Finite Group Databasebeta

← #35561264

Subgroups — #35561264

Order 21248.

580 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24919866C211yesmaximal, characteristic, complemented
2#24909694C211yesmaximal, characteristic, complemented
3#24909663C211yesmaximal, characteristic, complemented
4#24909651C211yesmaximal, characteristic, complemented
5#24909620C211yesmaximal, characteristic, complemented
6#24909561C211yesmaximal, characteristic, complemented
7#24909547C211yesmaximal, characteristic, complemented
8#24909515C211yesmaximal, characteristic, complemented
9#24909386C211yesmaximal, characteristic, complemented
10#24909375C211yesmaximal, characteristic, complemented
11C4×#14581733C211yesmaximal, characteristic, complemented
12D4×C83⋊C16C211yesmaximal, characteristic, complemented
13#24909170C211yesmaximal, characteristic, complemented
14C2×#19127502C211yesmaximal, characteristic, complemented
15C83×#3280C211yesmaximal, characteristic, complemented
16#19128397—21no—
17C2×#14581735C2211yescharacteristic, complemented
18C2×#14581733C2211yescharacteristic
19C2×#14581733C2211yescharacteristic
20C22×C83⋊C16C2211yescharacteristic
21C22×C83⋊C16C2211yescharacteristic, complemented
22#19128103C2211yescharacteristic, complemented
23#19128026—21no—
24#19128020—21no—
25#19128002—21no—
26#19127937C2211yescharacteristic
27#19127937C2211yescharacteristic, complemented
28C4×D664C2211yescharacteristic, complemented
29#19127739C2211yescharacteristic, complemented
30C2×#14581726C2211yescharacteristic
31#19127528C2211yescharacteristic, complemented
32#19127526C2211yescharacteristic, complemented
33#19127526C2211yescharacteristic, complemented
34C4⋊(C83⋊C16)C2211yescharacteristic, complemented
35#19127505C2211yescharacteristic, complemented
36#19127505C2211yescharacteristic, complemented
37#19127505C2211yescharacteristic, complemented
38#19127502C2211yescharacteristic
39#19127502C2211yescharacteristic
40#19127502C2211yescharacteristic
41#19127502C2211yescharacteristic
42#19127502C2211yescharacteristic
43#19127502C2211yescharacteristic
44#19127501C2211yescharacteristic, complemented
45C22⋊(C83⋊C16)C2211yescharacteristic
46C22⋊(C83⋊C16)C2211yescharacteristic, complemented
47C4×C83⋊C16C2211yescharacteristic, complemented
48C2×#135×C83C2211yescharacteristic
49C22×D8×C83C2211yescharacteristic
50#546×C83—21no—
51#495×C83C2211yescharacteristic
52#467×C83C2211yescharacteristic
53#466×C83C2211yescharacteristic
54#463×C83—21no—
55#450×C83—21no—
56#449×C83—21no—
57C4×D8×C83C2211yescharacteristic
58C8×D4×C83C2211yescharacteristic
59C2×D664C2311yescharacteristic
60#14581787—41no—
61#14581784—21no—
62#14581774—41no—
63#14581766—21no—
64#14581766—21no—
65#14581764—41no—
66C4×D332—21no—
67(C22⋊D83)⋊C4—41no—
68#14581751—21no—
69C2×C4⋊Dic83—21no—
70#14581735—41no—
71#14581733—22no—
72#14581733—41no—
73C2×C83⋊C16—41no—
74C2×C83⋊C16—41no—
75C2×C83⋊C16C2311yescharacteristic
76C2×C83⋊C16C2311yescharacteristic
77C2×C83⋊C16C2311yescharacteristic
78C2×C83⋊C16C2311yescharacteristic
79#14581726—21no—
80#14581726C2311yescharacteristic
81#14581726C2311yescharacteristic
82#14581726C2311yescharacteristic
83#14581724—21no—
84#14581724—21no—
85#14581714—21no—
86#14581714—21no—
87#14581714—21no—
88C2×#41×C83—21no—
89C22×D4×C83—21no—
90#135×C83—22no—
91#135×C83—41no—
92C2×Q16×C83C2311yescharacteristic
93C2×SD16×C83—21no—
94C2×SD16×C83—21no—
95C2×D8×C83—22no—
96C2×D8×C83—21no—
97C2×D8×C83—41no—
98C2×D8×C83C2311yescharacteristic
99C2×D8×C83C2311yescharacteristic
100C22×C8×C83C2311yescharacteristic
101C22×C8×C83C2311yescharacteristic
102#124×C83—21no—
103#123×C83—41no—
104C22⋊Q8×C83—21no—
105#121×C83—21no—
106#121×C83—41no—
107#120×C83—41no—
108C4×D4×C83—21no—
109C4×D4×C83D411yescharacteristic
110C2×C22⋊C4×C83—41no—
111#107×C83C2311yescharacteristic
112C4⋊C8×C83D411yescharacteristic
113#103×C83—21no—
114#102×C83—21no—
115#102×C83—21no—
116C22⋊C8×C83D411yescharacteristic
117C22⋊C8×C83D411yescharacteristic
118C4×C8×C83C2311yescharacteristic
119C2×D332—21no—
120C2×C22⋊D83—41no—
121C2×C22⋊D83—41no—
122C22×Dic83—41no—
123C2×C4×D83—41no—
124D664—21no—
125D664—21no—
126C4⋊Dic83—21no—
127C4⋊Dic83—21no—
128C4⋊Dic83—21no—
129C4⋊Dic83—41no—
130Dic83⋊C4—41no—
131Dic83⋊C4—41no—
132C22⋊Dic83—41no—
133(C2×D83)⋊C4—41no—
134(C2×D83)⋊C4—41no—
135(C2×D83)⋊C4—41no—
136C4×Dic83—41no—
137C83⋊C16—21no—
138C83⋊C16—21no—
139C83⋊C16—21no—
140C83⋊C16—21no—
141C24×C83—41no—
142#41×C83—42no—
143#41×C83—81no—
144C2×Q8×C83—21no—
145C2×D4×C83—21no—
146C2×D4×C83—21no—
147C2×D4×C83—42no—
148C2×D4×C83—41no—
149C2×D4×C83—41no—
150C2×D4×C83—41no—
151C2×D4×C83—81no—
152C2×D4×C83D811yescharacteristic
153C22×C4×C83—41no—
154C22×C4×C83—41no—
155C22×C4×C83C2×D411yescharacteristic
156C22×C4×C83C2×D411yescharacteristic
157Q16×C83—21no—
158Q16×C83—21no—
159SD16×C83—42no—
160SD16×C83—81no—
161D8×C83—21no—
162D8×C83—21no—
163D8×C83—21no—
164D8×C83—42no—
165D8×C83#4112yes—
166C2×C8×C83—21no—
167C2×C8×C83—21no—
168C2×C8×C83—21no—
169C2×C8×C83—21no—
170C2×C8×C83C2×D411yescharacteristic
171C2×C8×C83C2×D411yescharacteristic
172C2×C8×C83C2×D411yescharacteristic
173C2×C8×C83C2411yescharacteristic
174C4⋊C4×C83—21no—
175C4⋊C4×C83—41no—
176C4⋊C4×C83D811yescharacteristic
177C22⋊C4×C83—41no—
178C22⋊C4×C83—41no—
179C22⋊C4×C83—41no—
180C22⋊C4×C83—41no—
181C22⋊C4×C83—41no—
182C22⋊C4×C83D811yescharacteristic
183C22⋊C4×C83D811yescharacteristic
184C42×C83C2×D411yescharacteristic
185C23×C83—41no—
186C23×C83—41no—
187C23×C83—82no—
188C23×C83—82no—
189C23×C83—161no—
190C23×C83C2×D811yescharacteristic
191C23×C83C2×D811yescharacteristic
192C22×D83—41no—
193Q8×C83—41no—
194Q8×C83—41no—
195D4×C83—22no—
196D4×C83—22no—
197D4×C83—41no—
198D4×C83—41no—
199D4×C83—41no—
200D4×C83—82no—
201D4×C83—82no—
202D4×C83—161no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—41no—
208C2×C4×C83—41no—
209C2×C4×C83—82no—
210C2×C4×C83—161no—
211C2×C4×C83C2×D811yescharacteristic
212C2×C4×C83C2×D811yescharacteristic
213C2×C4×C83C2×D811yescharacteristic
214C2×C4×C83C2×D811yescharacteristic
215C2×C4×C83C22×D411yescharacteristic
216C22⋊D83—82no—
217C22⋊D83—161no—
218C2×Dic83—41no—
219C2×Dic83—41no—
220C2×Dic83—41no—
221C2×Dic83—161no—
222D332—41no—
223D332—41no—
224C4×D83—161no—
225C8×C83—21no—
226C8×C83—21no—
227C8×C83—21no—
228C8×C83C2×#4111yescharacteristic
229C8×C83H(5,𝔽2)11yescharacteristic
230C22×C83—21no—
231C22×C83—21no—
232C22×C83—21no—
233C22×C83—21no—
234C22×C83—21no—
235C22×C83—42no—
236C22×C83—81no—
237C22×C83—81no—
238C22×C83—81no—
239C22×C83—81no—
240C22×C83—81no—
241C22×C83—164no—
242C22×C83#50812yes—
243C22×C83C22×D811yescharacteristic
244C2×D83—81no—
245C2×D83—81no—
246C2×D83—81no—
247C4×C83—21no—
248C4×C83—21no—
249C4×C83—21no—
250C4×C83—21no—
251C4×C83—81no—
252C4×C83—81no—
253C4×C83#54611yescharacteristic
254C4×C83#57711yescharacteristic
255Dic83—81no—
256Dic83—81no—
257Dic83—81no—
258#33864—831nomaximal, Sylow
259C2×C83—22no—
260C2×C83—41no—
261C2×C83—82no—
262C2×C83—161no—
263C2×C83#329411yescharacteristic
264C2×C83C2×#50811yescharacteristic
265C2×C83#341511yescharacteristic
266D83—161no—
267#3339—831no—
268#3280D8311yescharacteristic, complemented
269#2242—831no—
270#2236—831no—
271#2234—831no—
272#2228—831no—
273#2217—831no—
274#2215—831no—
275#2210—831no—
276#2186—831no—
277#2184—831no—
278C4×D16—831no—
279D4×C16—831no—
280#2138—831no—
281C2×#357—831no—
282C83#3386411yescharacteristic, complemented, Sylow
283C2×#135C2×D8311yescharacteristic
284C22×D8C2×D8311yescharacteristic
285#553—1661no—
286#546—21no—
287C2×SD32—831no—
288C2×D16—831no—
289C2×D16—831no—
290C22×C16—831no—
291C22×C16—831no—
292#499—831no—
293#495C2×D8311yescharacteristic
294#482—1661no—
295#481—1661no—
296#478—1661no—
297#467C2×D8311yescharacteristic
298#466—831no—
299#466—831no—
300#466C2×D8311yescharacteristic
301#463—21no—
302#450—21no—
303#449—21no—
304C4×D8—831no—
305C4×D8C2×D8311yescharacteristic
306C8×D4C2×D8311yescharacteristic
307#427—831no—
308C2×#107—831no—
309#367—831no—
310#366—831no—
311#366—831no—
312C4⋊C16—831no—
313#358—831no—
314#358—831no—
315#358—831no—
316#357—831no—
317#357—831no—
318#357—831no—
319#357—831no—
320#357—831no—
321#357—831no—
322#357—831no—
323C22⋊C16—831no—
324C22⋊C16—831no—
325C4×C16—831no—
326C2×#41—21no—
327C22×D4—21no—
328#135—22no—
329#135—41no—
330C2×Q16C22×D8311yescharacteristic
331C2×SD16—21no—
332C2×SD16—21no—
333C2×D8—22no—
334C2×D8—21no—
335C2×D8—41no—
336C2×D8—831no—
337C2×D8C22×D8311yescharacteristic
338C2×D8C22×D8311yescharacteristic
339C22×C8C22×D8311yescharacteristic
340C22×C8C22×D8311yescharacteristic
341#126—3321no—
342#125—1661no—
343#124—21no—
344#123—41no—
345#123—3321no—
346C22⋊Q8—21no—
347#121—21no—
348#121—41no—
349#121—3321no—
350#121—1661no—
351#121—1661no—
352#120—41no—
353C4×D4—21no—
354C4×D4—3321no—
355C4×D4—1661no—
356C4×D4C22⋊D8311yescharacteristic
357#117—1661no—
358C2×C4⋊C4—1661no—
359C2×C22⋊C4—41no—
360SD32—3321no—
361D16—3321no—
362D16—1662no—
363C2×C16—3321no—
364C2×C16—3321no—
365C2×C16—831no—
366C2×C16—831no—
367C2×C16—831no—
368C2×C16—831no—
369#107—831no—
370#107—831no—
371#107—831no—
372#107—1661no—
373#107C22×D8311yescharacteristic
374#106—1661no—
375#106—1661no—
376C4⋊C8C22⋊D8311yescharacteristic
377#103—21no—
378#102—21no—
379#102—21no—
380#102—1661no—
381#102—1661no—
382#102—1661no—
383C22⋊C8C22⋊D8311yescharacteristic
384C22⋊C8C22⋊D8311yescharacteristic
385C4×C8C22×D8311yescharacteristic
386C24—41no—
387#41—42no—
388#41—81no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—42no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—81no—
397C2×D4—3321no—
398C2×D4—3321no—
399C2×D4—1661no—
400C2×D4#1305057511yescharacteristic
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—3321no—
404C22×C4—3321no—
405C22×C4C2×C22⋊D8311yescharacteristic
406C22×C4C2×C22⋊D8311yescharacteristic
407Q16—21no—
408Q16—21no—
409SD16—42no—
410SD16—81no—
411D8—21no—
412D8—21no—
413D8—21no—
414D8—42no—
415D8—1661no—
416D8—1661no—
417D8#1305058912yes—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8C2×C22⋊D8311yescharacteristic
423C2×C8C2×C22⋊D8311yescharacteristic
424C2×C8C2×C22⋊D8311yescharacteristic
425C2×C8C23×D8311yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—3321no—
429C4⋊C4—3321no—
430C4⋊C4—3321no—
431C4⋊C4—1661no—
432C4⋊C4—1661no—
433C4⋊C4—1661no—
434C4⋊C4#1305057511yescharacteristic
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—3321no—
441C22⋊C4—3321no—
442C22⋊C4—3321no—
443C22⋊C4—3321no—
444C22⋊C4#1305057511yescharacteristic
445C22⋊C4#1305057511yescharacteristic
446C42—3321no—
447C42C2×C22⋊D8311yescharacteristic
448C16—1661no—
449C16—1661no—
450C16—1661no—
451C16—1661no—
452C23—41no—
453C23—41no—
454C23—82no—
455C23—82no—
456C23—161no—
457C23—3321no—
458C23C2×#1305057511yescharacteristic
459C23C2×#1305057511yescharacteristic
460Q8—41no—
461Q8—41no—
462D4—22no—
463D4—22no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—82no—
468D4—82no—
469D4—161no—
470D4—13281no—
471D4—3321no—
472D4—3321no—
473D4—6642no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—161no—
482C2×C4—13281no—
483C2×C4—13281no—
484C2×C4—3321no—
485C2×C4—3321no—
486C2×C4—3321no—
487C2×C4C2×#1305057511yescharacteristic
488C2×C4C2×#1305057511yescharacteristic
489C2×C4C2×#1305057511yescharacteristic
490C2×C4C2×#1305057511yescharacteristic
491C2×C4C22×C22⋊D8311yescharacteristic
492C8—21no—
493C8—21no—
494C8—21no—
495C8C2×#1305058911yescharacteristic
496C8#1458183711yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—164no—
509C22—6641no—
510C22—6641no—
511C22—6641no—
512C22#1912813112yes—
513C22C22×#1305057511yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—81no—
519C4—81no—
520C4—6641no—
521C4—6641no—
522C4—6641no—
523C4#1912834511yescharacteristic
524C4#1912851211yescharacteristic
525C2—22no—
526C2—41no—
527C2—82no—
528C2—161no—
529C2—13281no—
530C2#2491927611yescharacteristic
531C2C2×#1912813111yescharacteristic
532C2#2492060211yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#3280C2
socleC22×C83C22×D8
hypercenter#3280D83
hypocenterC83#33864
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #24919866C2semidirect
2. #24909694C2semidirect
3. #24909663C2semidirect
4. #24909651C2semidirect
5. #24909620C2semidirect
6. #24909561C2semidirect
7. #24909547C2semidirect
8. #24909515C2semidirect
9. #24909386C2semidirect
10. #24909375C2semidirect
11. C4×#14581733C2semidirect
12. D4×C83⋊C16C2semidirect
13. #24909170C2semidirect
14. C2×#19127502C2semidirect
15. C83×#3280C2semidirect
17. C2×#14581735C22semidirect
21. C22×C83⋊C16C22semidirect
22. #19128103C22semidirect
27. #19127937C22semidirect
28. C4×D664C22semidirect
29. #19127739C22semidirect
31. #19127528C22semidirect
32. #19127526C22semidirect
33. #19127526C22semidirect
34. C4⋊(C83⋊C16)C22semidirect
35. #19127505C22semidirect
36. #19127505C22semidirect
37. #19127505C22semidirect
44. #19127501C22semidirect
46. C22⋊(C83⋊C16)C22semidirect
47. C4×C83⋊C16C22semidirect
268. #3280D83semidirect
282. C83#33864semidirect

Maximal subgroups

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

subgroupindex
#249198662
#249096942
#249096632
#249096512
#249096202
#249095612
#249095472
#249095152
#249093862
#249093752
C4×#145817332
D4×C83⋊C162
#249091702
C2×#191275022
C83×#32802
#3386483