Finite Group Databasebeta

← #38754476

Subgroups — #38754476

Order 25216.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161890C211yesmaximal, characteristic, complemented
2#27161886C211yesmaximal, characteristic, complemented
3#27161882C211yesmaximal, characteristic, complemented
4#27161467C211yesmaximal, characteristic, complemented
5#27161347C211yesmaximal, characteristic, complemented
6#27161346C211yesmaximal, characteristic, complemented
7#27161342C211yesmaximal, characteristic, complemented
8#27161340C211yesmaximal, characteristic, complemented
9#27161257C211yesmaximal, characteristic, complemented
10#27161250C211yesmaximal, characteristic, complemented
11#27161247C211yesmaximal, characteristic, complemented
12#27161245C211yesmaximal, characteristic, complemented
13#27161201C211yesmaximal, characteristic, complemented
14#27161199C211yesmaximal, characteristic, complemented
15#576×C197C211yesmaximal, characteristic, complemented
16#20263373C2211yescharacteristic, complemented
17#20263372C2211yescharacteristic, complemented
18#20263358—21no—
19#20263356—21no—
20#20263356C2211yescharacteristic, complemented
21#20263356C2211yescharacteristic, complemented
22#20263352—21no—
23#20263352—21no—
24#20263352C2211yescharacteristic, complemented
25#20263352C2211yescharacteristic, complemented
26#20263351—21no—
27#20263351C2211yescharacteristic
28#20263351C2211yescharacteristic, complemented
29#20263350—21no—
30#20263347—21no—
31#20263346—21no—
32#20263346—21no—
33#20263346C2211yescharacteristic
34C2×#15691347C2211yescharacteristic, complemented
35C2×#15691346—21no—
36C2×#15691346C2211yescharacteristic, complemented
37SD16×D197—21no—
38C2×#15691343—21no—
39C2×#15691343C2211yescharacteristic, complemented
40#20263333C2211yescharacteristic, complemented
41#20263333C2211yescharacteristic, complemented
42#20263332C2211yescharacteristic, complemented
43#20263322C2211yescharacteristic, complemented
44#20263313C2211yescharacteristic, complemented
45#20263262C2211yescharacteristic, complemented
46#20263261C2211yescharacteristic, complemented
47#20263253C2211yescharacteristic, complemented
48#20263253C2211yescharacteristic, complemented
49#20263253C2211yescharacteristic, complemented
50#20263252C2211yescharacteristic, complemented
51#20263244C2211yescharacteristic, complemented
52#20263243C2211yescharacteristic, complemented
53#20263243C2211yescharacteristic, complemented
54#20263242C2211yescharacteristic, complemented
55#20263236C2211yescharacteristic, complemented
56H(5,𝔽2)×C197C2211yescharacteristic
57H(5,𝔽2)×C197C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C197—21no—
59AGL(1,ℤ/8)×C197—21no—
60AGL(1,ℤ/8)×C197C2211yescharacteristic
61AGL(1,ℤ/8)×C197C2211yescharacteristic, complemented
62#135×C197—21no—
63#135×C197C2211yescharacteristic
64C2×D8×C197—21no—
65C2×D8×C197C2211yescharacteristic, complemented
66#131×C197C2211yescharacteristic, complemented
67#15691359—21no—
68#15691358—21no—
69#15691358—21no—
70#15691358C2311yescharacteristic
71#15691357—21no—
72#15691357—41no—
73C2×Dic394—21no—
74C2×Dic394C2311yescharacteristic
75Q8×D197—21no—
76C2×D788—21no—
77C2×D788C2311yescharacteristic
78C2×C22⋊D197—41no—
79D4×D197—21no—
80D4×D197—41no—
81#15691348—21no—
82#15691348—21no—
83#15691348D411yescharacteristic, complemented
84#15691347—21no—
85#15691347—21no—
86#15691346—21no—
87#15691346—21no—
88#15691346—21no—
89#15691346—21no—
90#15691346—21no—
91#15691346—21no—
92#15691346D411yescharacteristic, complemented
93#15691345—21no—
94#15691345—21no—
95#15691345D411yescharacteristic, complemented
96#15691343—21no—
97#15691343—21no—
98#15691343—21no—
99#15691343—21no—
100#15691343—21no—
101#15691343—21no—
102#15691343D411yescharacteristic, complemented
103#15691342—21no—
104#15691342—21no—
105#15691342C2311yescharacteristic
106#15691342C2311yescharacteristic
107#15691342C2311yescharacteristic
108#15691341—21no—
109C2×#13096802—21no—
110C2×#13096802—21no—
111C2×#13096802C2311yescharacteristic
112C8×D197—21no—
113C22⋊Dic197—41no—
114C4×Dic197C2311yescharacteristic
115#41×C197—21no—
116#41×C197—21no—
117#41×C197—21no—
118#41×C197—21no—
119#41×C197C2311yescharacteristic
120#41×C197C2311yescharacteristic
121#41×C197C2311yescharacteristic
122C2×D4×C197—21no—
123C2×D4×C197—21no—
124C2×D4×C197—21no—
125C2×D4×C197—21no—
126C2×D4×C197—41no—
127C2×D4×C197—41no—
128C2×D4×C197C2311yescharacteristic
129C2×D4×C197C2311yescharacteristic
130Q16×C197D411yescharacteristic, complemented
131SD16×C197—21no—
132SD16×C197—21no—
133SD16×C197D411yescharacteristic, complemented
134SD16×C197D411yescharacteristic, complemented
135D8×C197—21no—
136D8×C197—21no—
137D8×C197—21no—
138D8×C197—21no—
139D8×C197D411yescharacteristic, complemented
140M16×C197—21no—
141M16×C197C2311yescharacteristic
142C2×C8×C197—21no—
143C2×C8×C197C2311yescharacteristic
144C22×D197—21no—
145C22×D197—41no—
146Dic394—21no—
147Dic394—21no—
148Dic394—41no—
149Dic394C2×D411yescharacteristic
150D788—21no—
151D788—21no—
152D788—41no—
153D788C2×D411yescharacteristic
154C22⋊D197—21no—
155C22⋊D197—41no—
156C22⋊D197—41no—
157C22⋊D197—82no—
158C2×Dic197—21no—
159C2×Dic197—21no—
160C2×Dic197—41no—
161C4×D197—21no—
162C4×D197—41no—
163C4×D197—41no—
164#13096802—21no—
165#13096802—21no—
166#13096802—21no—
167#13096802C2×D411yescharacteristic
168#13096802C2×D411yescharacteristic
169C23×C197—21no—
170C23×C197—21no—
171C23×C197—41no—
172C23×C197—41no—
173Q8×C197C2×D411yescharacteristic
174Q8×C197C2×D411yescharacteristic
175Q8×C197C2×D411yescharacteristic
176D4×C197—21no—
177D4×C197—21no—
178D4×C197—21no—
179D4×C197—21no—
180D4×C197—21no—
181D4×C197—21no—
182D4×C197—21no—
183D4×C197—41no—
184D4×C197—41no—
185D4×C197—41no—
186D4×C197—41no—
187D4×C197C2×D411yescharacteristic
188D4×C197C2×D411yescharacteristic
189D4×C197C2×D411yescharacteristic
190C2×C4×C197—21no—
191C2×C4×C197—21no—
192C2×C4×C197—21no—
193C2×C4×C197—41no—
194C2×C4×C197—41no—
195C2×C4×C197C2411yescharacteristic
196C8×C197—21no—
197C8×C197C2×D411yescharacteristic
198C8×C197C2×D411yescharacteristic
199C22×C197—21no—
200C22×C197—21no—
201C22×C197—21no—
202C22×C197—41no—
203C22×C197—41no—
204C22×C197—81no—
205C22×C197—81no—
206C22×C197—161no—
207C22×C197—161no—
208C22×C197H(5,𝔽2)11yescharacteristic
209C2×D197—21no—
210C2×D197—41no—
211C2×D197—81no—
212C2×D197—161no—
213C4×C197—21no—
214C4×C197—21no—
215C4×C197—21no—
216C4×C197C22×D411yescharacteristic
217C4×C197C22×D411yescharacteristic
218Dic197—21no—
219Dic197—41no—
220Dic197—41no—
221C2×C197—21no—
222C2×C197—41no—
223C2×C197—41no—
224C2×C197—41no—
225C2×C197—81no—
226C2×C197—81no—
227C2×C197D4211yescharacteristic
228D197—41no—
229D197—81no—
230C197#328811yescharacteristic, complemented, Sylow
231#3288—1971nomaximal, Sylow
232#577—1971no—
233#577—1971no—
234#576—1971no—
235#576D19711yescharacteristic, complemented
236#496—1971no—
237#472—1971no—
238#472—1971no—
239#471—1971no—
240#471—1971no—
241#455—1971no—
242#454—1971no—
243#453—1971no—
244#453—1971no—
245#444—1971no—
246#444—1971no—
2472−1+4—1971no—
248H(5,𝔽2)—1971no—
249H(5,𝔽2)C2×D19711yescharacteristic
250H(5,𝔽2)C2×D19711yescharacteristic, complemented
251#137—3941no—
252#137—3941no—
253#137—1971no—
254#137—1971no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—3941no—
258AGL(1,ℤ/8)—3941no—
259AGL(1,ℤ/8)—3941no—
260AGL(1,ℤ/8)—3941no—
261AGL(1,ℤ/8)—1971no—
262AGL(1,ℤ/8)—1971no—
263AGL(1,ℤ/8)—1971no—
264AGL(1,ℤ/8)—1971no—
265AGL(1,ℤ/8)C2×D19711yescharacteristic
266AGL(1,ℤ/8)C2×D19711yescharacteristic, complemented
267#135—21no—
268#135—3941no—
269#135—3941no—
270#135—3941no—
271#135—1971no—
272#135C2×D19711yescharacteristic
273C2×SD16—3941no—
274C2×SD16—3941no—
275C2×SD16—1971no—
276C2×SD16—1971no—
277C2×D8—21no—
278C2×D8—3941no—
279C2×D8—1971no—
280C2×D8C2×D19711yescharacteristic, complemented
281#131—1971no—
282#131—1971no—
283#131—1971no—
284#131C2×D19711yescharacteristic, complemented
285#127—1971no—
286#124—1971no—
287#108—1971no—
288#108—1971no—
289#104—1971no—
290#104—1971no—
291#104—1971no—
292#104—1971no—
293#101—1971no—
294#100—1971no—
295#100—1971no—
296#100—1971no—
297#97—1971no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—7881no—
303#41—3941no—
304#41—3941no—
305#41—3941no—
306#41—3941no—
307#41—1971no—
308#41C22×D19711yescharacteristic
309#41C22×D19711yescharacteristic
310#41C22×D19711yescharacteristic
311C2×Q8—3941no—
312C2×Q8—3941no—
313C2×Q8—1971no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—7881no—
321C2×D4—7881no—
322C2×D4—3941no—
323C2×D4—3941no—
324C2×D4—1971no—
325C2×D4C22×D19711yescharacteristic
326C2×D4C22×D19711yescharacteristic
327Q16—3941no—
328Q16—3941no—
329Q16—1971no—
330Q16C22⋊D19711yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—3941no—
334SD16—3941no—
335SD16—3941no—
336SD16—3941no—
337SD16—3941no—
338SD16—3941no—
339SD16—3941no—
340SD16—3941no—
341SD16—3941no—
342SD16—3941no—
343SD16—1971no—
344SD16—1971no—
345SD16C22⋊D19711yescharacteristic, complemented
346SD16C22⋊D19711yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—3941no—
352D8—3941no—
353D8—3941no—
354D8—3941no—
355D8—3941no—
356D8—3941no—
357D8—1971no—
358D8C22⋊D19711yescharacteristic, complemented
359M16—21no—
360M16—3941no—
361M16—3941no—
362M16—3941no—
363M16—1971no—
364M16—1971no—
365M16—1971no—
366M16C22×D19711yescharacteristic
367C2×C8—21no—
368C2×C8—3941no—
369C2×C8—3941no—
370C2×C8—3941no—
371C2×C8—1971no—
372C2×C8C22×D19711yescharacteristic
373C22⋊C4—7881no—
374C42—1971no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—7881no—
380C23—3941no—
381Q8—7881no—
382Q8—3941no—
383Q8—3941no—
384Q8—1971no—
385Q8D4×D19711yescharacteristic
386Q8C2×C22⋊D19711yescharacteristic
387Q8C2×C22⋊D19711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—7881no—
400D4—7881no—
401D4—7881no—
402D4—15762no—
403D4—3941no—
404D4—3941no—
405D4—3941no—
406D4—1971no—
407D4D4×D19711yescharacteristic
408D4C2×C22⋊D19711yescharacteristic
409D4C2×C22⋊D19711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—7881no—
416C2×C4—7881no—
417C2×C4—7881no—
418C2×C4—3941no—
419C2×C4—3941no—
420C2×C4—3941no—
421C2×C4C23×D19711yescharacteristic
422C8—21no—
423C8—3941no—
424C8—3941no—
425C8—3941no—
426C8—1971no—
427C8—1971no—
428C8C2×C22⋊D19711yescharacteristic
429C8C2×C22⋊D19711yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—7881no—
440C22—15761no—
441C22—31521no—
442C22—3941no—
443C22#2026337111yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—7881no—
448C4—7881no—
449C4—3941no—
450C4C2×D4×D19711yescharacteristic
451C4C22×C22⋊D19711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—7881no—
459C2—15761no—
460C2D4×C22⋊D19711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#576×C197C2
socleC2×C197D42
hypercenter#576D197
hypocenterC197#3288
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #27161890C2semidirect
2. #27161886C2semidirect
3. #27161882C2semidirect
4. #27161467C2semidirect
5. #27161347C2semidirect
6. #27161346C2semidirect
7. #27161342C2semidirect
8. #27161340C2semidirect
9. #27161257C2semidirect
10. #27161250C2semidirect
11. #27161247C2semidirect
12. #27161245C2semidirect
13. #27161201C2semidirect
14. #27161199C2semidirect
15. #576×C197C2semidirect
16. #20263373C22semidirect
17. #20263372C22semidirect
20. #20263356C22semidirect
21. #20263356C22semidirect
24. #20263352C22semidirect
25. #20263352C22semidirect
28. #20263351C22semidirect
34. C2×#15691347C22semidirect
36. C2×#15691346C22semidirect
39. C2×#15691343C22semidirect
40. #20263333C22semidirect
41. #20263333C22semidirect
42. #20263332C22semidirect
43. #20263322C22semidirect
44. #20263313C22semidirect
45. #20263262C22semidirect
46. #20263261C22semidirect
47. #20263253C22semidirect
48. #20263253C22semidirect
49. #20263253C22semidirect
50. #20263252C22semidirect
51. #20263244C22semidirect
52. #20263243C22semidirect
53. #20263243C22semidirect
54. #20263242C22semidirect
55. #20263236C22semidirect
57. H(5,𝔽2)×C197C22semidirect
61. AGL(1,ℤ/8)×C197C22semidirect
65. C2×D8×C197C22semidirect
66. #131×C197C22semidirect
83. #15691348D4semidirect
92. #15691346D4semidirect
95. #15691345D4semidirect
102. #15691343D4semidirect
130. Q16×C197D4semidirect
133. SD16×C197D4semidirect
134. SD16×C197D4semidirect
139. D8×C197D4semidirect
230. C197#3288semidirect
235. #576D197semidirect
250. H(5,𝔽2)C2×D197semidirect
266. AGL(1,ℤ/8)C2×D197semidirect
280. C2×D8C2×D197semidirect
284. #131C2×D197semidirect
330. Q16C22⋊D197semidirect
345. SD16C22⋊D197semidirect
346. SD16C22⋊D197semidirect
358. D8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271618902
#271618862
#271618822
#271614672
#271613472
#271613462
#271613422
#271613402
#271612572
#271612502
#271612472
#271612452
#271612012
#271611992
#576×C1972
#3288197