Finite Group Databasebeta

← #35561233

Subgroups — #35561233

Order 21248.

622 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24919852C211yesmaximal, characteristic, complemented
2#24919137C211yesmaximal, characteristic
3#24909685C211yesmaximal, characteristic, complemented
4#24909650C211yesmaximal, characteristic, complemented
5#24909646C211yesmaximal, characteristic, complemented
6#24909609C211yesmaximal, characteristic, complemented
7#24909562C211yesmaximal, characteristic, complemented
8#24909501C211yesmaximal, characteristic, complemented
9#24909391C211yesmaximal, characteristic, complemented
10#24909390C211yesmaximal, characteristic, complemented
11#24909312C211yesmaximal, characteristic, complemented
12C16×C22⋊D83C211yesmaximal, characteristic, complemented
13C2×#19127503C211yesmaximal, characteristic, complemented
14#357×D83C211yesmaximal, characteristic, complemented
15C83×#2217C211yesmaximal, characteristic, complemented
16C22×D664C2211yescharacteristic
17C2×D8×D83C2211yescharacteristic
18#19128409—21no—
19#19128335—21no—
20C2×#14581735C2211yescharacteristic
21C2×#14581735C2211yescharacteristic, complemented
22C2×C16×D83C2211yescharacteristic, complemented
23#19128105C2211yescharacteristic, complemented
24#19128076C2211yescharacteristic
25#19128035—21no—
26#19128034—21no—
27#19127994—21no—
28#19127939C2211yescharacteristic, complemented
29#19127938C2211yescharacteristic
30#19127895—21no—
31#19127836—21no—
32#19127834—21no—
33Q16×Dic83C2211yescharacteristic, complemented
34#19127783C2211yescharacteristic
35C8×C22⋊D83C2211yescharacteristic
36C2×#14581726C2211yescharacteristic, complemented
37#107×D83C2211yescharacteristic, complemented
38#19127528C2211yescharacteristic, complemented
39#19127528C2211yescharacteristic, complemented
40Dic83⋊C16C2211yescharacteristic, complemented
41#19127503C2211yescharacteristic
42#19127503C2211yescharacteristic
43#19127503C2211yescharacteristic, complemented
44#19127503C2211yescharacteristic, complemented
45#19127502C2211yescharacteristic, complemented
46#19127502C2211yescharacteristic, complemented
47#19127501C2211yescharacteristic
48#19127501C2211yescharacteristic, complemented
49C22⋊(C83⋊C16)C2211yescharacteristic, complemented
50(C2×D83)⋊C16C2211yescharacteristic, complemented
51C16×Dic83C2211yescharacteristic, complemented
52C2×SD32×C83C2211yescharacteristic, complemented
53C22×C16×C83C2211yescharacteristic, complemented
54#467×C83C2211yescharacteristic, complemented
55#466×C83C2211yescharacteristic
56#367×C83C2211yescharacteristic, complemented
57#358×C83C2211yescharacteristic, complemented
58#357×C83C2211yescharacteristic, complemented
59C22×D332—21no—
60C2×D4×D83—21no—
61C2×D4×D83—41no—
62C2×D664—22no—
63C2×D664—21no—
64C2×D664—41no—
65C2×D664C2311yescharacteristic
66C2×D664C2311yescharacteristic
67C2×#13050575—21no—
68C2×#13050575—21no—
69D8×D83—22no—
70D8×D83—41no—
71C2×C8×D83C2311yescharacteristic
72#14581791—21no—
73#14581788—21no—
74#14581782—41no—
75#14581772—41no—
76#14581769—41no—
77#14581768—21no—
78#14581767—21no—
79#14581765—41no—
80#14581762—41no—
81#14581760—41no—
82Q8×Dic83—21no—
83#14581754—21no—
84C4×C22⋊D83D411yescharacteristic
85(C22⋊D83)⋊C4—41no—
86C2×C4⋊Dic83—21no—
87C4⋊C4×D83—21no—
88#14581735—22no—
89#14581735—41no—
90C2×C83⋊C16C2311yescharacteristic
91C2×C83⋊C16C2311yescharacteristic
92C16×D83—41no—
93#14581726—21no—
94#14581726C2311yescharacteristic
95#14581726C2311yescharacteristic
96#14581725—21no—
97#14581725—21no—
98Dic83⋊C8D411yescharacteristic
99#14581716—21no—
100#14581716—21no—
101#14581712—21no—
102#14581712—21no—
103C22⋊(C83⋊C8)D411yescharacteristic
104(C2×D83)⋊C8D411yescharacteristic
105C8×Dic83C2311yescharacteristic
106C2×Q16×C83C2311yescharacteristic
107C2×D8×C83C2311yescharacteristic
108C22×C8×C83C2311yescharacteristic
109C22⋊Q8×C83—21no—
110#121×C83—21no—
111SD32×C83—41no—
112C2×C16×C83—41no—
113C2×C16×C83C2311yescharacteristic
114C2×C16×C83C2311yescharacteristic
115#107×C83C2311yescharacteristic
116#107×C83C2311yescharacteristic
117#103×C83—21no—
118#102×C83—21no—
119C23×D83—41no—
120C23×D83—41no—
121C2×Dic166—41no—
122C2×D332—21no—
123C2×D332—21no—
124C2×D332—42no—
125C2×D332—41no—
126C2×D332—41no—
127C2×D332—81no—
128C2×C22⋊D83—41no—
129C2×C22⋊D83—41no—
130C2×C22⋊D83—41no—
131C2×C22⋊D83D811yescharacteristic
132D4×D83—42no—
133D4×D83—81no—
134D4×D83—84no—
135C22×Dic83—41no—
136C2×C4×D83—41no—
137C2×C4×D83—41no—
138C2×C4×D83C2×D411yescharacteristic
139D664—21no—
140D664—21no—
141D664—21no—
142D664—42no—
143D664#4112yes—
144#13050575—42no—
145#13050575—81no—
146C2×C83⋊C8C2×D411yescharacteristic
147C2×C83⋊C8C2×D411yescharacteristic
148C8×D83—21no—
149C8×D83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—41no—
153C4⋊Dic83—41no—
154Dic83⋊C4—41no—
155Dic83⋊C4—41no—
156Dic83⋊C4—41no—
157Dic83⋊C4D811yescharacteristic
158C22⋊Dic83D811yescharacteristic
159(C2×D83)⋊C4—41no—
160(C2×D83)⋊C4—41no—
161(C2×D83)⋊C4—41no—
162(C2×D83)⋊C4D811yescharacteristic
163C4×Dic83—41no—
164C4×Dic83C2×D411yescharacteristic
165C83⋊C16—21no—
166C83⋊C16—21no—
167C2×Q8×C83—21no—
168C2×D4×C83—21no—
169C2×D4×C83—41no—
170C22×C4×C83C2×D411yescharacteristic
171Q16×C83—21no—
172Q16×C83—21no—
173D8×C83—21no—
174D8×C83—21no—
175C2×C8×C83—21no—
176C2×C8×C83—21no—
177C2×C8×C83C2×D411yescharacteristic
178C2×C8×C83C2411yescharacteristic
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—41no—
182C22⋊C4×C83—41no—
183C22⋊C4×C83—41no—
184C16×C83—21no—
185C16×C83—21no—
186C23×C83—41no—
187C23×C83C2×D811yescharacteristic
188C22×D83—41no—
189C22×D83—41no—
190C22×D83—82no—
191C22×D83—82no—
192C22×D83—82no—
193C22×D83—82no—
194C22×D83—161no—
195C22×D83—161no—
196C22×D83C2×D811yescharacteristic
197Q8×C83—41no—
198Q8×C83—41no—
199D4×C83—41no—
200D4×C83—41no—
201D4×C83—82no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—41no—
205C2×C4×C83—41no—
206C2×C4×C83—41no—
207C2×C4×C83C2×D811yescharacteristic
208C2×C4×C83C22×D411yescharacteristic
209C22⋊D83—82no—
210C22⋊D83—82no—
211C22⋊D83—161no—
212C22⋊D83SD3214yescomplemented
213C2×Dic83—41no—
214C2×Dic83—41no—
215C2×Dic83—161no—
216C2×Dic83C2×D811yescharacteristic
217C2×Dic83C2×D811yescharacteristic
218C2×Dic83C2×D811yescharacteristic
219D332—22no—
220D332—41no—
221D332—41no—
222D332—41no—
223D332—82no—
224D332—82no—
225C4×D83—21no—
226C4×D83—21no—
227C4×D83—82no—
228C4×D83—161no—
229Dic166—161no—
230C8×C83—21no—
231C8×C83C2×#4111yescharacteristic
232C8×C83H(5,𝔽2)11yescharacteristic
233C83⋊C8—21no—
234C83⋊C8—21no—
235C22×C83—21no—
236C22×C83—21no—
237C22×C83—81no—
238C22×C83—81no—
239C22×C83—81no—
240C22×C83C2×SD3212yes—
241C22×C83C22×D811yescharacteristic
242C2×D83—21no—
243C2×D83—21no—
244C2×D83—42no—
245C2×D83—81no—
246C2×D83—81no—
247C2×D83—81no—
248C2×D83—81no—
249C2×D83—81no—
250C2×D83—164no—
251C2×D83—164no—
252C2×D83C2×SD3212yes—
253C4×C83—21no—
254C4×C83—81no—
255C4×C83—81no—
256C4×C83—81no—
257C4×C83#54611yescharacteristic
258C4×C83#57711yescharacteristic
259Dic83—21no—
260Dic83—21no—
261Dic83—81no—
262Dic83—81no—
263Dic83C2×SD3212yes—
264#33862—831nomaximal, Sylow
265C2×C83—22no—
266C2×C83—161no—
267C2×C83#329411yescharacteristic
268C2×C83C22×SD3211yescharacteristic
269C2×C83#341511yescharacteristic
270D83—22no—
271D83—82no—
272D83—161no—
273#3338—831no—
274D4×D8—831no—
275#2240—831no—
276#2233—831no—
277#2233—831no—
278#2226—831no—
279#2217—831no—
280#2217D8311yescharacteristic, complemented
281#2207—831no—
282#2187—831no—
283#2187—831no—
284C4×SD32—831no—
285D4×C16—831no—
286C2×#357—831no—
287C2×#357—831no—
288C83#3386211yescharacteristic, complemented, Sylow
289C22×D8—831no—
290C22×D8—831no—
291#554—1661no—
292D42—1661no—
293C2×SD32—831no—
294C2×SD32—831no—
295C2×SD32C2×D8311yescharacteristic, complemented
296C22×C16—831no—
297C22×C16C2×D8311yescharacteristic, complemented
298#500—831no—
299#493—831no—
300#484—1661no—
301#484—1661no—
302#477—1661no—
303#467—831no—
304#467C2×D8311yescharacteristic, complemented
305#466—831no—
306#466C2×D8311yescharacteristic
307#459—1661no—
308#447—1661no—
309#447—1661no—
310C4×Q16—831no—
311C4×D8—831no—
312C8×D4—831no—
313C2×#107—831no—
314C2×#107—831no—
315#367—831no—
316#367—831no—
317#367C2×D8311yescharacteristic, complemented
318C4⋊C16—831no—
319#358C2×D8311yescharacteristic, complemented
320#357—831no—
321#357—831no—
322#357—831no—
323#357—831no—
324#357—831no—
325#357—831no—
326#357—831no—
327#357—831no—
328#357C2×D8311yescharacteristic, complemented
329C22⋊C16—831no—
330C22⋊C16—831no—
331C4×C16—831no—
332C22×D4—3321no—
333C22×D4—1661no—
334C22×D4—1661no—
335C2×Q16C22×D8311yescharacteristic
336C2×D8—3321no—
337C2×D8—3321no—
338C2×D8—831no—
339C2×D8—831no—
340C2×D8—1662no—
341C2×D8—1661no—
342C2×D8—1661no—
343C2×D8—1662no—
344C2×D8—1661no—
345C2×D8C22×D8311yescharacteristic
346C22×C8—831no—
347C22×C8C22×D8311yescharacteristic
348C4⋊Q8—1661no—
349#127—1661no—
350#125—3321no—
351C22⋊Q8—21no—
352C22⋊Q8—3321no—
353C22⋊Q8—3321no—
354C22⋊Q8—1661no—
355#121—21no—
356#121—3321no—
357#121—1661no—
358#120—3321no—
359#120—3321no—
360C4×Q8—1661no—
361C4×D4—3321no—
362C4×D4—831no—
363C4×D4—1661no—
364C2×C4⋊C4—1661no—
365C2×C4⋊C4—1661no—
366SD32—41no—
367SD32—3321no—
368SD32—1662no—
369C2×C16—41no—
370C2×C16—3321no—
371C2×C16—831no—
372C2×C16—831no—
373C2×C16C22×D8311yescharacteristic
374C2×C16C22×D8311yescharacteristic
375#107—831no—
376#107—831no—
377#107—1661no—
378#107—1661no—
379#107—1661no—
380#107C22×D8311yescharacteristic
381#107C22×D8311yescharacteristic
382C4⋊C8—831no—
383#103—21no—
384#103—1661no—
385#103—1661no—
386#102—21no—
387#102—1661no—
388#102—1661no—
389C22⋊C8—831no—
390C22⋊C8—831no—
391C4×C8—831no—
392C24—3321no—
393C24—3321no—
394C2×Q8—21no—
395C2×Q8—3321no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—3322no—
399C2×D4—3321no—
400C2×D4—3321no—
401C2×D4—3321no—
402C2×D4—3321no—
403C2×D4—3322no—
404C2×D4—3321no—
405C2×D4—831no—
406C2×D4—6641no—
407C2×D4—6644no—
408C2×D4—6641no—
409C2×D4—1661no—
410C2×D4—1661no—
411C22×C4—3321no—
412C22×C4—3321no—
413C22×C4—3321no—
414C22×C4—831no—
415C22×C4D4×D8311yescharacteristic
416Q16—21no—
417Q16—21no—
418D8—21no—
419D8—21no—
420D8—3322no—
421D8—3322no—
422D8—832no—
423D8—6641no—
424D8—1661no—
425D8—1661no—
426D8—1661no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—831no—
430C2×C8—831no—
431C2×C8—1661no—
432C2×C8—1661no—
433C2×C8D4×D8311yescharacteristic
434C2×C8C23×D8311yescharacteristic
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—41no—
438C4⋊C4—3321no—
439C4⋊C4—3321no—
440C4⋊C4—3321no—
441C4⋊C4—3321no—
442C4⋊C4—3321no—
443C4⋊C4—831no—
444C4⋊C4—1661no—
445C4⋊C4—1661no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—3321no—
449C22⋊C4—3321no—
450C22⋊C4—3321no—
451C22⋊C4—831no—
452C22⋊C4—831no—
453C42—3321no—
454C42—831no—
455C16—21no—
456C16—21no—
457C16—1661no—
458C16—1661no—
459C23—41no—
460C23—13281no—
461C23—13281no—
462C23—3321no—
463C23—3321no—
464C23—831no—
465C23—6642no—
466C23—6642no—
467C23—6642no—
468C23—6642no—
469C23D8×D8311yescharacteristic
470Q8—41no—
471Q8—41no—
472Q8—13281no—
473D4—41no—
474D4—41no—
475D4—82no—
476D4—13281no—
477D4—3321no—
478D4—3321no—
479D4—3321no—
480D4—834no—
481D4—6642no—
482D4—6642no—
483D4—6642no—
484D4—6642no—
485D4—1662no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—13281no—
492C2×C4—13281no—
493C2×C4—3321no—
494C2×C4—3321no—
495C2×C4—831no—
496C2×C4—831no—
497C2×C4—831no—
498C2×C4—6642no—
499C2×C4—1661no—
500C2×C4—1661no—
501C2×C4D8×D8311yescharacteristic
502C2×C4C2×D4×D8311yescharacteristic
503C8—21no—
504C8—1661no—
505C8—1661no—
506C8#41×D8311yescharacteristic
507C8#1458183711yescharacteristic
508C22—21no—
509C22—21no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—13284no—
514C22—13284no—
515C22—3322no—
516C22—832no—
517C22—6641no—
518C22—6641no—
519C22—6641no—
520C22—6641no—
521C22—6641no—
522C22—1661no—
523C22—1661no—
524C22SD32×D8312yes—
525C22C2×D8×D8311yescharacteristic
526C4—21no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—832no—
531C4—6641no—
532C4—6641no—
533C4—1661no—
534C4—1661no—
535C4#1912834011yescharacteristic
536C4#1912850811yescharacteristic
537C2—22no—
538C2—161no—
539C2—13281no—
540C2—6642no—
541C2—1662no—
542C2#2491927111yescharacteristic
543C2C2×SD32×D8311yescharacteristic
544C2#2492060111yescharacteristic
subgroupG/H
centerC22C2×D8×D83
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#2217C2
socleC22×C83C22×D8
prime radical#24919137C2
hypercenter#2217D83
hypocenterC83#33862
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. #24919852C2semidirect
3. #24909685C2semidirect
4. #24909650C2semidirect
5. #24909646C2semidirect
6. #24909609C2semidirect
7. #24909562C2semidirect
8. #24909501C2semidirect
9. #24909391C2semidirect
10. #24909390C2semidirect
11. #24909312C2semidirect
12. C16×C22⋊D83C2semidirect
13. C2×#19127503C2semidirect
14. #357×D83C2semidirect
15. C83×#2217C2semidirect
21. C2×#14581735C22semidirect
22. C2×C16×D83C22semidirect
23. #19128105C22semidirect
28. #19127939C22semidirect
33. Q16×Dic83C22semidirect
36. C2×#14581726C22semidirect
37. #107×D83C22semidirect
38. #19127528C22semidirect
39. #19127528C22semidirect
40. Dic83⋊C16C22semidirect
43. #19127503C22semidirect
44. #19127503C22semidirect
45. #19127502C22semidirect
46. #19127502C22semidirect
48. #19127501C22semidirect
49. C22⋊(C83⋊C16)C22semidirect
50. (C2×D83)⋊C16C22semidirect
51. C16×Dic83C22semidirect
52. C2×SD32×C83C22semidirect
53. C22×C16×C83C22semidirect
54. #467×C83C22semidirect
56. #367×C83C22semidirect
57. #358×C83C22semidirect
58. #357×C83C22semidirect
212. C22⋊D83SD32semidirect
280. #2217D83semidirect
288. C83#33862semidirect
295. C2×SD32C2×D83semidirect
297. C22×C16C2×D83semidirect
304. #467C2×D83semidirect
317. #367C2×D83semidirect
319. #358C2×D83semidirect
328. #357C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249198522
#249191372
#249096852
#249096502
#249096462
#249096092
#249095622
#249095012
#249093912
#249093902
#249093122
C16×C22⋊D832
C2×#191275032
#357×D832
C83×#22172
#3386283