Finite Group Databasebeta

← #35559506

Subgroups — #35559506

Order 21248.

568 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916316C211yesmaximal, characteristic, complemented
2#24909668C211yesmaximal, characteristic, complemented
3#24909651C211yesmaximal, characteristic, complemented
4#24909607C211yesmaximal, characteristic, complemented
5#24909597C211yesmaximal, characteristic, complemented
6#24909515C211yesmaximal, characteristic, complemented
7#24909515C211yesmaximal, characteristic, complemented
8#24909391C211yesmaximal, characteristic, complemented
9#24909374C211yesmaximal, characteristic
10#24909337C211yesmaximal, characteristic, complemented
11#24909325C211yesmaximal, characteristic, complemented
12C4×#14581735C211yesmaximal, characteristic, complemented
13C4×#14581733C211yesmaximal, characteristic, complemented
14#24909270C211yesmaximal, characteristic, complemented
15C83×#3175C211yesmaximal, characteristic, complemented
16C22×D664C2211yescharacteristic
17#19128175—21no—
18C2×#14581735C2211yescharacteristic
19C2×#14581735C2211yescharacteristic
20C2×#14581733C2211yescharacteristic
21C2×#14581733C2211yescharacteristic
22#19127814—21no—
23#19127814—21no—
24C4×D664C2211yescharacteristic
25C4×D664C2211yescharacteristic
26C4×D664C2211yescharacteristic
27C4×D664C2211yescharacteristic
28#19127739C2211yescharacteristic
29#19127707—21no—
30#19127528C2211yescharacteristic, complemented
31#19127528C2211yescharacteristic, complemented
32#19127526C2211yescharacteristic, complemented
33#19127526C2211yescharacteristic, complemented
34C4⋊(C83⋊C16)C2211yescharacteristic, complemented
35C4⋊(C83⋊C16)C2211yescharacteristic, complemented
36#19127505C2211yescharacteristic, complemented
37#19127505C2211yescharacteristic, complemented
38#19127502C2211yescharacteristic, complemented
39#19127502C2211yescharacteristic, complemented
40#19127501C2211yescharacteristic
41#19127501C2211yescharacteristic
42#19127501C2211yescharacteristic
43#19127501C2211yescharacteristic
44C22⋊(C83⋊C16)C2211yescharacteristic
45C22⋊(C83⋊C16)C2211yescharacteristic
46#19127477C2211yescharacteristic, complemented
47C4×C83⋊C16C2211yescharacteristic, complemented
48#542×C83—21no—
49#500×C83C2211yescharacteristic, complemented
50#495×C83C2211yescharacteristic, complemented
51#489×C83—21no—
52#468×C83—21no—
53#467×C83C2211yescharacteristic
54#466×C83C2211yescharacteristic
55#440×C83—21no—
56C4×Q16×C83C2211yescharacteristic, complemented
57C4×D8×C83C2211yescharacteristic, complemented
58#405×C83C2211yescharacteristic
59C22×D332—21no—
60C2×D664—22no—
61C2×D664—22no—
62C2×D664—21no—
63C2×D664C2311yescharacteristic
64C2×D664C2311yescharacteristic
65C4×D332—21no—
66C4×D332—21no—
67C4×D332—21no—
68C4×D332—21no—
69#14581754—41no—
70#14581754—41no—
71#14581751—21no—
72C22⋊C4×D83—41no—
73C22⋊C4×D83—41no—
74#14581735—22no—
75#14581735—22no—
76#14581733—22no—
77#14581733—22no—
78C2×C83⋊C16C2311yescharacteristic
79C2×C83⋊C16C2311yescharacteristic
80C2×C83⋊C16C2311yescharacteristic
81C2×C83⋊C16C2311yescharacteristic
82#14581726C2311yescharacteristic
83#14581726C2311yescharacteristic
84#14581724—21no—
85#14581714—21no—
86#14581714—21no—
87#14581714—21no—
88#14581714—21no—
89C2×Q16×C83C2311yescharacteristic
90C2×SD16×C83—21no—
91C2×D8×C83C2311yescharacteristic
92C2×M16×C83D411yescharacteristic
93C22×C8×C83C2311yescharacteristic
94C4⋊Q8×C83—21no—
95#126×C83—41no—
96#124×C83—21no—
97#124×C83—41no—
98#123×C83—41no—
99C22⋊Q8×C83—21no—
100C22⋊Q8×C83—41no—
101#121×C83—21no—
102C4×Q8×C83—21no—
103C4×D4×C83—21no—
104#117×C83D411yescharacteristic
105#107×C83—21no—
106#107×C83C2311yescharacteristic
107#107×C83C2311yescharacteristic
108#103×C83—21no—
109#103×C83—21no—
110#102×C83—21no—
111#102×C83—21no—
112#97×C83D411yescharacteristic
113#97×C83D411yescharacteristic
114C4×C8×C83C2311yescharacteristic
115C4×C8×C83C2311yescharacteristic
116C23×D83—41no—
117C2×D332—24no—
118C2×D332—21no—
119C2×D332—21no—
120C2×D332—42no—
121C2×D332—41no—
122C2×C4×D83—41no—
123C2×C4×D83—41no—
124C2×C4×D83—41no—
125C2×C4×D83—41no—
126D664—24no—
127D664—21no—
128D664—21no—
129D664#4112yes—
130D664#4112yes—
131C4⋊Dic83—21no—
132C4⋊Dic83—21no—
133C22⋊Dic83—41no—
134(C2×D83)⋊C4—41no—
135(C2×D83)⋊C4—41no—
136(C2×D83)⋊C4—41no—
137(C2×D83)⋊C4—41no—
138C4×Dic83—41no—
139C83⋊C16—21no—
140C83⋊C16—21no—
141C83⋊C16—21no—
142C83⋊C16—21no—
143C2×Q8×C83—21no—
144C2×Q8×C83—41no—
145C2×D4×C83—21no—
146C2×D4×C83—41no—
147C22×C4×C83—41no—
148C22×C4×C83C2×D411yescharacteristic
149Q16×C83—21no—
150Q16×C83—21no—
151SD16×C83—42no—
152D8×C83—21no—
153D8×C83—21no—
154M16×C83—41no—
155C2×C8×C83—21no—
156C2×C8×C83—21no—
157C2×C8×C83C2×D411yescharacteristic
158C2×C8×C83C2×D411yescharacteristic
159C2×C8×C83C2×D411yescharacteristic
160C2×C8×C83C2411yescharacteristic
161C4⋊C4×C83—21no—
162C4⋊C4×C83—21no—
163C4⋊C4×C83—41no—
164C4⋊C4×C83—41no—
165C4⋊C4×C83—41no—
166C4⋊C4×C83D811yescharacteristic
167C4⋊C4×C83D811yescharacteristic
168C22⋊C4×C83—41no—
169C22⋊C4×C83—41no—
170C22⋊C4×C83—41no—
171C22⋊C4×C83—41no—
172C22⋊C4×C83—41no—
173C22⋊C4×C83D811yescharacteristic
174C22⋊C4×C83D811yescharacteristic
175C42×C83—41no—
176C42×C83C2×D411yescharacteristic
177C42×C83C2×D411yescharacteristic
178C23×C83—41no—
179C23×C83C2×D811yescharacteristic
180C22×D83—41no—
181C22×D83—41no—
182C22×D83—44no—
183C22×D83—161no—
184C22×D83—161no—
185Q8×C83—41no—
186Q8×C83—41no—
187Q8×C83—161no—
188D4×C83—41no—
189D4×C83—41no—
190D4×C83—161no—
191C2×C4×C83—21no—
192C2×C4×C83—21no—
193C2×C4×C83—41no—
194C2×C4×C83—41no—
195C2×C4×C83—41no—
196C2×C4×C83—161no—
197C2×C4×C83C2×D811yescharacteristic
198C2×C4×C83C2×D811yescharacteristic
199C2×C4×C83C2×D811yescharacteristic
200C2×C4×C83C2×D811yescharacteristic
201C2×C4×C83C2×D811yescharacteristic
202C2×C4×C83C22×D411yescharacteristic
203C2×Dic83—41no—
204C2×Dic83—41no—
205D332—22no—
206D332—22no—
207D332—44no—
208D332—41no—
209D332—41no—
210C4×D83—161no—
211C4×D83—161no—
212C4×D83—161no—
213C4×D83—161no—
214C8×C83—21no—
215C8×C83—21no—
216C8×C83—21no—
217C8×C83C2×#4111yescharacteristic
218C8×C83C2×#4111yescharacteristic
219C22×C83—21no—
220C22×C83—21no—
221C22×C83—21no—
222C22×C83—81no—
223C22×C83—81no—
224C22×C83—81no—
225C22×C83C22×D811yescharacteristic
226C2×D83—42no—
227C2×D83—42no—
228C2×D83—81no—
229C2×D83—81no—
230C2×D83—81no—
231C2×D83—81no—
232C2×D83—164no—
233C4×C83—21no—
234C4×C83—21no—
235C4×C83—21no—
236C4×C83—21no—
237C4×C83—21no—
238C4×C83—81no—
239C4×C83—81no—
240C4×C83—81no—
241C4×C83#52511yescharacteristic
242C4×C83#57511yescharacteristic
243Dic83—81no—
244Dic83—81no—
245#33678—831nomaximal, Sylow
246C2×C83—41no—
247C2×C83—161no—
248C2×C83#310011yescharacteristic
249C2×C83#341511yescharacteristic
250C2×C83#341611yescharacteristic
251D83—82no—
252D83—82no—
253#3175D8311yescharacteristic, complemented
254#2948—831no—
255#2237—831no—
256#2234—831no—
257#2226—831no—
258#2224—831no—
259#2210—831no—
260#2210—831no—
261#2187—831no—
262#2184—831no—
263#2177—831no—
264#2175—831no—
265C4×SD32—831no—
266C4×D16—831no—
267#2164—831no—
268C83#3367811yescharacteristic, complemented, Sylow
269C22×D8—831no—
270#542—21no—
271#518—1661no—
272C2×SD32—831no—
273C2×SD32—831no—
274C2×D16—831no—
275C2×D16—831no—
276#500C2×D8311yescharacteristic, complemented
277#495C2×D8311yescharacteristic, complemented
278#489—21no—
279#468—21no—
280#467C2×D8311yescharacteristic
281#466C2×D8311yescharacteristic
282#442—1661no—
283#442—1661no—
284#440—21no—
285C4×Q16C2×D8311yescharacteristic, complemented
286C4×D8—831no—
287C4×D8—831no—
288C4×D8—831no—
289C4×D8—831no—
290C4×D8C2×D8311yescharacteristic, complemented
291#427—831no—
292#418—1661no—
293#405C2×D8311yescharacteristic
294#367—831no—
295#367—831no—
296#366—831no—
297#366—831no—
298C4⋊C16—831no—
299C4⋊C16—831no—
300#358—831no—
301#358—831no—
302#357—831no—
303#357—831no—
304#357—831no—
305#357—831no—
306#357—831no—
307#357—831no—
308C22⋊C16—831no—
309C22⋊C16—831no—
310#346—831no—
311C4×C16—831no—
312C22×D4—1661no—
313C2×Q16C22×D8311yescharacteristic
314C2×SD16—21no—
315C2×D8—831no—
316C2×D8—831no—
317C2×D8—1662no—
318C2×D8—1662no—
319C2×D8—1661no—
320C2×D8C22×D8311yescharacteristic
321C2×M16C22⋊D8311yescharacteristic
322C22×C8C22×D8311yescharacteristic
323C4⋊Q8—21no—
324#126—41no—
325#124—21no—
326#124—41no—
327#123—41no—
328C22⋊Q8—21no—
329C22⋊Q8—41no—
330#121—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—3321no—
334C4×D4—3321no—
335C4×D4—1661no—
336C4×D4—1661no—
337C4×D4—1661no—
338C4×D4—1661no—
339#117—1661no—
340#117C22⋊D8311yescharacteristic
341C2×C22⋊C4—3321no—
342C2×C22⋊C4—3321no—
343SD32—1662no—
344SD32—1662no—
345D16—1662no—
346D16—1662no—
347C2×C16—831no—
348C2×C16—831no—
349C2×C16—831no—
350C2×C16—831no—
351#107—21no—
352#107—831no—
353#107—831no—
354#107C22×D8311yescharacteristic
355#107C22×D8311yescharacteristic
356#106—1661no—
357#103—21no—
358#103—21no—
359#102—21no—
360#102—21no—
361#102—1661no—
362#102—1661no—
363#102—1661no—
364#102—1661no—
365#97C22⋊D8311yescharacteristic
366#97C22⋊D8311yescharacteristic
367C4×C8C22×D8311yescharacteristic
368C4×C8C22×D8311yescharacteristic
369C24—3321no—
370C2×Q8—21no—
371C2×Q8—41no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—3322no—
375C2×D4—3321no—
376C2×D4—1664no—
377C2×D4—1661no—
378C2×D4—1661no—
379C22×C4—41no—
380C22×C4—3321no—
381C22×C4—3321no—
382C22×C4—3321no—
383C22×C4—3321no—
384C22×C4C2×C22⋊D8311yescharacteristic
385Q16—21no—
386Q16—21no—
387SD16—42no—
388D8—21no—
389D8—21no—
390D8—832no—
391D8—832no—
392D8—1664no—
393D8—1661no—
394D8—1661no—
395M16—41no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8C2×C22⋊D8311yescharacteristic
399C2×C8C2×C22⋊D8311yescharacteristic
400C2×C8C2×C22⋊D8311yescharacteristic
401C2×C8C23×D8311yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—1661no—
408C4⋊C4—1661no—
409C4⋊C4#1305057511yescharacteristic
410C4⋊C4#1305057511yescharacteristic
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—3321no—
417C22⋊C4—3321no—
418C22⋊C4—3321no—
419C22⋊C4—3321no—
420C22⋊C4—3321no—
421C22⋊C4#1305057511yescharacteristic
422C22⋊C4#1305057511yescharacteristic
423C42—41no—
424C42—3321no—
425C42C2×C22⋊D8311yescharacteristic
426C42C2×C22⋊D8311yescharacteristic
427C16—1661no—
428C16—1661no—
429C16—1661no—
430C16—1661no—
431C23—41no—
432C23—13281no—
433C23—13281no—
434C23—3324no—
435C23—3321no—
436C23—3321no—
437C23C2×#1305057511yescharacteristic
438Q8—41no—
439Q8—41no—
440Q8—161no—
441D4—41no—
442D4—41no—
443D4—161no—
444D4—3324no—
445D4—3321no—
446D4—3321no—
447D4—1662no—
448D4—1662no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—161no—
455C2×C4—13281no—
456C2×C4—13281no—
457C2×C4—13281no—
458C2×C4—13281no—
459C2×C4—3321no—
460C2×C4—3321no—
461C2×C4C2×#1305057511yescharacteristic
462C2×C4C2×#1305057511yescharacteristic
463C2×C4C2×#1305057511yescharacteristic
464C2×C4C2×#1305057511yescharacteristic
465C2×C4C2×#1305057511yescharacteristic
466C2×C4C22×C22⋊D8311yescharacteristic
467C8—21no—
468C8—21no—
469C8—21no—
470C8#41×D8311yescharacteristic
471C8#41×D8311yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—13284no—
479C22—3322no—
480C22—3322no—
481C22—6641no—
482C22—6641no—
483C22—6641no—
484C22—6641no—
485C22C22×#1305057511yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—6641no—
495C4—6641no—
496C4#1912820811yescharacteristic
497C4#1912849911yescharacteristic
498C2—41no—
499C2—161no—
500C2—6642no—
501C2—6642no—
502C2#2491745911yescharacteristic
503C2#2492060211yescharacteristic
504C2#2492060711yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#3175C2
socleC22×C83C22×D8
prime radical#24909374C2
hypercenter#3175D83
hypocenterC83#33678
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #24916316C2semidirect
2. #24909668C2semidirect
3. #24909651C2semidirect
4. #24909607C2semidirect
5. #24909597C2semidirect
6. #24909515C2semidirect
7. #24909515C2semidirect
8. #24909391C2semidirect
10. #24909337C2semidirect
11. #24909325C2semidirect
12. C4×#14581735C2semidirect
13. C4×#14581733C2semidirect
14. #24909270C2semidirect
15. C83×#3175C2semidirect
30. #19127528C22semidirect
31. #19127528C22semidirect
32. #19127526C22semidirect
33. #19127526C22semidirect
34. C4⋊(C83⋊C16)C22semidirect
35. C4⋊(C83⋊C16)C22semidirect
36. #19127505C22semidirect
37. #19127505C22semidirect
38. #19127502C22semidirect
39. #19127502C22semidirect
46. #19127477C22semidirect
47. C4×C83⋊C16C22semidirect
49. #500×C83C22semidirect
50. #495×C83C22semidirect
56. C4×Q16×C83C22semidirect
57. C4×D8×C83C22semidirect
253. #3175D83semidirect
268. C83#33678semidirect
276. #500C2×D83semidirect
277. #495C2×D83semidirect
285. C4×Q16C2×D83semidirect
290. C4×D8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249163162
#249096682
#249096512
#249096072
#249095972
#249095152
#249093912
#249093742
#249093372
#249093252
C4×#145817352
C4×#145817332
#249092702
C83×#31752
#3367883