Finite Group Databasebeta

← #35561369

Subgroups — #35561369

Order 21248.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24919150C211yesmaximal, characteristic, complemented
2#24909702C211yesmaximal, characteristic, complemented
3#24909660C211yesmaximal, characteristic, complemented
4#24909648C211yesmaximal, characteristic, complemented
5#24909607C211yesmaximal, characteristic, complemented
6#24909564C211yesmaximal, characteristic, complemented
7#24909557C211yesmaximal, characteristic, complemented
8#24909518C211yesmaximal, characteristic, complemented
9#24909384C211yesmaximal, characteristic, complemented
10#24909374C211yesmaximal, characteristic, complemented
11#24909337C211yesmaximal, characteristic, complemented
12#24909289C211yesmaximal, characteristic, complemented
13#24909173C211yesmaximal, characteristic, complemented
14C2×#19127501C211yesmaximal, characteristic, complemented
15C83×#3334C211yesmaximal, characteristic, complemented
16C2×#14581814C2211yescharacteristic
17C22×D664C2211yescharacteristic
18#19128347—21no—
19C2×#14581736C2211yescharacteristic, complemented
20C2×#14581733C2211yescharacteristic
21C2×#14581733C2211yescharacteristic
22C2×#14581732C2211yescharacteristic, complemented
23C22×C83⋊C16C2211yescharacteristic
24#19128085C2211yescharacteristic
25#19127942C2211yescharacteristic
26#19127937C2211yescharacteristic
27#19127917—21no—
28#19127854—21no—
29#19127848—21no—
30C4×D664C2211yescharacteristic
31#19127528C2211yescharacteristic, complemented
32#19127528C2211yescharacteristic, complemented
33#19127528C2211yescharacteristic, complemented
34C4⋊(C83⋊C16)C2211yescharacteristic, complemented
35#19127504C2211yescharacteristic, complemented
36#19127504C2211yescharacteristic, complemented
37#19127504C2211yescharacteristic, complemented
38#19127502C2211yescharacteristic, complemented
39#19127501C2211yescharacteristic
40#19127501C2211yescharacteristic
41#19127501C2211yescharacteristic
42#19127501C2211yescharacteristic
43#19127501C2211yescharacteristic
44#19127501C2211yescharacteristic
45C22⋊(C83⋊C16)C2211yescharacteristic
46C22⋊(C83⋊C16)C2211yescharacteristic, complemented
47#19127477C2211yescharacteristic, complemented
48#547×C83—21no—
49#500×C83C2211yescharacteristic, complemented
50#480×C83—21no—
51#480×C83—21no—
52#474×C83—21no—
53#466×C83C2211yescharacteristic
54#466×C83C2211yescharacteristic, complemented
55C4×D8×C83C2211yescharacteristic, complemented
56C8×D4×C83C2211yescharacteristic
57C2×#107×C83C2211yescharacteristic
58C2×#107×C83C2211yescharacteristic, complemented
59C2×#13050590—21no—
60C22×D332—21no—
61#14581814—22no—
62#14581814—41no—
63C2×Dic332C2311yescharacteristic
64C2×#13050579—21no—
65C2×#13050579—21no—
66C2×D664—22no—
67C2×D664—21no—
68C2×D664—41no—
69C2×D664C2311yescharacteristic
70C2×D664C2311yescharacteristic
71#14581780—21no—
72#14581777—41no—
73#14581771—21no—
74#14581766—21no—
75#14581765—41no—
76#14581760—41no—
77C4×D332—21no—
78C22⋊C4×D83—41no—
79#14581736—41no—
80#14581733—22no—
81#14581733—41no—
82#14581732—41no—
83C2×C83⋊C16—41no—
84C2×C83⋊C16C2311yescharacteristic
85C2×C83⋊C16C2311yescharacteristic
86C2×C83⋊C16C2311yescharacteristic
87C2×C83⋊C16C2311yescharacteristic
88#14581726C2311yescharacteristic
89#14581717—21no—
90#14581714—21no—
91#14581714—21no—
92C2×#41×C83—41no—
93C2×D8×C83C2311yescharacteristic
94C22×C8×C83C2311yescharacteristic
95C22×C8×C83C2311yescharacteristic
96C4⋊Q8×C83—21no—
97#123×C83—41no—
98#123×C83—41no—
99C22⋊Q8×C83—41no—
100#121×C83—21no—
101#121×C83—21no—
102C4×D4×C83—21no—
103C4×D4×C83D411yescharacteristic
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106#107×C83—21no—
107#107×C83—21no—
108#107×C83—21no—
109#107×C83C2311yescharacteristic
110#107×C83C2311yescharacteristic
111#107×C83C2311yescharacteristic
112C4⋊C8×C83D411yescharacteristic
113#102×C83—21no—
114#102×C83—21no—
115#102×C83—21no—
116C22⋊C8×C83D411yescharacteristic
117C22⋊C8×C83D411yescharacteristic
118C4×C8×C83C2311yescharacteristic
119C23×D83—41no—
120#13050590—42no—
121#13050590—81no—
122C2×Dic166—21no—
123C2×D332—21no—
124C2×D332—21no—
125C2×D332—42no—
126C2×D332—41no—
127C2×D332—81no—
128C2×C22⋊D83—41no—
129C2×C22⋊D83—41no—
130C2×C4×D83—41no—
131C2×C4×D83—41no—
132Dic332—21no—
133Dic332—21no—
134#13050579—42no—
135#13050579—81no—
136D664—21no—
137D664—21no—
138D664—21no—
139D664—42no—
140D664#4112yescomplemented
141C4⋊Dic83—21no—
142Dic83⋊C4—41no—
143C22⋊Dic83—41no—
144(C2×D83)⋊C4—41no—
145(C2×D83)⋊C4—41no—
146(C2×D83)⋊C4—41no—
147(C2×D83)⋊C4—41no—
148C83⋊C16—21no—
149C83⋊C16—21no—
150C83⋊C16—21no—
151C83⋊C16—21no—
152#41×C83—84no—
153C2×Q8×C83—41no—
154C2×D4×C83—21no—
155C2×D4×C83—41no—
156C2×D4×C83—41no—
157C2×D4×C83D811yescharacteristic
158C22×C4×C83—41no—
159C22×C4×C83—41no—
160C22×C4×C83—41no—
161C22×C4×C83C2×D411yescharacteristic
162C22×C4×C83C2×D411yescharacteristic
163D8×C83—21no—
164D8×C83—21no—
165C2×C8×C83—21no—
166C2×C8×C83—21no—
167C2×C8×C83—21no—
168C2×C8×C83—21no—
169C2×C8×C83C2×D411yescharacteristic
170C2×C8×C83C2×D411yescharacteristic
171C2×C8×C83C2×D411yescharacteristic
172C2×C8×C83C2411yescharacteristic
173C4⋊C4×C83—21no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—41no—
177C4⋊C4×C83—41no—
178C4⋊C4×C83—41no—
179C4⋊C4×C83D811yescharacteristic
180C22⋊C4×C83—41no—
181C22⋊C4×C83—41no—
182C22⋊C4×C83—41no—
183C22⋊C4×C83D811yescharacteristic
184C22⋊C4×C83D811yescharacteristic
185C42×C83C2×D411yescharacteristic
186C23×C83—41no—
187C23×C83C2×D811yescharacteristic
188C23×C83C2×D811yescharacteristic
189C22×D83—41no—
190C22×D83—41no—
191C22×D83—82no—
192C22×D83—82no—
193C22×D83—161no—
194Q8×C83—82no—
195D4×C83—22no—
196D4×C83—41no—
197D4×C83—41no—
198D4×C83—82no—
199D4×C83—82no—
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—41no—
205C2×C4×C83—41no—
206C2×C4×C83—41no—
207C2×C4×C83—82no—
208C2×C4×C83—82no—
209C2×C4×C83—82no—
210C2×C4×C83C2×D811yescharacteristic
211C2×C4×C83C2×D811yescharacteristic
212C2×C4×C83C2×D811yescharacteristic
213C2×C4×C83C2×D811yescharacteristic
214C2×C4×C83C22×D411yescharacteristic
215C22⋊D83—82no—
216C22⋊D83—161no—
217C2×Dic83—41no—
218C2×Dic83—41no—
219D332—22no—
220D332—41no—
221D332—41no—
222D332—41no—
223D332—82no—
224C4×D83—82no—
225C4×D83—161no—
226Dic166—41no—
227Dic166—41no—
228C8×C83—21no—
229C8×C83—21no—
230C8×C83—21no—
231C8×C83C2×#4111yescharacteristic
232C8×C83H(5,𝔽2)11yescharacteristic
233C22×C83—21no—
234C22×C83—21no—
235C22×C83—21no—
236C22×C83—21no—
237C22×C83—21no—
238C22×C83—81no—
239C22×C83—81no—
240C22×C83—81no—
241C22×C83#50912yes—
242C22×C83C22×D811yescharacteristic
243C2×D83—42no—
244C2×D83—81no—
245C2×D83—81no—
246C2×D83—81no—
247C2×D83—81no—
248C2×D83—81no—
249C2×D83—164no—
250C4×C83—21no—
251C4×C83—21no—
252C4×C83—21no—
253C4×C83—21no—
254C4×C83—81no—
255C4×C83—81no—
256C4×C83—81no—
257C4×C83#54611yescharacteristic
258C4×C83#57711yescharacteristic
259Dic83—81no—
260Dic83—81no—
261#33873—831nomaximal, Sylow
262C2×C83—22no—
263C2×C83—41no—
264C2×C83—161no—
265C2×C83#329411yescharacteristic
266C2×C83C2×#50911yescharacteristic
267C2×C83#341611yescharacteristic
268D83—82no—
269D83—161no—
270#3334D8311yescharacteristic, complemented
271#3280—831no—
272#2243—831no—
273#2235—831no—
274#2233—831no—
275#2226—831no—
276#2218—831no—
277#2217—831no—
278#2210—831no—
279#2186—831no—
280#2184—831no—
281#2177—831no—
282#2168—831no—
283#2139—831no—
284C2×#357—831no—
285C83#3387311yescharacteristic, complemented, Sylow
286C2×#135—831no—
287C22×D8—831no—
288#547—21no—
289#546—1661no—
290C2×SD32—831no—
291C2×D16—831no—
292C2×D16—831no—
293C2×M32—831no—
294C22×C16—831no—
295#500C2×D8311yescharacteristic, complemented
296#495—831no—
297#480—21no—
298#480—21no—
299#474—21no—
300#467—831no—
301#466—831no—
302#466C2×D8311yescharacteristic
303#466C2×D8311yescharacteristic, complemented
304#463—1661no—
305#450—1661no—
306#449—1661no—
307C4×D8—831no—
308C4×D8C2×D8311yescharacteristic, complemented
309C8×D4C2×D8311yescharacteristic
310C2×#107C2×D8311yescharacteristic
311C2×#107C2×D8311yescharacteristic, complemented
312#367—831no—
313#367—831no—
314#367—831no—
315C4⋊C16—831no—
316#358—831no—
317#358—831no—
318#358—831no—
319#357—831no—
320#357—831no—
321#357—831no—
322#357—831no—
323#357—831no—
324#357—831no—
325#357—831no—
326C22⋊C16—831no—
327C22⋊C16—831no—
328#346—831no—
329C2×#41—41no—
330C2×#41—1661no—
331C22×D4—1661no—
332#135—3321no—
333#135—1662no—
334C2×Q16—831no—
335C2×SD16—1661no—
336C2×SD16—1661no—
337C2×D8—3321no—
338C2×D8—831no—
339C2×D8—831no—
340C2×D8—1662no—
341C2×D8—1661no—
342C2×D8C22×D8311yescharacteristic
343C22×C8C22×D8311yescharacteristic
344C22×C8C22×D8311yescharacteristic
345C4⋊Q8—21no—
346#124—1661no—
347#123—41no—
348#123—41no—
349#123—3321no—
350C22⋊Q8—41no—
351C22⋊Q8—1661no—
352#121—21no—
353#121—21no—
354#121—3321no—
355#121—1661no—
356#120—3321no—
357C4×D4—21no—
358C4×D4—1661no—
359C4×D4C22⋊D8311yescharacteristic
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C22⋊C4—3321no—
363SD32—3321no—
364D16—3321no—
365D16—1662no—
366M32—3321no—
367C2×C16—3321no—
368C2×C16—831no—
369C2×C16—831no—
370C2×C16—831no—
371C2×C16—831no—
372#107—21no—
373#107—21no—
374#107—21no—
375#107—831no—
376#107C22×D8311yescharacteristic
377#107C22×D8311yescharacteristic
378#107C22×D8311yescharacteristic
379C4⋊C8C22⋊D8311yescharacteristic
380#103—1661no—
381#102—21no—
382#102—21no—
383#102—21no—
384#102—1661no—
385#102—1661no—
386C22⋊C8C22⋊D8311yescharacteristic
387C22⋊C8C22⋊D8311yescharacteristic
388C4×C8C22×D8311yescharacteristic
389C24—3321no—
390#41—84no—
391#41—3322no—
392#41—6641no—
393C2×Q8—41no—
394C2×Q8—1661no—
395C2×D4—21no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—3322no—
399C2×D4—3321no—
400C2×D4—3321no—
401C2×D4—3321no—
402C2×D4—6641no—
403C2×D4—1661no—
404C2×D4—1661no—
405C2×D4#1305057511yescharacteristic
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—3321no—
410C22×C4—3321no—
411C22×C4C2×C22⋊D8311yescharacteristic
412C22×C4C2×C22⋊D8311yescharacteristic
413Q16—1661no—
414Q16—1661no—
415SD16—3322no—
416SD16—6641no—
417D8—21no—
418D8—21no—
419D8—3322no—
420D8—832no—
421D8—1661no—
422D8—1661no—
423D8—1661no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×C22⋊D8311yescharacteristic
429C2×C8C2×C22⋊D8311yescharacteristic
430C2×C8C2×C22⋊D8311yescharacteristic
431C2×C8C23×D8311yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—3321no—
439C4⋊C4—1661no—
440C4⋊C4#1305057511yescharacteristic
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—3321no—
445C22⋊C4—3321no—
446C22⋊C4—3321no—
447C22⋊C4—3321no—
448C22⋊C4—3321no—
449C22⋊C4#1305057511yescharacteristic
450C22⋊C4#1305057511yescharacteristic
451C42C2×C22⋊D8311yescharacteristic
452C16—1661no—
453C16—1661no—
454C16—1661no—
455C16—1661no—
456C23—41no—
457C23—13281no—
458C23—3321no—
459C23—3321no—
460C23—6642no—
461C23—6642no—
462C23C2×#1305057511yescharacteristic
463C23C2×#1305057511yescharacteristic
464Q8—82no—
465Q8—3321no—
466Q8—3321no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—82no—
471D4—82no—
472D4—13281no—
473D4—3321no—
474D4—3321no—
475D4—3321no—
476D4—6642no—
477D4—6642no—
478D4—1662no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—13281no—
490C2×C4—3321no—
491C2×C4—3321no—
492C2×C4—6642no—
493C2×C4C2×#1305057511yescharacteristic
494C2×C4C2×#1305057511yescharacteristic
495C2×C4C2×#1305057511yescharacteristic
496C2×C4C2×#1305057511yescharacteristic
497C2×C4C22×C22⋊D8311yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8#41×D8311yescharacteristic
502C8#1458183811yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—13284no—
512C22—3322no—
513C22—6641no—
514C22—6641no—
515C22—6641no—
516C22—6641no—
517C22—6641no—
518C22#1912813712yes—
519C22C22×#1305057511yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—6641no—
528C4—6641no—
529C4#1912833911yescharacteristic
530C4#1912850911yescharacteristic
531C2—22no—
532C2—41no—
533C2—161no—
534C2—13281no—
535C2—6642no—
536C2#2491927011yescharacteristic
537C2C2×#1912813711yescharacteristic
538C2#2492060711yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#3334C2
socleC22×C83C22×D8
hypercenter#3334D83
hypocenterC83#33873
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #24919150C2semidirect
2. #24909702C2semidirect
3. #24909660C2semidirect
4. #24909648C2semidirect
5. #24909607C2semidirect
6. #24909564C2semidirect
7. #24909557C2semidirect
8. #24909518C2semidirect
9. #24909384C2semidirect
10. #24909374C2semidirect
11. #24909337C2semidirect
12. #24909289C2semidirect
13. #24909173C2semidirect
14. C2×#19127501C2semidirect
15. C83×#3334C2semidirect
19. C2×#14581736C22semidirect
22. C2×#14581732C22semidirect
31. #19127528C22semidirect
32. #19127528C22semidirect
33. #19127528C22semidirect
34. C4⋊(C83⋊C16)C22semidirect
35. #19127504C22semidirect
36. #19127504C22semidirect
37. #19127504C22semidirect
38. #19127502C22semidirect
46. C22⋊(C83⋊C16)C22semidirect
47. #19127477C22semidirect
49. #500×C83C22semidirect
54. #466×C83C22semidirect
55. C4×D8×C83C22semidirect
58. C2×#107×C83C22semidirect
140. D664#41semidirect
270. #3334D83semidirect
285. C83#33873semidirect
295. #500C2×D83semidirect
303. #466C2×D83semidirect
308. C4×D8C2×D83semidirect
311. C2×#107C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249191502
#249097022
#249096602
#249096482
#249096072
#249095642
#249095572
#249095182
#249093842
#249093742
#249093372
#249092892
#249091732
C2×#191275012
C83×#33342
#3387383