Finite Group Databasebeta

← #38754549

Subgroups — #38754549

Order 25216.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161769C211yesmaximal, characteristic, complemented
2#27161725C211yesmaximal, characteristic
3#27161434C211yesmaximal, characteristic, complemented
4#27161410C211yesmaximal, characteristic, complemented
5#27161404C211yesmaximal, characteristic, complemented
6#27161380C211yesmaximal, characteristic, complemented
7#27161320C211yesmaximal, characteristic, complemented
8#27161308C211yesmaximal, characteristic, complemented
9#27161302C211yesmaximal, characteristic, complemented
10#27161243C211yesmaximal, characteristic, complemented
11#27161232C211yesmaximal, characteristic, complemented
12C4×#15691347C211yesmaximal, characteristic, complemented
13#27161076C211yesmaximal, characteristic, complemented
14C2×#20263248C211yesmaximal, characteristic, complemented
15C8×D4×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×Dic788C2211yescharacteristic, complemented
19C2×#15691347C2211yescharacteristic
20C2×#15691347C2211yescharacteristic
21#20263319—21no—
22#20263318C2211yescharacteristic, complemented
23#20263316—21no—
24#20263314C2211yescharacteristic
25#20263311—21no—
26#20263307—21no—
27#20263305C2211yescharacteristic
28#20263305C2211yescharacteristic
29#20263305C2211yescharacteristic, complemented
30#20263300C2211yescharacteristic
31#20263299—21no—
32#20263294—21no—
33C4×Dic394C2211yescharacteristic, complemented
34C4×D788C2211yescharacteristic
35#20263285C2211yescharacteristic, complemented
36#20263282—21no—
37C2×C4⋊Dic197C2211yescharacteristic
38C22⋊C4×D197—21no—
39#20263260C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41#20263258C2211yescharacteristic, complemented
42#20263251C2211yescharacteristic, complemented
43#20263251C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263251C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic
47#20263248C2211yescharacteristic
48#20263248C2211yescharacteristic
49#20263248C2211yescharacteristic
50#20263248C2211yescharacteristic
51#20263248C2211yescharacteristic
52C22×C8×C197C2211yescharacteristic
53C22×C8×C197C2211yescharacteristic, complemented
54C4×D4×C197C2211yescharacteristic
55C4⋊C8×C197C2211yescharacteristic, complemented
56C22⋊C8×C197C2211yescharacteristic
57C22⋊C8×C197C2211yescharacteristic, complemented
58C4×C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691358—22no—
61#15691358—41no—
62C2×Dic394C2311yescharacteristic
63C2×Dic394C2311yescharacteristic
64C2×D788—22no—
65C2×D788—21no—
66C2×D788—41no—
67C2×D788C2311yescharacteristic
68C2×D788C2311yescharacteristic
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71C22×Dic197—21no—
72C2×C4×D197—21no—
73C2×C4×D197—21no—
74Dic788—41no—
75#15691347—22no—
76#15691347—41no—
77C4⋊Dic197—21no—
78C4⋊Dic197C2311yescharacteristic
79C4⋊Dic197C2311yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82#15691337—21no—
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86#15691337—21no—
87#15691337—21no—
88C22⋊Dic197—21no—
89C22⋊Dic197—21no—
90C22⋊Dic197—21no—
91C22⋊Dic197—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96C4×Dic197—21no—
97C4×Dic197—21no—
98C2×D4×C197D411yescharacteristic
99C22×C4×C197C2311yescharacteristic
100C22×C4×C197C2311yescharacteristic
101C2×C8×C197—41no—
102C2×C8×C197—41no—
103C2×C8×C197C2311yescharacteristic
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197D411yescharacteristic
108C22⋊C4×C197D411yescharacteristic
109C22⋊C4×C197D411yescharacteristic
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—81no—
116Dic394—21no—
117Dic394—21no—
118Dic394—21no—
119Dic394—21no—
120D788—21no—
121D788—21no—
122D788—21no—
123D788—42no—
124D788#4112yes—
125C22⋊D197—42no—
126C22⋊D197—81no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197—21no—
130C2×Dic197—21no—
131C2×Dic197—21no—
132C2×Dic197—21no—
133C2×Dic197—42no—
134C4×D197—42no—
135C4×D197—81no—
136C23×C197C2×D411yescharacteristic
137C23×C197C2×D411yescharacteristic
138D4×C197—22no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2411yescharacteristic
148C8×C197—21no—
149C8×C197—21no—
150C8×C197—21no—
151C8×C197—21no—
152C22×C197—21no—
153C22×C197—21no—
154C22×C197—21no—
155C22×C197—21no—
156C22×C197—21no—
157C22×C197#13512yes—
158C22×C197C22×D411yescharacteristic
159C2×D197—22no—
160C2×D197—41no—
161C2×D197—41no—
162C2×D197—41no—
163C2×D197—41no—
164C2×D197—41no—
165C2×D197—84no—
166C4×C197—21no—
167C4×C197—21no—
168C4×C197—21no—
169C4×C197—21no—
170C4×C197C2×#4111yescharacteristic
171C4×C197H(5,𝔽2)11yescharacteristic
172Dic197—41no—
173Dic197—41no—
174Dic197—41no—
175Dic197—41no—
176Dic197—41no—
177Dic197—41no—
178C2×C197—22no—
179C2×C197—41no—
180C2×C197#54611yescharacteristic
181C2×C197C2×#13511yescharacteristic
182C2×C197#57711yescharacteristic
183D197—42no—
184D197—81no—
185C197#329611yescharacteristic, complemented, Sylow
186#3296—1971nomaximal, Sylow
187#552—1971no—
188#546—1971no—
189#488—1971no—
190#483—1971no—
191#482—1971no—
192#478—1971no—
193#467—1971no—
194#465—1971no—
195#464—1971no—
196#452—1971no—
197#450—1971no—
198C4×SD16—1971no—
199C8×D4D19711yescharacteristic, complemented
200#416—1971no—
201C2×#102—1971no—
202C2×#41—1971no—
203C22×D4—1971no—
204C2×Q16—1971no—
205C2×SD16—1971no—
206C2×SD16—1971no—
207C22×C8C2×D19711yescharacteristic
208C22×C8C2×D19711yescharacteristic, complemented
209#126—3941no—
210#125—3941no—
211#125—1971no—
212#124—1971no—
213#123—3941no—
214#123—3941no—
215C22⋊Q8—1971no—
216C22⋊Q8—1971no—
217C22⋊Q8—1971no—
218#121—3941no—
219#121—1971no—
220#120—3941no—
221C4×Q8—1971no—
222C4×D4—1971no—
223C4×D4C2×D19711yescharacteristic
224#117—3941no—
225#117—1971no—
226C2×C4⋊C4—1971no—
227C2×C22⋊C4—3941no—
228#107—1971no—
229#106—1971no—
230#106—1971no—
231C4⋊C8C2×D19711yescharacteristic, complemented
232#103—1971no—
233#103—1971no—
234#103—1971no—
235#103—1971no—
236#102—1971no—
237#102—1971no—
238#102—1971no—
239#102—1971no—
240#102—1971no—
241#102—1971no—
242C22⋊C8C2×D19711yescharacteristic
243C22⋊C8C2×D19711yescharacteristic, complemented
244C4×C8C2×D19711yescharacteristic, complemented
245C24—3941no—
246#41—7881no—
247#41—3942no—
248C2×Q8—1971no—
249C2×Q8—1971no—
250C2×D4—7881no—
251C2×D4—3942no—
252C2×D4—3941no—
253C2×D4—3941no—
254C2×D4—3941no—
255C2×D4—1971no—
256C2×D4—1971no—
257C2×D4D78811yescharacteristic
258C22×C4—3941no—
259C22×C4—3941no—
260C22×C4—3941no—
261C22×C4C22×D19711yescharacteristic
262C22×C4C22×D19711yescharacteristic
263Q16—7881no—
264SD16—7881no—
265SD16—3942no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D19711yescharacteristic
269C2×C8C22×D19711yescharacteristic
270C2×C8C22×D19711yescharacteristic
271C2×C8C22×D19711yescharacteristic
272C4⋊C4—3941no—
273C4⋊C4—3941no—
274C4⋊C4—3941no—
275C4⋊C4—3941no—
276C4⋊C4—3941no—
277C4⋊C4—3941no—
278C4⋊C4—3941no—
279C4⋊C4—1971no—
280C4⋊C4—1971no—
281C4⋊C4—1971no—
282C4⋊C4—1971no—
283C4⋊C4D78811yescharacteristic
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C22⋊C4—3941no—
287C22⋊C4—3941no—
288C22⋊C4—3941no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—3941no—
292C22⋊C4D78811yescharacteristic
293C22⋊C4D78811yescharacteristic
294C42—3941no—
295C42—3941no—
296C42C22×D19711yescharacteristic
297C23—7882no—
298C23—7882no—
299C23—15761no—
300C23—3941no—
301C23—3941no—
302C23C2×D78811yescharacteristic
303C23C2×D78811yescharacteristic
304Q8—3941no—
305Q8—3941no—
306Q8—3941no—
307Q8—3941no—
308D4—22no—
309D4—7882no—
310D4—7882no—
311D4—15761no—
312D4—3941no—
313D4—3941no—
314D4—3941no—
315D4—1972no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—7882no—
321C2×C4—7882no—
322C2×C4—15761no—
323C2×C4—3941no—
324C2×C4—3941no—
325C2×C4—3941no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4—3941no—
329C2×C4C2×D78811yescharacteristic
330C2×C4C2×D78811yescharacteristic
331C2×C4C2×D78811yescharacteristic
332C2×C4C2×D78811yescharacteristic
333C2×C4C23×D19711yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—7881no—
344C22—7881no—
345C22—7881no—
346C22—7881no—
347C22—7881no—
348C22—15764no—
349C22—3942no—
350C22#2026334812yes—
351C22C22×D78811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—7881no—
357C4—7881no—
358C4—7881no—
359C4—7881no—
360C4—7881no—
361C4—7881no—
362C4#41×D19711yescharacteristic
363C4#2026337111yescharacteristic
364C2—22no—
365C2—41no—
366C2—7882no—
367C2—15761no—
368C2#2716172411yescharacteristic
369C2C2×#2026334811yescharacteristic
370C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC8×D4×C197C2
socleC22×C197C22×D4
prime radical#27161725C2
hypercenterC8×D4D197
hypocenterC197#3296
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #27161769C2semidirect
3. #27161434C2semidirect
4. #27161410C2semidirect
5. #27161404C2semidirect
6. #27161380C2semidirect
7. #27161320C2semidirect
8. #27161308C2semidirect
9. #27161302C2semidirect
10. #27161243C2semidirect
11. #27161232C2semidirect
12. C4×#15691347C2semidirect
13. #27161076C2semidirect
14. C2×#20263248C2semidirect
15. C8×D4×C197C2semidirect
18. C2×Dic788C22semidirect
22. #20263318C22semidirect
29. #20263305C22semidirect
33. C4×Dic394C22semidirect
35. #20263285C22semidirect
39. #20263260C22semidirect
40. #20263258C22semidirect
41. #20263258C22semidirect
42. #20263251C22semidirect
43. #20263251C22semidirect
44. #20263251C22semidirect
45. #20263251C22semidirect
53. C22×C8×C197C22semidirect
55. C4⋊C8×C197C22semidirect
57. C22⋊C8×C197C22semidirect
58. C4×C8×C197C22semidirect
185. C197#3296semidirect
199. C8×D4D197semidirect
208. C22×C8C2×D197semidirect
231. C4⋊C8C2×D197semidirect
243. C22⋊C8C2×D197semidirect
244. C4×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617692
#271617252
#271614342
#271614102
#271614042
#271613802
#271613202
#271613082
#271613022
#271612432
#271612322
C4×#156913472
#271610762
C2×#202632482
C8×D4×C1972
#3296197