Finite Group Databasebeta

← #35561249

Subgroups — #35561249

Order 21248.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24919832C211yesmaximal, characteristic, complemented
2#24909694C211yesmaximal, characteristic, complemented
3#24909658C211yesmaximal, characteristic, complemented
4#24909651C211yesmaximal, characteristic, complemented
5#24909620C211yesmaximal, characteristic, complemented
6#24909557C211yesmaximal, characteristic, complemented
7#24909552C211yesmaximal, characteristic, complemented
8#24909522C211yesmaximal, characteristic, complemented
9#24909403C211yesmaximal, characteristic, complemented
10#24909392C211yesmaximal, characteristic, complemented
11C4×#14581736C211yesmaximal, characteristic, complemented
12D4×C83⋊C16C211yesmaximal, characteristic, complemented
13#24909170C211yesmaximal, characteristic, complemented
14C2×#19127502C211yesmaximal, characteristic, complemented
15C83×#3280C211yesmaximal, characteristic
16#19128391—21no—
17C2×#14581738C2211yescharacteristic, complemented
18C2×#14581736C2211yescharacteristic
19C2×#14581736C2211yescharacteristic
20C22×C83⋊C16C2211yescharacteristic
21C22×C83⋊C16C2211yescharacteristic, complemented
22#19128103C2211yescharacteristic, complemented
23#19128043—21no—
24#19128032—21no—
25#19128009—21no—
26#19127942C2211yescharacteristic
27#19127942C2211yescharacteristic, complemented
28C4×Dic332C2211yescharacteristic, complemented
29#19127739C2211yescharacteristic, complemented
30C2×#14581726C2211yescharacteristic
31#19127528C2211yescharacteristic, complemented
32#19127528C2211yescharacteristic, complemented
33#19127526C2211yescharacteristic, complemented
34C4⋊(C83⋊C16)C2211yescharacteristic, complemented
35#19127505C2211yescharacteristic, complemented
36#19127505C2211yescharacteristic, complemented
37#19127505C2211yescharacteristic, complemented
38#19127504C2211yescharacteristic, complemented
39#19127502C2211yescharacteristic
40#19127502C2211yescharacteristic
41#19127502C2211yescharacteristic
42#19127502C2211yescharacteristic
43#19127502C2211yescharacteristic
44#19127502C2211yescharacteristic
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×Dic332C2311yescharacteristic
60#14581785—41no—
61#14581784—21no—
62#14581782—41no—
63#14581773—41no—
64#14581771—21no—
65#14581771—21no—
66C4×Dic166—21no—
67#14581751—21no—
68#14581748—41no—
69C2×C4⋊Dic83—21no—
70#14581738—41no—
71#14581736—22no—
72#14581736—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#14581717—21no—
86#14581717—21no—
87#14581717—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×Dic166—21no—
120C22×Dic83—41no—
121Dic332—21no—
122Dic332—21no—
123C4⋊Dic83—21no—
124C4⋊Dic83—21no—
125C4⋊Dic83—21no—
126C4⋊Dic83—41no—
127Dic83⋊C4—41no—
128Dic83⋊C4—41no—
129Dic83⋊C4—41no—
130Dic83⋊C4—41no—
131Dic83⋊C4—41no—
132C22⋊Dic83—41no—
133C22⋊Dic83—41no—
134C22⋊Dic83—41no—
135C4×Dic83—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
192Q8×C83—41no—
193Q8×C83—41no—
194D4×C83—22no—
195D4×C83—22no—
196D4×C83—41no—
197D4×C83—41no—
198D4×C83—41no—
199D4×C83—82no—
200D4×C83—82no—
201D4×C83—161no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—41no—
207C2×C4×C83—41no—
208C2×C4×C83—82no—
209C2×C4×C83—161no—
210C2×C4×C83C2×D811yescharacteristic
211C2×C4×C83C2×D811yescharacteristic
212C2×C4×C83C2×D811yescharacteristic
213C2×C4×C83C2×D811yescharacteristic
214C2×C4×C83C22×D411yescharacteristic
215C2×Dic83—41no—
216C2×Dic83—41no—
217C2×Dic83—41no—
218C2×Dic83—41no—
219C2×Dic83—82no—
220Dic166—41no—
221Dic166—41no—
222C8×C83—21no—
223C8×C83—21no—
224C8×C83—21no—
225C8×C83C2×#4111yescharacteristic
226C8×C83H(5,𝔽2)11yescharacteristic
227C22×C83—21no—
228C22×C83—21no—
229C22×C83—21no—
230C22×C83—21no—
231C22×C83—21no—
232C22×C83—42no—
233C22×C83—81no—
234C22×C83—81no—
235C22×C83—81no—
236C22×C83—81no—
237C22×C83—81no—
238C22×C83—164no—
239C22×C83#50812yes—
240C22×C83C22×D811yescharacteristic
241C4×C83—21no—
242C4×C83—21no—
243C4×C83—21no—
244C4×C83—21no—
245C4×C83—81no—
246C4×C83—81no—
247C4×C83#54611yescharacteristic
248C4×C83#57711yescharacteristic
249Dic83—81no—
250Dic83—81no—
251Dic83—81no—
252Dic83—81no—
253#33863—831nomaximal, Sylow
254C2×C83—22no—
255C2×C83—41no—
256C2×C83—82no—
257C2×C83—161no—
258C2×C83#329411yescharacteristic
259C2×C83C2×#50811yescharacteristic
260C2×C83#341611yescharacteristic
261#3336—831no—
262#3280D8311yescharacteristic
263#2242—831no—
264#2235—831no—
265#2234—831no—
266#2228—831no—
267#2217—831no—
268#2216—831no—
269#2211—831no—
270#2189—831no—
271#2187—831no—
272C4×SD32—831no—
273D4×C16—831no—
274#2138—831no—
275C2×#357—831no—
276C83#3386311yescharacteristic, complemented, Sylow
277C2×#135C2×D8311yescharacteristic
278C22×D8C2×D8311yescharacteristic
279#552—1661no—
280#546—21no—
281C2×Q32—831no—
282C2×SD32—831no—
283C2×SD32—831no—
284C22×C16—831no—
285C22×C16—831no—
286#499—831no—
287#495C2×D8311yescharacteristic
288#485—1661no—
289#483—1661no—
290#479—1661no—
291#467—831no—
292#467—831no—
293#467C2×D8311yescharacteristic
294#466C2×D8311yescharacteristic
295#463—21no—
296#450—21no—
297#449—21no—
298C4×Q16—831no—
299C4×D8C2×D8311yescharacteristic
300C8×D4C2×D8311yescharacteristic
301#427—831no—
302C2×#107—831no—
303#367—831no—
304#367—831no—
305#366—831no—
306C4⋊C16—831no—
307#358—831no—
308#358—831no—
309#358—831no—
310#358—831no—
311#357—831no—
312#357—831no—
313#357—831no—
314#357—831no—
315#357—831no—
316#357—831no—
317C22⋊C16—831no—
318C22⋊C16—831no—
319C4×C16—831no—
320C2×#41—21no—
321C22×D4—21no—
322#135—22no—
323#135—41no—
324C2×Q16—831no—
325C2×Q16C22×D8311yescharacteristic
326C2×SD16—21no—
327C2×SD16—21no—
328C2×D8—22no—
329C2×D8—21no—
330C2×D8—41no—
331C2×D8C22×D8311yescharacteristic
332C2×D8C22×D8311yescharacteristic
333C22×C8C22×D8311yescharacteristic
334C22×C8C22×D8311yescharacteristic
335#126—3321no—
336#125—3321no—
337#125—1661no—
338#124—21no—
339#123—41no—
340#123—3321no—
341C22⋊Q8—21no—
342C22⋊Q8—1661no—
343C22⋊Q8—1661no—
344#121—21no—
345#121—41no—
346#120—41no—
347C4×Q8—1661no—
348C4×D4—21no—
349C4×D4C22⋊D8311yescharacteristic
350#117—3321no—
351#117—1661no—
352C2×C4⋊C4—1661no—
353C2×C22⋊C4—41no—
354Q32—3321no—
355SD32—3321no—
356SD32—1662no—
357C2×C16—3321no—
358C2×C16—3321no—
359C2×C16—831no—
360C2×C16—831no—
361C2×C16—831no—
362C2×C16—831no—
363#107—831no—
364#107—831no—
365#107—831no—
366#107—1661no—
367#107C22×D8311yescharacteristic
368#106—1661no—
369#106—1661no—
370C4⋊C8C22⋊D8311yescharacteristic
371#103—21no—
372#103—1661no—
373#103—1661no—
374#103—1661no—
375#102—21no—
376#102—21no—
377C22⋊C8C22⋊D8311yescharacteristic
378C22⋊C8C22⋊D8311yescharacteristic
379C4×C8C22×D8311yescharacteristic
380C24—41no—
381#41—42no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—1661no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—81no—
392C2×D4#1305057511yescharacteristic
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—3321no—
396C22×C4C2×C22⋊D8311yescharacteristic
397C22×C4C2×C22⋊D8311yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—1661no—
401Q16—1661no—
402SD16—42no—
403SD16—81no—
404D8—21no—
405D8—21no—
406D8—21no—
407D8—42no—
408D8#1305058912yes—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C22⋊D8311yescharacteristic
414C2×C8C2×C22⋊D8311yescharacteristic
415C2×C8C2×C22⋊D8311yescharacteristic
416C2×C8C23×D8311yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—3321no—
420C4⋊C4—3321no—
421C4⋊C4—3321no—
422C4⋊C4—3321no—
423C4⋊C4—3321no—
424C4⋊C4—3321no—
425C4⋊C4—1661no—
426C4⋊C4—1661no—
427C4⋊C4—1661no—
428C4⋊C4#1305057511yescharacteristic
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—3321no—
435C22⋊C4—3321no—
436C22⋊C4—3321no—
437C22⋊C4#1305057511yescharacteristic
438C22⋊C4#1305057511yescharacteristic
439C42—3321no—
440C42—3321no—
441C42C2×C22⋊D8311yescharacteristic
442C16—1661no—
443C16—1661no—
444C16—1661no—
445C16—1661no—
446C23—41no—
447C23—41no—
448C23—82no—
449C23—82no—
450C23—161no—
451C23C2×#1305057511yescharacteristic
452C23C2×#1305057511yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—3321no—
456Q8—3321no—
457D4—22no—
458D4—22no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—82no—
463D4—82no—
464D4—161no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—161no—
473C2×C4—3321no—
474C2×C4—3321no—
475C2×C4—3321no—
476C2×C4—3321no—
477C2×C4—6642no—
478C2×C4C2×#1305057511yescharacteristic
479C2×C4C2×#1305057511yescharacteristic
480C2×C4C2×#1305057511yescharacteristic
481C2×C4C2×#1305057511yescharacteristic
482C2×C4C22×C22⋊D8311yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8C2×#1305058911yescharacteristic
487C8#1458183711yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—164no—
500C22#1912813112yes—
501C22C22×#1305057511yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—6641no—
509C4—6641no—
510C4—6641no—
511C4—6641no—
512C4#1912834511yescharacteristic
513C4#1912851211yescharacteristic
514C2—22no—
515C2—41no—
516C2—82no—
517C2—161no—
518C2#2491927611yescharacteristic
519C2C2×#1912813111yescharacteristic
520C2#2492061011yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
Fitting, prime radicalC83×#3280C2
socleC22×C83C22×D8
hypercenter#3280D83
hypocenterC83#33863
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. #24919832C2semidirect
2. #24909694C2semidirect
3. #24909658C2semidirect
4. #24909651C2semidirect
5. #24909620C2semidirect
6. #24909557C2semidirect
7. #24909552C2semidirect
8. #24909522C2semidirect
9. #24909403C2semidirect
10. #24909392C2semidirect
11. C4×#14581736C2semidirect
12. D4×C83⋊C16C2semidirect
13. #24909170C2semidirect
14. C2×#19127502C2semidirect
17. C2×#14581738C22semidirect
21. C22×C83⋊C16C22semidirect
22. #19128103C22semidirect
27. #19127942C22semidirect
28. C4×Dic332C22semidirect
29. #19127739C22semidirect
31. #19127528C22semidirect
32. #19127528C22semidirect
33. #19127526C22semidirect
34. C4⋊(C83⋊C16)C22semidirect
35. #19127505C22semidirect
36. #19127505C22semidirect
37. #19127505C22semidirect
38. #19127504C22semidirect
46. C22⋊(C83⋊C16)C22semidirect
47. C4×C83⋊C16C22semidirect
276. C83#33863semidirect

Maximal subgroups

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

subgroupindex
#249198322
#249096942
#249096582
#249096512
#249096202
#249095572
#249095522
#249095222
#249094032
#249093922
C4×#145817362
D4×C83⋊C162
#249091702
C2×#191275022
C83×#32802
#3386383