊C8—1221no—
411C4⋊C8—1221no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8C22×D6111yescharacteristic
416C4⋊C8C22×D6111yescharacteristic
417#103—611no—
418#103—611no—
419#103—611no—
420#103C4×D6111yescharacteristic
421#102—611no—
422#102—611no—
423#102—611no—
424#102C4×D6111yescharacteristic
425C22⋊C8—21no—
426C22⋊C8—611no—
427C22⋊C8—611no—
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432#97—21no—
433#97—1221no—
434#97—1221no—
435#97—1221no—
436C4×C8—611no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8C22×D6111yescharacteristic
442C4×C8C22×D6111yescharacteristic
443C4×C8C22×D6111yescharacteristic
444#41—2442no—
445#41—2442no—
446C2×Q8—611no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—2441no—
450C2×Q8—2441no—
451C2×Q8C2×C4×D6111yescharacteristic
452C2×D4—611no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4C2×C4×D6111yescharacteristic
458C22×C4—21no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465SD16—41no—
466SD16—2441no—
467SD16—2441no—
468SD16—2441no—
469M16—42no—
470M16—2442no—
471M16—2442no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—611no—
476C2×C8—614no—
477C2×C8—611no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—1222no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—2442no—
485C2×C8—2442no—
486C2×C8—2442no—
487C2×C8C2×C4×D6111yescharacteristic
488C2×C8C2×C4×D6111yescharacteristic
489C2×C8D4×D6111yescharacteristic
490C2×C8D4×D6111yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—611no—
493C4⋊C4—611no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4C2×C4×D6111yescharacteristic
500C4⋊C4C2×C4×D6111yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C22⋊C4—1221no—
505C42—21no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—1221no—
510C42—1221no—
511C42—1221no—
512C42C23×D6111yescharacteristic
513C23—21no—
514C23—611no—
515C23—1221no—
516Q8—21no—
517Q8—21no—
518Q8—1221no—
519Q8—1221no—
520Q8—2442no—
521Q8—2442no—
522D4—21no—
523D4—21no—
524D4—1221no—
525D4—1221no—
526D4—2442no—
527D4—2442no—
528D4—2442no—
529D4—2442no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—611no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—2441no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4C22×C4×D6111yescharacteristic
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4#41×D6111yescharacteristic
553C8—21no—
554C8—21no—
555C8—21no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—612no—
560C8—1221no—
561C8—1221no—
562C8—1221no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C8SD16×D6112yes—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—612no—
571C22—1221no—
572C22—1221no—
573C22—2441no—
574C22—2441no—
575C22—2441no—
576C22C4×D4×D6111yescharacteristic
577C4—21no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—612no—
582C4—1221no—
583C4—1221no—
584C4—2441no—
585C4—2441no—
586C4—2441no—
587C4#1684052711yescharacteristic
588C4#1684052811yescharacteristic
589C4C2×SD16×D6111yescharacteristic
590C4#135×D6111yescharacteristic
591C2—81no—
592C2—1222no—
593C2—4881no—
594C2#2145451911yescharacteristic
595C2C4×SD16×D6111yescharacteristic
596C2#2145471711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1590C2
socleC22×C61C2×C4×D4
prime radical#16840309C22
cosocleC42×C61C24
hypercenter#1590D61
hypocenterC61#17929
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #446×D61C2semidirect
4. #21444711C2semidirect
5. #21444709C2semidirect
6. #21444707C2semidirect
7. #21444658C2semidirect
8. #21444656C2semidirect
9. #21444654C2semidirect
10. #21444652C2semidirect
11. #21444588C2semidirect
12. #21444246C2semidirect
14. #21444101C2semidirect
15. C61×#1590C2semidirect
213. C8×D61SD16semidirect
299. #1590D61semidirect
365. C61#17929semidirect

Maximal subgroups

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

subgroupindex
#214558482
#446×D612
#214545282
#214447112
#214447092
#214447072
#214446582
#214446562
#214446542
#214446522
#214445882
#214442462
#214442162
#214441012
C61×#15902
#1792961