Finite Group Databasebeta

← #35561243

Subgroups — #35561243

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#24919150C211yesmaximal, characteristic
2#24909697C211yesmaximal, characteristic, complemented
3#24909660C211yesmaximal, characteristic, complemented
4#24909654C211yesmaximal, characteristic, complemented
5#24909617C211yesmaximal, characteristic, complemented
6#24909561C211yesmaximal, characteristic, complemented
7#24909552C211yesmaximal, characteristic, complemented
8#24909525C211yesmaximal, characteristic, complemented
9#24909401C211yesmaximal, characteristic, complemented
10#24909391C211yesmaximal, characteristic, complemented
11C4×#14581735C211yesmaximal, characteristic, complemented
12D4×C83⋊C16C211yesmaximal, characteristic, complemented
13#24909168C211yesmaximal, characteristic, complemented
14C2×#19127501C211yesmaximal, characteristic, complemented
15C83×#3336C211yesmaximal, characteristic, complemented
16C2×#14581814C2211yescharacteristic
17C22×D664C2211yescharacteristic
18#19128347—21no—
19C2×#14581738C2211yescharacteristic, complemented
20C2×#14581735C2211yescharacteristic
21C2×#14581735C2211yescharacteristic
22C22×C83⋊C16C2211yescharacteristic
23C22×C83⋊C16C2211yescharacteristic, complemented
24#19128085C2211yescharacteristic
25#19127942C2211yescharacteristic
26#19127937C2211yescharacteristic
27#19127917—21no—
28#19127854—21no—
29#19127848—21no—
30C4×D664C2211yescharacteristic
31#19127528C2211yescharacteristic, complemented
32#19127528C2211yescharacteristic, complemented
33#19127526C2211yescharacteristic, complemented
34C4⋊(C83⋊C16)C2211yescharacteristic, complemented
35#19127505C2211yescharacteristic, complemented
36#19127504C2211yescharacteristic, complemented
37#19127504C2211yescharacteristic, complemented
38#19127504C2211yescharacteristic, complemented
39#19127501C2211yescharacteristic
40#19127501C2211yescharacteristic
41#19127501C2211yescharacteristic
42#19127501C2211yescharacteristic
43#19127501C2211yescharacteristic
44#19127501C2211yescharacteristic
45C22⋊(C83⋊C16)C2211yescharacteristic
46C22⋊(C83⋊C16)C2211yescharacteristic, complemented
47C4×C83⋊C16C2211yescharacteristic, complemented
48#552×C83—21no—
49#499×C83C2211yescharacteristic, complemented
50#485×C83—21no—
51#483×C83—21no—
52#479×C83—21no—
53#467×C83C2211yescharacteristic
54#467×C83C2211yescharacteristic, complemented
55C4×Q16×C83C2211yescharacteristic, complemented
56C8×D4×C83C2211yescharacteristic
57#427×C83C2211yescharacteristic, complemented
58C2×#107×C83C2211yescharacteristic
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#14581738—41no—
80#14581735—22no—
81#14581735—41no—
82C2×C83⋊C16—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×Q16×C83C2311yescharacteristic
93C22×C8×C83C2311yescharacteristic
94C22×C8×C83C2311yescharacteristic
95#126×C83—41no—
96#125×C83—21no—
97#125×C83—41no—
98#123×C83—41no—
99C22⋊Q8×C83—21no—
100C22⋊Q8×C83—21no—
101C4×Q8×C83—21no—
102C4×D4×C83D411yescharacteristic
103#117×C83—21no—
104#117×C83—41no—
105C2×C4⋊C4×C83—21no—
106#107×C83—21no—
107#107×C83C2311yescharacteristic
108#107×C83C2311yescharacteristic
109#107×C83C2311yescharacteristic
110#106×C83—21no—
111#106×C83—21no—
112C4⋊C8×C83D411yescharacteristic
113#103×C83—21no—
114#103×C83—21no—
115#103×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#4112yes—
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—
152C2×Q8×C83—21no—
153C2×D4×C83D811yescharacteristic
154C22×C4×C83—41no—
155C22×C4×C83C2×D411yescharacteristic
156C22×C4×C83C2×D411yescharacteristic
157Q16×C83—21no—
158Q16×C83—21no—
159C2×C8×C83—21no—
160C2×C8×C83—21no—
161C2×C8×C83—21no—
162C2×C8×C83—21no—
163C2×C8×C83C2×D411yescharacteristic
164C2×C8×C83C2×D411yescharacteristic
165C2×C8×C83C2×D411yescharacteristic
166C2×C8×C83C2411yescharacteristic
167C4⋊C4×C83—21no—
168C4⋊C4×C83—21no—
169C4⋊C4×C83—21no—
170C4⋊C4×C83—41no—
171C4⋊C4×C83—41no—
172C4⋊C4×C83—41no—
173C4⋊C4×C83—41no—
174C4⋊C4×C83—41no—
175C4⋊C4×C83—41no—
176C4⋊C4×C83D811yescharacteristic
177C22⋊C4×C83—41no—
178C22⋊C4×C83—41no—
179C22⋊C4×C83—41no—
180C22⋊C4×C83D811yescharacteristic
181C22⋊C4×C83D811yescharacteristic
182C42×C83—41no—
183C42×C83—41no—
184C42×C83C2×D411yescharacteristic
185C23×C83C2×D811yescharacteristic
186C23×C83C2×D811yescharacteristic
187C22×D83—41no—
188C22×D83—41no—
189C22×D83—82no—
190C22×D83—82no—
191C22×D83—161no—
192Q8×C83—41no—
193Q8×C83—41no—
194D4×C83—22no—
195C2×C4×C83—21no—
196C2×C4×C83—21no—
197C2×C4×C83—21no—
198C2×C4×C83—21no—
199C2×C4×C83—41no—
200C2×C4×C83—41no—
201C2×C4×C83—41no—
202C2×C4×C83—41no—
203C2×C4×C83—82no—
204C2×C4×C83C2×D811yescharacteristic
205C2×C4×C83C2×D811yescharacteristic
206C2×C4×C83C2×D811yescharacteristic
207C2×C4×C83C2×D811yescharacteristic
208C2×C4×C83C22×D411yescharacteristic
209C22⋊D83—82no—
210C22⋊D83—161no—
211C2×Dic83—41no—
212C2×Dic83—41no—
213D332—22no—
214D332—41no—
215D332—41no—
216D332—41no—
217D332—82no—
218C4×D83—82no—
219C4×D83—161no—
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#50812yes—
233C22×C83C22×D811yescharacteristic
234C2×D83—42no—
235C2×D83—81no—
236C2×D83—81no—
237C2×D83—81no—
238C2×D83—81no—
239C2×D83—81no—
240C2×D83—164no—
241C4×C83—21no—
242C4×C83—21no—
243C4×C83—21no—
244C4×C83—21no—
245C4×C83—81no—
246C4×C83—81no—
247C4×C83—81no—
248C4×C83—81no—
249C4×C83#54611yescharacteristic
250C4×C83#57711yescharacteristic
251Dic83—81no—
252Dic83—81no—
253#33863—831nomaximal, Sylow
254C2×C83—22no—
255C2×C83—41no—
256C2×C83#329411yescharacteristic
257C2×C83C2×#50811yescharacteristic
258C2×C83#341611yescharacteristic
259D83—82no—
260D83—161no—
261#3336D8311yescharacteristic, complemented
262#3280—831no—
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×#135—831no—
278C22×D8—831no—
279#552—21no—
280#546—1661no—
281C2×Q32—831no—
282C2×SD32—831no—
283C2×SD32—831no—
284C22×C16—831no—
285C22×C16—831no—
286#499C2×D8311yescharacteristic, complemented
287#495—831no—
288#485—21no—
289#483—21no—
290#479—21no—
291#467—831no—
292#467C2×D8311yescharacteristic
293#467C2×D8311yescharacteristic, complemented
294#466—831no—
295#463—1661no—
296#450—1661no—
297#449—1661no—
298C4×Q16C2×D8311yescharacteristic, complemented
299C4×D8—831no—
300C8×D4C2×D8311yescharacteristic
301#427C2×D8311yescharacteristic, complemented
302C2×#107C2×D8311yescharacteristic
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—1661no—
321C22×D4—1661no—
322#135—3321no—
323#135—1662no—
324C2×Q16—831no—
325C2×Q16C22×D8311yescharacteristic
326C2×SD16—1661no—
327C2×SD16—1661no—
328C2×D8—3321no—
329C2×D8—831no—
330C2×D8—831no—
331C2×D8—1662no—
332C2×D8—1661no—
333C22×C8C22×D8311yescharacteristic
334C22×C8C22×D8311yescharacteristic
335#126—41no—
336#125—21no—
337#125—41no—
338#124—1661no—
339#123—41no—
340#123—3321no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—1661no—
344#121—3321no—
345#121—1661no—
346#120—3321no—
347C4×Q8—21no—
348C4×D4—1661no—
349C4×D4C22⋊D8311yescharacteristic
350#117—21no—
351#117—41no—
352C2×C4⋊C4—21no—
353C2×C22⋊C4—3321no—
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—21no—
364#107—831no—
365#107C22×D8311yescharacteristic
366#107C22×D8311yescharacteristic
367#107C22×D8311yescharacteristic
368#106—21no—
369#106—21no—
370C4⋊C8C22⋊D8311yescharacteristic
371#103—21no—
372#103—21no—
373#103—21no—
374#103—1661no—
375#102—1661no—
376#102—1661no—
377C22⋊C8C22⋊D8311yescharacteristic
378C22⋊C8C22⋊D8311yescharacteristic
379C4×C8C22×D8311yescharacteristic
380C24—3321no—
381#41—3322no—
382#41—6641no—
383C2×Q8—21no—
384C2×Q8—1661no—
385C2×D4—3322no—
386C2×D4—3321no—
387C2×D4—3321no—
388C2×D4—3321no—
389C2×D4—6641no—
390C2×D4—1661no—
391C2×D4—1661no—
392C2×D4#1305057511yescharacteristic
393C22×C4—41no—
394C22×C4—3321no—
395C22×C4—3321no—
396C22×C4C2×C22⋊D8311yescharacteristic
397C22×C4C2×C22⋊D8311yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—1661no—
401Q16—1661no—
402SD16—3322no—
403SD16—6641no—
404D8—3322no—
405D8—832no—
406D8—1661no—
407D8—1661no—
408D8—1661no—
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—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—3321no—
427C4⋊C4—1661no—
428C4⋊C4#1305057511yescharacteristic
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—3321no—
433C22⋊C4—3321no—
434C22⋊C4—3321no—
435C22⋊C4—3321no—
436C22⋊C4—3321no—
437C22⋊C4#1305057511yescharacteristic
438C22⋊C4#1305057511yescharacteristic
439C42—41no—
440C42—41no—
441C42C2×C22⋊D8311yescharacteristic
442C16—1661no—
443C16—1661no—
444C16—1661no—
445C16—1661no—
446C23—13281no—
447C23—3321no—
448C23—3321no—
449C23—6642no—
450C23—6642no—
451C23C2×#1305057511yescharacteristic
452C23C2×#1305057511yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—3321no—
456Q8—3321no—
457D4—22no—
458D4—13281no—
459D4—3321no—
460D4—3321no—
461D4—3321no—
462D4—6642no—
463D4—6642no—
464D4—1662no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—13281no—
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—
486C8#41×D8311yescharacteristic
487C8#1458183811yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—13284no—
494C22—3322no—
495C22—6641no—
496C22—6641no—
497C22—6641no—
498C22—6641no—
499C22—6641no—
500C22#1912813112yes—
501C22C22×#1305057511yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—6641no—
511C4—6641no—
512C4#1912833911yescharacteristic
513C4#1912850911yescharacteristic
514C2—22no—
515C2—41no—
516C2—13281no—
517C2—6642no—
518C2#2491927011yescharacteristic
519C2C2×#1912813111yescharacteristic
520C2#2492060711yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#3336C2
socleC22×C83C22×D8
prime radical#24919150C2
hypercenter#3336D83
hypocenterC83#33863
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #24909697C2semidirect
3. #24909660C2semidirect
4. #24909654C2semidirect
5. #24909617C2semidirect
6. #24909561C2semidirect
7. #24909552C2semidirect
8. #24909525C2semidirect
9. #24909401C2semidirect
10. #24909391C2semidirect
11. C4×#14581735C2semidirect
12. D4×C83⋊C16C2semidirect
13. #24909168C2semidirect
14. C2×#19127501C2semidirect
15. C83×#3336C2semidirect
19. C2×#14581738C22semidirect
23. C22×C83⋊C16C22semidirect
31. #19127528C22semidirect
32. #19127528C22semidirect
33. #19127526C22semidirect
34. C4⋊(C83⋊C16)C22semidirect
35. #19127505C22semidirect
36. #19127504C22semidirect
37. #19127504C22semidirect
38. #19127504C22semidirect
46. C22⋊(C83⋊C16)C22semidirect
47. C4×C83⋊C16C22semidirect
49. #499×C83C22semidirect
54. #467×C83C22semidirect
55. C4×Q16×C83C22semidirect
57. #427×C83C22semidirect
261. #3336D83semidirect
276. C83#33863semidirect
286. #499C2×D83semidirect
293. #467C2×D83semidirect
298. C4×Q16C2×D83semidirect
301. #427C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249191502
#249096972
#249096602
#249096542
#249096172
#249095612
#249095522
#249095252
#249094012
#249093912
C4×#145817352
D4×C83⋊C162
#249091682
C2×#191275012
C83×#33362
#3386383