Finite Group Databasebeta

← #38754479

Subgroups — #38754479

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#27161891C211yesmaximal, characteristic, complemented
2#27161886C211yesmaximal, characteristic, complemented
3#27161884C211yesmaximal, characteristic, complemented
4#27161881C211yesmaximal, characteristic, complemented
5#27161468C211yesmaximal, characteristic, complemented
6#27161350C211yesmaximal, characteristic, complemented
7#27161349C211yesmaximal, characteristic, complemented
8#27161343C211yesmaximal, characteristic, complemented
9#27161342C211yesmaximal, characteristic, complemented
10#27161258C211yesmaximal, characteristic, complemented
11#27161253C211yesmaximal, characteristic, complemented
12#27161248C211yesmaximal, characteristic, complemented
13#27161248C211yesmaximal, characteristic, complemented
14#27161202C211yesmaximal, characteristic, complemented
15#444×C197C211yesmaximal, characteristic, complemented
16#20263373C2211yescharacteristic, complemented
17#20263372C2211yescharacteristic
18#20263372C2211yescharacteristic, complemented
19#20263372C2211yescharacteristic, complemented
20#20263359—21no—
21#20263359C2211yescharacteristic, complemented
22#20263358—21no—
23#20263356C2211yescharacteristic, complemented
24#20263354—21no—
25#20263354—21no—
26#20263354—21no—
27#20263354C2211yescharacteristic
28#20263354C2211yescharacteristic, complemented
29#20263354C2211yescharacteristic, complemented
30#20263353—21no—
31#20263353—21no—
32#20263351C2211yescharacteristic
33#20263351C2211yescharacteristic, complemented
34#20263351C2211yescharacteristic, complemented
35#20263350—21no—
36#20263349—21no—
37#20263348—21no—
38#20263348—21no—
39#20263348C2211yescharacteristic
40#20263347—21no—
41C2×#15691347—21no—
42C2×#15691347C2211yescharacteristic, complemented
43C2×#15691347C2211yescharacteristic, complemented
44SD16×D197—21no—
45C2×D1576—21no—
46C2×D1576C2211yescharacteristic, complemented
47C2×D1576C2211yescharacteristic, complemented
48D8×D197—21no—
49#20263332C2211yescharacteristic, complemented
50#20263332C2211yescharacteristic, complemented
51#20263323C2211yescharacteristic, complemented
52#20263314C2211yescharacteristic, complemented
53#20263262C2211yescharacteristic, complemented
54#20263254C2211yescharacteristic, complemented
55#20263254C2211yescharacteristic, complemented
56#20263244C2211yescharacteristic, complemented
57#20263242C2211yescharacteristic, complemented
58#20263242C2211yescharacteristic, complemented
59#20263242C2211yescharacteristic, complemented
60#135×C197C2211yescharacteristic
61#131×C197C2211yescharacteristic, complemented
62#131×C197C2211yescharacteristic, complemented
63#108×C197C2211yescharacteristic, complemented
64#104×C197C2211yescharacteristic, complemented
65#104×C197C2211yescharacteristic, complemented
66#97×C197C2211yescharacteristic, complemented
67#15691359—21no—
68#15691359—21no—
69#15691359—21no—
70#15691358—21no—
71#15691358—21no—
72#15691358—21no—
73#15691358—21no—
74#15691358C2311yescharacteristic
75#15691358C2311yescharacteristic
76#15691357—21no—
77#15691357—41no—
78C2×Dic394—21no—
79C2×Dic394C2311yescharacteristic
80Q8×D197—21no—
81C2×D788—21no—
82C2×D788—21no—
83C2×D788—21no—
84C2×D788—41no—
85C2×D788C2311yescharacteristic
86C2×D788C2311yescharacteristic
87C2×D788C2311yescharacteristic
88D4×D197—21no—
89D4×D197—21no—
90D4×D197—21no—
91D4×D197—41no—
92D4×D197—41no—
93D4×D197—41no—
94Dic788—21no—
95Dic788D411yescharacteristic, complemented
96#15691348—21no—
97#15691347—21no—
98#15691347—21no—
99#15691347—21no—
100#15691347—21no—
101#15691347—21no—
102#15691347—21no—
103#15691347—21no—
104#15691347—21no—
105#15691347D411yescharacteristic, complemented
106#15691347D411yescharacteristic, complemented
107#15691346—21no—
108#15691345—21no—
109#15691345—21no—
110#15691345—21no—
111D1576—21no—
112D1576—21no—
113D1576—21no—
114D1576—21no—
115D1576—21no—
116D1576—21no—
117D1576—21no—
118D1576D411yescharacteristic, complemented
119#15691343—21no—
120#15691343—21no—
121#15691343—21no—
122#15691342C2311yescharacteristic
123#15691342C2311yescharacteristic
124#15691341—21no—
125#15691341—21no—
126C8×D197—21no—
127C8×D197—21no—
128#15691335—41no—
129#41×C197C2311yescharacteristic
130#41×C197C2311yescharacteristic
131Q16×C197D411yescharacteristic, complemented
132SD16×C197D411yescharacteristic, complemented
133SD16×C197D411yescharacteristic, complemented
134D8×C197D411yescharacteristic, complemented
135M16×C197—21no—
136M16×C197—21no—
137M16×C197C2311yescharacteristic
138M16×C197C2311yescharacteristic
139C2×C8×C197—21no—
140C2×C8×C197—21no—
141C2×C8×C197C2311yescharacteristic
142C2×C8×C197C2311yescharacteristic
143C42×C197C2311yescharacteristic
144C22×D197—21no—
145C22×D197—21no—
146C22×D197—21no—
147C22×D197—41no—
148C22×D197—41no—
149C22×D197—41no—
150Dic394—21no—
151Dic394—21no—
152Dic394—41no—
153Dic394C2×D411yescharacteristic
154Dic394C2×D411yescharacteristic
155D788—21no—
156D788—21no—
157D788—21no—
158D788—21no—
159D788—21no—
160D788—21no—
161D788—41no—
162D788—41no—
163D788—41no—
164D788—82no—
165D788C2×D411yescharacteristic
166D788C2×D411yescharacteristic
167C22⋊D197—21no—
168C22⋊D197—21no—
169C22⋊D197—41no—
170C22⋊D197—41no—
171C22⋊D197—41no—
172C22⋊D197—41no—
173C2×Dic197—21no—
174C2×Dic197—41no—
175C4×D197—21no—
176C4×D197—21no—
177C4×D197—41no—
178C4×D197—41no—
179C4×D197—41no—
180C4×D197—41no—
181#13096802—21no—
182#13096802—21no—
183Q8×C197C2×D411yescharacteristic
184Q8×C197C2×D411yescharacteristic
185D4×C197—21no—
186D4×C197—21no—
187D4×C197C2×D411yescharacteristic
188D4×C197C2×D411yescharacteristic
189C2×C4×C197—21no—
190C2×C4×C197—21no—
191C2×C4×C197—21no—
192C2×C4×C197C2411yescharacteristic
193C8×C197—21no—
194C8×C197—21no—
195C8×C197C2×D411yescharacteristic
196C8×C197C2×D411yescharacteristic
197C8×C197C2×D411yescharacteristic
198C8×C197C2×D411yescharacteristic
199C22×C197—21no—
200C22×C197—21no—
201C22×C197H(5,𝔽2)11yescharacteristic
202C2×D197—21no—
203C2×D197—21no—
204C2×D197—41no—
205C2×D197—41no—
206C2×D197—41no—
207C2×D197—81no—
208C2×D197—81no—
209C2×D197—81no—
210C2×D197—161no—
211C2×D197—161no—
212C2×D197—161no—
213C4×C197—21no—
214C4×C197—21no—
215C4×C197—41no—
216C4×C197C22×D411yescharacteristic
217C4×C197C22×D411yescharacteristic
218Dic197—21no—
219Dic197—21no—
220Dic197—41no—
221C2×C197—21no—
222C2×C197—41no—
223C2×C197—41no—
224C2×C197D4211yescharacteristic
225D197—41no—
226D197—41no—
227D197—81no—
228D197—81no—
229D197—81no—
230C197#328811yescharacteristic, complemented, Sylow
231#3288—1971nomaximal, Sylow
232#577—1971no—
233#577—1971no—
234#576—1971no—
235#576—1971no—
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#444D19711yescharacteristic, complemented
2472−1+4—1971no—
248H(5,𝔽2)—1971no—
249H(5,𝔽2)—1971no—
250H(5,𝔽2)—1971no—
251#137—3941no—
252#137—3941no—
253#137—1971no—
254#137—1971no—
255AGL(1,ℤ/8)—3941no—
256AGL(1,ℤ/8)—3941no—
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)—1971no—
266AGL(1,ℤ/8)—1971no—
267#135—3941no—
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—3941no—
278C2×D8—3941no—
279C2×D8—1971no—
280C2×D8—1971no—
281#131—1971no—
282#131—1971no—
283#131C2×D19711yescharacteristic, complemented
284#131C2×D19711yescharacteristic, complemented
285#127—1971no—
286#124—1971no—
287#108—1971no—
288#108C2×D19711yescharacteristic, complemented
289#104—1971no—
290#104—1971no—
291#104C2×D19711yescharacteristic, complemented
292#104C2×D19711yescharacteristic, complemented
293#101—1971no—
294#100—1971no—
295#100—1971no—
296#100—1971no—
297#97C2×D19711yescharacteristic, complemented
298#41—7881no—
299#41—3941no—
300#41—3941no—
301#41—3941no—
302#41—3941no—
303#41—3941no—
304#41—3941no—
305#41—3941no—
306#41—3941no—
307#41—1971no—
308#41—1971no—
309#41C22×D19711yescharacteristic
310#41C22×D19711yescharacteristic
311C2×Q8—3941no—
312C2×Q8—3941no—
313C2×Q8—1971no—
314C2×D4—7881no—
315C2×D4—7881no—
316C2×D4—7881no—
317C2×D4—7881no—
318C2×D4—3941no—
319C2×D4—3941no—
320C2×D4—3941no—
321C2×D4—3941no—
322C2×D4—3941no—
323C2×D4—3941no—
324C2×D4—1971no—
325C2×D4—1971no—
326C2×D4—1971no—
327Q16—3941no—
328Q16—3941no—
329Q16—1971no—
330Q16D78811yescharacteristic, complemented
331SD16—3941no—
332SD16—3941no—
333SD16—3941no—
334SD16—3941no—
335SD16—3941no—
336SD16—3941no—
337SD16—3941no—
338SD16—3941no—
339SD16—3941no—
340SD16—3941no—
341SD16—3941no—
342SD16—3941no—
343SD16—1971no—
344SD16—1971no—
345SD16D78811yescharacteristic, complemented
346SD16D78811yescharacteristic, complemented
347D8—3941no—
348D8—3941no—
349D8—3941no—
350D8—3941no—
351D8—3941no—
352D8—3941no—
353D8—3941no—
354D8—3941no—
355D8—3941no—
356D8—3941no—
357D8—1971no—
358D8D78811yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—3941no—
362M16—3941no—
363M16—1971no—
364M16—1971no—
365M16C22×D19711yescharacteristic
366M16C22×D19711yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—3941no—
370C2×C8—3941no—
371C2×C8C22×D19711yescharacteristic
372C2×C8C22×D19711yescharacteristic
373C22⋊C4—7881no—
374C42C22×D19711yescharacteristic
375C23—7881no—
376C23—7881no—
377C23—7881no—
378C23—3941no—
379C23—3941no—
380C23—3941no—
381Q8—7881no—
382Q8—3941no—
383Q8—3941no—
384Q8—1971no—
385Q8—1971no—
386Q8C2×D78811yescharacteristic
387Q8C2×D78811yescharacteristic
388D4—21no—
389D4—21no—
390D4—7881no—
391D4—7881no—
392D4—7881no—
393D4—7881no—
394D4—7881no—
395D4—7881no—
396D4—7881no—
397D4—15762no—
398D4—3941no—
399D4—3941no—
400D4—3941no—
401D4—3941no—
402D4—3941no—
403D4—3941no—
404D4—3941no—
405D4—3941no—
406D4—1971no—
407D4—1971no—
408D4C2×D78811yescharacteristic
409D4C2×D78811yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—7881no—
414C2×C4—7881no—
415C2×C4—7881no—
416C2×C4—7881no—
417C2×C4—7881no—
418C2×C4—3941no—
419C2×C4—3941no—
420C2×C4—3941no—
421C2×C4C23×D19711yescharacteristic
422C8—21no—
423C8—21no—
424C8—3941no—
425C8—3941no—
426C8D4×D19711yescharacteristic
427C8D4×D19711yescharacteristic
428C8C2×D78811yescharacteristic
429C8C2×D78811yescharacteristic
430C22—21no—
431C22—21no—
432C22—7881no—
433C22—7881no—
434C22—7881no—
435C22—15761no—
436C22—15761no—
437C22—15761no—
438C22—31521no—
439C22—31521no—
440C22—31521no—
441C22—3941no—
442C22—3941no—
443C22#2026337211yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—7881no—
448C4—3941no—
449C4—3941no—
450C4C2×D4×D19711yescharacteristic
451C4C22×D78811yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—7881no—
456C2—7881no—
457C2—15761no—
458C2—15761no—
459C2—15761no—
460C2D4×D78811yescharacteristic
subgroupG/H
centerC2D4×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#444×C197C2
socleC2×C197D42
hypercenter#444D197
hypocenterC197#3288
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #27161891C2semidirect
2. #27161886C2semidirect
3. #27161884C2semidirect
4. #27161881C2semidirect
5. #27161468C2semidirect
6. #27161350C2semidirect
7. #27161349C2semidirect
8. #27161343C2semidirect
9. #27161342C2semidirect
10. #27161258C2semidirect
11. #27161253C2semidirect
12. #27161248C2semidirect
13. #27161248C2semidirect
14. #27161202C2semidirect
15. #444×C197C2semidirect
16. #20263373C22semidirect
18. #20263372C22semidirect
19. #20263372C22semidirect
21. #20263359C22semidirect
23. #20263356C22semidirect
28. #20263354C22semidirect
29. #20263354C22semidirect
33. #20263351C22semidirect
34. #20263351C22semidirect
42. C2×#15691347C22semidirect
43. C2×#15691347C22semidirect
46. C2×D1576C22semidirect
47. C2×D1576C22semidirect
49. #20263332C22semidirect
50. #20263332C22semidirect
51. #20263323C22semidirect
52. #20263314C22semidirect
53. #20263262C22semidirect
54. #20263254C22semidirect
55. #20263254C22semidirect
56. #20263244C22semidirect
57. #20263242C22semidirect
58. #20263242C22semidirect
59. #20263242C22semidirect
61. #131×C197C22semidirect
62. #131×C197C22semidirect
63. #108×C197C22semidirect
64. #104×C197C22semidirect
65. #104×C197C22semidirect
66. #97×C197C22semidirect
95. Dic788D4semidirect
105. #15691347D4semidirect
106. #15691347D4semidirect
118. D1576D4semidirect
131. Q16×C197D4semidirect
132. SD16×C197D4semidirect
133. SD16×C197D4semidirect
134. D8×C197D4semidirect
230. C197#3288semidirect
246. #444D197semidirect
283. #131C2×D197semidirect
284. #131C2×D197semidirect
288. #108C2×D197semidirect
291. #104C2×D197semidirect
292. #104C2×D197semidirect
297. #97C2×D197semidirect
330. Q16D788semidirect
345. SD16D788semidirect
346. SD16D788semidirect
358. D8D788semidirect

Maximal subgroups

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

subgroupindex
#271618912
#271618862
#271618842
#271618812
#271614682
#271613502
#271613492
#271613432
#271613422
#271612582
#271612532
#271612482
#271612022
#444×C1972
#3288197