Finite Group Databasebeta

← #29662757

Subgroups — #29662757

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456394C211yesmaximal, characteristic, complemented
2#21455880C211yesmaximal, characteristic, complemented
3#21455873C211yesmaximal, characteristic, complemented
4#21454780C211yesmaximal, characteristic, complemented
5#21445362C211yesmaximal, characteristic, complemented
6#21445358C211yesmaximal, characteristic, complemented
7#21445355C211yesmaximal, characteristic, complemented
8#21445351C211yesmaximal, characteristic, complemented
9#21445346C211yesmaximal, characteristic, complemented
10#21445341C211yesmaximal, characteristic, complemented
11#21445302C211yesmaximal, characteristic, complemented
12#21445297C211yesmaximal, characteristic, complemented
13#21444565C211yesmaximal, characteristic, complemented
14#21444479C211yesmaximal, characteristic, complemented
15C61×#1543C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17C4×#11758222C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840138C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840108C2211yescharacteristic
22#16840087—21no—
23#16840083—21no—
24#16840058C2211yescharacteristic
25#16840054C2211yescharacteristic
26#16840053C2211yescharacteristic
27#16840049C2211yescharacteristic
28#16839996—21no—
29#16839996—21no—
30#16839963C2211yescharacteristic
31#16839959C2211yescharacteristic
32#16839958C2211yescharacteristic
33#16839954C2211yescharacteristic
34#16839909—21no—
35#16839905—21no—
36#16839889C2211yescharacteristic
37#16839889C2211yescharacteristic
38C4×#11758190—21no—
39C4×#11758200—21no—
40C4×D488—21no—
41C4×#11758199—21no—
42C4×#11758199—21no—
43C4×#11758199—21no—
44#16839839C2211yescharacteristic
45#16839839C2211yescharacteristic
46#16839830—21no—
47#16839830—21no—
48#16839815C2211yescharacteristic
49#16839813—21no—
50C4×#11758189—21no—
51C2×C4×#92418C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839512C2211yescharacteristic
55#16839512C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839510C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#532C2211yescharacteristic
61C61×#431C2211yescharacteristic
62C61×#431C2211yescharacteristic
63C61×#333C2211yescharacteristic
64C61×#333C2211yescharacteristic
65C61×#331C2211yescharacteristic
66C61×#331C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758222D411yescharacteristic
69C2×#11758190—21no—
70C2×#11758200—21no—
71C2×D488—21no—
72C2×#11758199—21no—
73C2×#11758199—21no—
74C2×#11758199—21no—
75C2×#11758189—21no—
76C2×#11758189—21no—
77C22×#92418D411yescharacteristic
78C22×#92418D411yescharacteristic
79#14519777C2311yescharacteristic
80#14519774C2311yescharacteristic
81#14519765—21no—
82#14519751—21no—
83#14519751—21no—
84C4×Dic122D411yescharacteristic
85C4×Dic122C2311yescharacteristic
86C4×D244—21no—
87C4×D244—21no—
88C4×D244D411yescharacteristic
89C4×D244C2311yescharacteristic
90C4×C22⋊D61—21no—
91C4×C22⋊D61—21no—
92#14519736D411yescharacteristic
93#14519734—21no—
94C42×D61—21no—
95#14519711—21no—
96#14519710—21no—
97#14519710—21no—
98#14519710—21no—
99#14519709—21no—
100#14519708—21no—
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519702—21no—
106#14519700—21no—
107#14519699—21no—
108#14519699—21no—
109#14519699—21no—
110#14519698—21no—
111#14519698—21no—
112#14519698—21no—
113#14519698—21no—
114#14519697—21no—
115#14519697—21no—
116#14519697—21no—
117C22⋊#92418—21no—
118#14519689—21no—
119#14519689—21no—
120#14519687—21no—
121#14519687—21no—
122C4×#92418—21no—
123C4×#92418D411yescharacteristic
124C4×#92418D411yescharacteristic
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C2×#41×C61—21no—
128C4⋊Q8×C61C2311yescharacteristic
129#127×C61C2311yescharacteristic
130#124×C61—21no—
131#121×C61—21no—
132#121×C61—21no—
133C4×D4×C61—21no—
134C4×D4×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C22⋊C8×C61—21no—
141C22⋊C8×C61—21no—
142C4×C8×C61—21no—
143C4×C8×C61—21no—
144#41×C61—82no—
145C2×Q8×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—41no—
150C22×C4×C61—41no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61#4111yescharacteristic
154C2×C22⋊D61—21no—
155C2×C22⋊D61—41no—
156#11758222—41no—
157#11758222—41no—
158#11758222—82no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—41no—
167C2×Dic122—21no—
168C2×Dic122C2×D411yescharacteristic
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C22⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61#4111yescharacteristic
180C42×C61C2411yescharacteristic
181C22⋊Dic61—21no—
182#11758200—42no—
183#11758199—42no—
184#11758199—42no—
185#11758199—42no—
186#11758198—21no—
187#11758198—21no—
188#11758198—41no—
189#11758198—41no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61C2×D411yescharacteristic
193C4⋊Dic61C2×D411yescharacteristic
194#11758196—21no—
195#11758196—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C2×#92418—22no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—41no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207D488—42no—
208#11758190—42no—
209#11758189—42no—
210#11758189—42no—
211C23×C61—41no—
212C23×C61#12111yescharacteristic
213C22×D61—21no—
214C22×D61—41no—
215Q8×C61—81no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—81no—
221D4×C61—82no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—82no—
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—42no—
236C22⋊D61—82no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—41no—
241D244—21no—
242D244—21no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—81no—
248C4×D61—42no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—42no—
252C4×D61—82no—
253Dic122—21no—
254Dic122—21no—
255Dic122—81no—
256C8×C61—41no—
257C8×C61—41no—
258C8×C61—41no—
259C8×C61—41no—
260#92418—21no—
261#92418—21no—
262#92418—21no—
263#92418—41no—
264#92418—41no—
265#92418#13512yes—
266#19160—611nomaximal, Sylow
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61C2×#12111yescharacteristic
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—81no—
284C4×C61#47212yes—
285C4×C61C2×#13511yescharacteristic
286C4×C61#57511yescharacteristic
287C4×C61#57511yescharacteristic
288C4×C61#57511yescharacteristic
289Dic61—41no—
290Dic61—41no—
291Dic61—41no—
292Dic61—81no—
293#3169—611no—
294#3106—611no—
295#3106—611no—
296#2965—611no—
297#1698—611no—
298#1698—611no—
299#1697—611no—
300#1697—611no—
301#1696—611no—
302#1695—611no—
303#1688—611no—
304#1687—611no—
305#1564—611no—
306#1543—611no—
307#1543D6111yescharacteristic, complemented
308C2×C61—41no—
309C2×C61—161no—
310C2×C61#303511yescharacteristic
311C2×C61#305411yescharacteristic
312C2×C61C2×#47211yescharacteristic
313D61—81no—
314D61—161no—
315#532—611no—
316#532C2×D6111yescharacteristic
317C4×#41—611no—
318#500—611no—
319#493—611no—
320#486—611no—
321#486—611no—
322#482—1221no—
323#482—1221no—
324#477—611no—
325#477—611no—
326#476—611no—
327#476—611no—
328#466—1221no—
329#466—1221no—
330#460—611no—
331#460—611no—
332#459—611no—
333#459—611no—
334#449—1221no—
335#449—1221no—
336#446—611no—
337#446—611no—
338C4×SD16—1221no—
339C4×SD16—1221no—
340C4×D8—1221no—
341C4×D8—1221no—
342C4×D8—1221no—
343C4×D8—1221no—
344#436—611no—
345#436—611no—
346#435—1221no—
347#435—1221no—
348#431—611no—
349#431—1221no—
350#431C2×D6111yescharacteristic
351#431C2×D6111yescharacteristic
352C4×M16—1221no—
353C2×C4×C8—611no—
354#335—611no—
355#335—611no—
356#334—611no—
357#334—611no—
358#333—611no—
359#333—611no—
360#333C2×D6111yescharacteristic
361#333C2×D6111yescharacteristic
362#331—611no—
363#331—611no—
364#331C2×D6111yescharacteristic
365#331C2×D6111yescharacteristic
366C61#1916011yescharacteristic, complemented, Sylow
367C2×#41—21no—
368C2×#41—611no—
369C2×#41—1221no—
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×D8—1221no—
373C2×D8—1221no—
374C2×D8—1221no—
375C2×D8—1221no—
376C2×M16—1221no—
377C2×M16—1221no—
378C22×C8—611no—
379C22×C8—611no—
380C4⋊Q8—611no—
381C4⋊Q8C22×D6111yescharacteristic
382#127—611no—
383#127C22×D6111yescharacteristic
384#124—21no—
385#124—1221no—
386#121—21no—
387#121—21no—
388#121—1221no—
389#121—1221no—
390C4×Q8—611no—
391C4×Q8—611no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—611no—
395C4×D4—611no—
396C4×D4—1221no—
397C4×D4—1221no—
398C4×D4—1221no—
399C4×D4—1221no—
400#117—611no—
401#117—1221no—
402C2×C42—1221no—
403C2×C42C22×D6111yescharacteristic
404#107—1221no—
405#107—1221no—
406#107—1221no—
407#107—1221no—
408#106—1221no—
409#106—1221no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8C22×D6111yescharacteristic
415C4⋊C8C22×D6111yescharacteristic
416C4⋊C8C22×D6111yescharacteristic
417C4⋊C8C22×D6111yescharacteristic
418#103—1221no—
419#103—1221no—
420#102—1221no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435#97—1221no—
436#97—1221no—
437C4×C8—21no—
438C4×C8—21no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—1221no—
444#41—82no—
445#41—4882no—
446#41—2441no—
447#41—2441no—
448C2×Q8—21no—
449C2×Q8—611no—
450C2×Q8—1221no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—41no—
455C2×D4—611no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—2441no—
461C22×C4—41no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2441no—
466C22×C4#1175822211yescharacteristic
467C22×C4D4×D6111yescharacteristic
468C22×C4D4×D6111yescharacteristic
469SD16—2442no—
470SD16—2442no—
471D8—2442no—
472D8—2442no—
473D8—2442no—
474D8—2442no—
475M16—2442no—
476M16—2442no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—1222no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—2441no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—611no—
492C4⋊C4—611no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—1221no—
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C42—1221no—
505C42—1221no—
506C42—1221no—
507C42#1175822211yescharacteristic
508C42D4×D6111yescharacteristic
509C42D4×D6111yescharacteristic
510C42C23×D6111yescharacteristic
511C23—41no—
512C23—1221no—
513C23—2441no—
514C23#1451975011yescharacteristic
515Q8—81no—
516Q8—1221no—
517Q8—1221no—
518Q8—4881no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—81no—
524D4—82no—
525D4—1221no—
526D4—1221no—
527D4—4881no—
528D4—4882no—
529D4—2442no—
530D4—2441no—
531D4—2441no—
532D4—2441no—
533D4—2441no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—82no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—4882no—
545C2×C4—2442no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4—2441no—
550C2×C4#1451975011yescharacteristic
551C2×C4#1451975011yescharacteristic
552C2×C4#1451975011yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4C2×#1175822211yescharacteristic
556C8—41no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—612no—
561C8—1221no—
562C8—1221no—
563C8—1221no—
564C8—2441no—
565C8—2441no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—4881no—
573C22—4881no—
574C22—4881no—
575C22—2441no—
576C22—2441no—
577C22—2441no—
578C22C2×#1451975011yescharacteristic
579C4—21no—
580C4—21no—
581C4—21no—
582C4—81no—
583C4—4881no—
584C4—2441no—
585C4—2441no—
586C4—2441no—
587C4#1684003112yes—
588C4#135×D6111yescharacteristic
589C4#1684055911yescharacteristic
590C4#1684055911yescharacteristic
591C4#1684055911yescharacteristic
592C2—41no—
593C2—161no—
594C2—9761no—
595C2—4881no—
596C2#2145527211yescharacteristic
597C2#2145542711yescharacteristic
598C2C2×#1684003111yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1543C2
socleC22×C61C2×#121
hypercenter#1543D61
hypocenterC61#19160
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1543

Complements

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

NH ≅ G/Ntype
1. #21456394C2semidirect
2. #21455880C2semidirect
3. #21455873C2semidirect
4. #21454780C2semidirect
5. #21445362C2semidirect
6. #21445358C2semidirect
7. #21445355C2semidirect
8. #21445351C2semidirect
9. #21445346C2semidirect
10. #21445341C2semidirect
11. #21445302C2semidirect
12. #21445297C2semidirect
13. #21444565C2semidirect
14. #21444479C2semidirect
15. C61×#1543C2semidirect
307. #1543D61semidirect
366. C61#19160semidirect

Maximal subgroups

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

subgroupindex
#214563942
#214558802
#214558732
#214547802
#214453622
#214453582
#214453552
#214453512
#214453462
#214453412
#214453022
#214452972
#214445652
#214444792
C61×#15432
#1916061