Finite Group Databasebeta

← #38754503

Subgroups — #38754503

Order 25216.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161890C211yesmaximal, characteristic, complemented
2#27161887C211yesmaximal, characteristic, complemented
3#27161886C211yesmaximal, characteristic, complemented
4#27161450C211yesmaximal, characteristic, complemented
5#27161344C211yesmaximal, characteristic, complemented
6#27161342C211yesmaximal, characteristic, complemented
7#27161340C211yesmaximal, characteristic, complemented
8#27161339C211yesmaximal, characteristic, complemented
9#27161262C211yesmaximal, characteristic, complemented
10#27161260C211yesmaximal, characteristic, complemented
11#27161247C211yesmaximal, characteristic, complemented
12#27161245C211yesmaximal, characteristic, complemented
13#27161196C211yesmaximal, characteristic, complemented
14#27161194C211yesmaximal, characteristic, complemented
15#577×C197C211yesmaximal, characteristic, complemented
16#20263373C2211yescharacteristic
17#20263372C2211yescharacteristic, complemented
18#20263358—21no—
19#20263357—21no—
20#20263357C2211yescharacteristic, complemented
21#20263356—21no—
22#20263356C2211yescharacteristic, complemented
23#20263356C2211yescharacteristic, complemented
24#20263352—21no—
25#20263352C2211yescharacteristic
26#20263351—21no—
27#20263351C2211yescharacteristic
28#20263351C2211yescharacteristic
29#20263350—21no—
30#20263347—21no—
31#20263346—21no—
32#20263346—21no—
33#20263346C2211yescharacteristic
34C2×#15691347C2211yescharacteristic, complemented
35C2×#15691346—21no—
36C2×#15691346C2211yescharacteristic, complemented
37C2×#15691345—21no—
38C2×#15691345C2211yescharacteristic, complemented
39SD16×D197—21no—
40#20263333C2211yescharacteristic, complemented
41#20263333C2211yescharacteristic, complemented
42#20263332C2211yescharacteristic, complemented
43#20263324C2211yescharacteristic, complemented
44#20263322C2211yescharacteristic
45#20263262C2211yescharacteristic, complemented
46#20263261C2211yescharacteristic, complemented
47#20263253C2211yescharacteristic, complemented
48#20263253C2211yescharacteristic, complemented
49#20263253C2211yescharacteristic, complemented
50#20263252C2211yescharacteristic, complemented
51#20263245C2211yescharacteristic, complemented
52#20263244C2211yescharacteristic, complemented
53#20263243C2211yescharacteristic, complemented
54#20263242C2211yescharacteristic, complemented
55C8×Dic197C2211yescharacteristic, complemented
562−1+4×C197C2211yescharacteristic
57H(5,𝔽2)×C197C2211yescharacteristic, complemented
58#137×C197—21no—
59#137×C197C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C197—21no—
61AGL(1,ℤ/8)×C197C2211yescharacteristic
62#135×C197—21no—
63#135×C197C2211yescharacteristic
64C2×SD16×C197—21no—
65C2×SD16×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#15691348—21no—
84#15691348D411yescharacteristic, complemented
85#15691347—21no—
86#15691347—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#15691345—21no—
96#15691345—21no—
97#15691345—21no—
98#15691345D411yescharacteristic, complemented
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—
113#15691337—41no—
114C4×Dic197C2311yescharacteristic
115#41×C197—21no—
116#41×C197—21no—
117#41×C197—21no—
118#41×C197—21no—
119#41×C197—41no—
120#41×C197C2311yescharacteristic
121#41×C197C2311yescharacteristic
122#41×C197C2311yescharacteristic
123C2×Q8×C197—21no—
124C2×Q8×C197—21no—
125C2×Q8×C197C2311yescharacteristic
126C2×D4×C197—21no—
127C2×D4×C197—21no—
128C2×D4×C197—41no—
129C2×D4×C197C2311yescharacteristic
130Q16×C197—21no—
131Q16×C197D411yescharacteristic, complemented
132SD16×C197—21no—
133SD16×C197—21no—
134SD16×C197—21no—
135SD16×C197—21no—
136SD16×C197D411yescharacteristic, complemented
137SD16×C197D411yescharacteristic, complemented
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—41no—
171Q8×C197—21no—
172Q8×C197—21no—
173Q8×C197—41no—
174Q8×C197C2×D411yescharacteristic
175Q8×C197C2×D411yescharacteristic
176Q8×C197C2×D411yescharacteristic
177D4×C197—21no—
178D4×C197—21no—
179D4×C197—21no—
180D4×C197—21no—
181D4×C197—21no—
182D4×C197—41no—
183D4×C197—41no—
184D4×C197—41no—
185D4×C197C2×D411yescharacteristic
186D4×C197C2×D411yescharacteristic
187D4×C197C2×D411yescharacteristic
188C2×C4×C197—21no—
189C2×C4×C197—21no—
190C2×C4×C197—21no—
191C2×C4×C197—21no—
192C2×C4×C197—41no—
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—81no—
204C22×C197—161no—
205C22×C197H(5,𝔽2)11yescharacteristic
206C2×D197—21no—
207C2×D197—41no—
208C2×D197—81no—
209C2×D197—161no—
210C4×C197—21no—
211C4×C197—21no—
212C4×C197—21no—
213C4×C197—41no—
214C4×C197C22×D411yescharacteristic
215C4×C197C22×D411yescharacteristic
216Dic197—22no—
217Dic197—21no—
218Dic197—41no—
219C2×C197—21no—
220C2×C197—41no—
221C2×C197—41no—
222C2×C197—41no—
223C2×C197—81no—
224C2×C197D4211yescharacteristic
225D197—41no—
226D197—81no—
227C197#329111yescharacteristic, complemented, Sylow
228#3291—1971nomaximal, Sylow
229#577—1971no—
230#577—1971no—
231#577—1971no—
232#577D19711yescharacteristic, complemented
233#492—1971no—
234#471—1971no—
235#471—1971no—
236#471—1971no—
237#471—1971no—
238#456—1971no—
239#456—1971no—
240#453—1971no—
241#453—1971no—
242#443—1971no—
243#443—1971no—
2442−1+4—1971no—
2452−1+4C2×D19711yescharacteristic
246H(5,𝔽2)—1971no—
247H(5,𝔽2)C2×D19711yescharacteristic, complemented
248#137—21no—
249#137—3941no—
250#137—3941no—
251#137—3941no—
252#137—1971no—
253#137—1971no—
254#137—1971no—
255#137C2×D19711yescharacteristic, complemented
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—3941no—
258AGL(1,ℤ/8)—3941no—
259AGL(1,ℤ/8)—3941no—
260AGL(1,ℤ/8)—1971no—
261AGL(1,ℤ/8)—1971no—
262AGL(1,ℤ/8)—1971no—
263AGL(1,ℤ/8)C2×D19711yescharacteristic
264#135—21no—
265#135—3941no—
266#135—3941no—
267#135—3941no—
268#135—1971no—
269#135C2×D19711yescharacteristic
270C2×SD16—21no—
271C2×SD16—3941no—
272C2×SD16—3941no—
273C2×SD16—3941no—
274C2×SD16—1971no—
275C2×SD16—1971no—
276C2×SD16—1971no—
277C2×SD16C2×D19711yescharacteristic, complemented
278#131—1971no—
279#131—1971no—
280#131—1971no—
281#131C2×D19711yescharacteristic, complemented
282C4⋊Q8—1971no—
283#127—1971no—
284#108—1971no—
285#108—1971no—
286#104—1971no—
287#104—1971no—
288#104—1971no—
289#104—1971no—
290#101—1971no—
291#101—1971no—
292#100—1971no—
293#100—1971no—
294C4×C8—1971no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—7881no—
301#41—3941no—
302#41—3941no—
303#41—3941no—
304#41—3941no—
305#41—1971no—
306#41C22×D19711yescharacteristic
307#41C22×D19711yescharacteristic
308#41C22×D19711yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—3941no—
312C2×Q8—3941no—
313C2×Q8—1971no—
314C2×Q8C22×D19711yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—7881no—
319C2×D4—7881no—
320C2×D4—3941no—
321C2×D4—3941no—
322C2×D4—1971no—
323C2×D4C22×D19711yescharacteristic
324Q16—21no—
325Q16—3941no—
326Q16—3941no—
327Q16—3941no—
328Q16—1971no—
329Q16C22⋊D19711yescharacteristic, complemented
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—3941no—
335SD16—3941no—
336SD16—3941no—
337SD16—3941no—
338SD16—3941no—
339SD16—3941no—
340SD16—3941no—
341SD16—3941no—
342SD16—3941no—
343SD16—3941no—
344SD16—3941no—
345SD16—3941no—
346SD16—1971no—
347SD16—1971no—
348SD16C22⋊D19711yescharacteristic, complemented
349SD16C22⋊D19711yescharacteristic, complemented
350D8—21no—
351D8—3941no—
352D8—3941no—
353D8—3941no—
354D8—1971no—
355D8C22⋊D19711yescharacteristic, complemented
356M16—21no—
357M16—3941no—
358M16—3941no—
359M16—3941no—
360M16—1971no—
361M16—1971no—
362M16—1971no—
363M16C22×D19711yescharacteristic
364C2×C8—21no—
365C2×C8—3941no—
366C2×C8—3941no—
367C2×C8—3941no—
368C2×C8—1971no—
369C2×C8C22×D19711yescharacteristic
370C4⋊C4—7881no—
371C42—1971no—
372C23—21no—
373C23—41no—
374C23—7881no—
375C23—3941no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—7881no—
380Q8—3941no—
381Q8—3941no—
382Q8—1971no—
383Q8D4×D19711yescharacteristic
384Q8C2×C22⋊D19711yescharacteristic
385Q8C2×C22⋊D19711yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—7881no—
395D4—7881no—
396D4—7881no—
397D4—15762no—
398D4—3941no—
399D4—3941no—
400D4—3941no—
401D4—1971no—
402D4D4×D19711yescharacteristic
403D4C2×C22⋊D19711yescharacteristic
404D4C2×C22⋊D19711yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—7881no—
413C2×C4—7881no—
414C2×C4—7881no—
415C2×C4—3941no—
416C2×C4—3941no—
417C2×C4—3941no—
418C2×C4C23×D19711yescharacteristic
419C8—21no—
420C8—3941no—
421C8—3941no—
422C8—3941no—
423C8—1971no—
424C8—1971no—
425C8C2×C22⋊D19711yescharacteristic
426C8C2×C22⋊D19711yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—7881no—
434C22—15761no—
435C22—31521no—
436C22—3941no—
437C22#2026337111yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—7881no—
443C4—3942no—
444C4—3941no—
445C4C2×D4×D19711yescharacteristic
446C4C22×C22⋊D19711yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—7881no—
453C2—15761no—
454C2D4×C22⋊D19711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#577×C197C2
socleC2×C197D42
hypercenter#577D197
hypocenterC197#3291
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #27161890C2semidirect
2. #27161887C2semidirect
3. #27161886C2semidirect
4. #27161450C2semidirect
5. #27161344C2semidirect
6. #27161342C2semidirect
7. #27161340C2semidirect
8. #27161339C2semidirect
9. #27161262C2semidirect
10. #27161260C2semidirect
11. #27161247C2semidirect
12. #27161245C2semidirect
13. #27161196C2semidirect
14. #27161194C2semidirect
15. #577×C197C2semidirect
17. #20263372C22semidirect
20. #20263357C22semidirect
22. #20263356C22semidirect
23. #20263356C22semidirect
34. C2×#15691347C22semidirect
36. C2×#15691346C22semidirect
38. C2×#15691345C22semidirect
40. #20263333C22semidirect
41. #20263333C22semidirect
42. #20263332C22semidirect
43. #20263324C22semidirect
45. #20263262C22semidirect
46. #20263261C22semidirect
47. #20263253C22semidirect
48. #20263253C22semidirect
49. #20263253C22semidirect
50. #20263252C22semidirect
51. #20263245C22semidirect
52. #20263244C22semidirect
53. #20263243C22semidirect
54. #20263242C22semidirect
55. C8×Dic197C22semidirect
57. H(5,𝔽2)×C197C22semidirect
59. #137×C197C22semidirect
65. C2×SD16×C197C22semidirect
66. #131×C197C22semidirect
84. #15691348D4semidirect
92. #15691346D4semidirect
98. #15691345D4semidirect
102. #15691343D4semidirect
131. Q16×C197D4semidirect
136. SD16×C197D4semidirect
137. SD16×C197D4semidirect
139. D8×C197D4semidirect
227. C197#3291semidirect
232. #577D197semidirect
247. H(5,𝔽2)C2×D197semidirect
255. #137C2×D197semidirect
277. C2×SD16C2×D197semidirect
281. #131C2×D197semidirect
329. Q16C22⋊D197semidirect
348. SD16C22⋊D197semidirect
349. SD16C22⋊D197semidirect
355. D8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271618902
#271618872
#271618862
#271614502
#271613442
#271613422
#271613402
#271613392
#271612622
#271612602
#271612472
#271612452
#271611962
#271611942
#577×C1972
#3291197