Finite Group Databasebeta

← #38754527

Subgroups — #38754527

Order 25216.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161734C211yesmaximal, characteristic, complemented
2D4×D788C211yesmaximal, characteristic, complemented
3#27161426C211yesmaximal, characteristic, complemented
4#27161394C212yesmaximal, complemented
5#27161359C211yesmaximal, characteristic, complemented
6#27161314C212yesmaximal, complemented
7#27161274C211yesmaximal, characteristic, complemented
8#27161211C212yesmaximal, complemented
9C8×D788C211yesmaximal, characteristic, complemented
10#102×D197C212yesmaximal, complemented
11C4×D8×C197C211yesmaximal, characteristic, complemented
12C2×#15691358—21no—
13C22×D788—21no—
14C2×D4×D197C2212yes—
15C2×#15691343C2212yescomplemented
16C2×C8×D197C2212yescomplemented
17#20263326C2211yescharacteristic, complemented
18#20263323C2211yescharacteristic
19#20263308—22no—
20#20263305—21no—
21#20263301C2212yes—
22#20263300—21no—
23#20263297C2212yescomplemented
24#20263296—22no—
25C4×D788C2211yescharacteristic
26C4⋊C4×D197C2212yescomplemented
27#20263259C2212yescomplemented
28C4⋊#13096802C2211yescharacteristic, complemented
29#20263248C2212yescomplemented
30#20263248C2212yescomplemented
31#20263247C2212yes—
32#20263246C2212yescomplemented
33#20263238C2212yescomplemented
34C2×D8×C197C2211yescharacteristic
35C4×D4×C197C2211yescharacteristic
36C4×D4×C197C2211yescharacteristic, complemented
37#107×C197C2211yescharacteristic, complemented
38#102×C197C2211yescharacteristic
39#102×C197C2211yescharacteristic, complemented
40C4×C8×C197C2211yescharacteristic, complemented
41C23×D197—22no—
42#15691358—44no—
43C2×Dic394—21no—
44C2×D788—22no—
45C2×D788—21no—
46C2×D788—21no—
47C2×D788—44no—
48C2×D788D411yescharacteristic
49C2×D788C2312yes—
50C2×C22⋊D197—22no—
51C2×C22⋊D197—22no—
52D4×D197—24no—
53D4×D197—42no—
54C2×C4×D197—22no—
55C2×C4×D197C2312yes—
56#15691343—42no—
57C2×#13096802C2312yes—
58C8×D197—42no—
59C4⋊Dic197—21no—
60C4⋊Dic197D411yescharacteristic
61C4⋊Dic197C2312yes—
62#15691337—22no—
63C22⋊Dic197—22no—
64#15691335—22no—
65#15691335D412yes—
66C2×D4×C197C2311yescharacteristic
67C2×D4×C197C2311yescharacteristic
68C22×C4×C197—21no—
69C22×C4×C197—21no—
70D8×C197—22no—
71C2×C8×C197C2311yescharacteristic
72C2×C8×C197C2311yescharacteristic
73C4⋊C4×C197C2311yescharacteristic
74C4⋊C4×C197C2311yescharacteristic
75C22⋊C4×C197—21no—
76C22⋊C4×C197—21no—
77C42×C197C2311yescharacteristic
78C22×D197—22no—
79C22×D197—44no—
80C22×D197—44no—
81C22×D197—82no—
82C22×D197C2×D412yes—
83Dic394—42no—
84D788—22no—
85D788—22no—
86D788—44no—
87D788—42no—
88D788—81no—
89D788D814yescomplemented
90C22⋊D197—44no—
91C22⋊D197—44no—
92C2×Dic197—22no—
93C2×Dic197C2×D412yes—
94C4×D197—22no—
95C4×D197—22no—
96C4×D197—44no—
97#13096802—22no—
98C23×C197—21no—
99C23×C197—21no—
100D4×C197—21no—
101D4×C197—21no—
102D4×C197—21no—
103D4×C197#4112yescomplemented
104C2×C4×C197—21no—
105C2×C4×C197—21no—
106C2×C4×C197—42no—
107C2×C4×C197—42no—
108C2×C4×C197C2×D411yescharacteristic
109C2×C4×C197C2×D411yescharacteristic
110C2×C4×C197C2411yescharacteristic
111C8×C197—21no—
112C8×C197—21no—
113C22×C197—22no—
114C22×C197—41no—
115C22×C197—41no—
116C22×C197—41no—
117C22×C197—41no—
118C22×C197—41no—
119C22×C197C22×D411yescharacteristic
120C2×D197—22no—
121C2×D197—22no—
122C2×D197—42no—
123C2×D197—42no—
124C2×D197—42no—
125C2×D197—88no—
126C2×D197C2×D814yes—
127C4×C197—21no—
128C4×C197—41no—
129C4×C197—41no—
130C4×C197C2×D812yes—
131C4×C197C2×#4111yescharacteristic
132C4×C197H(5,𝔽2)11yescharacteristic
133Dic197—22no—
134Dic197—42no—
135C2×C197—42no—
136C2×C197—81no—
137C2×C197#54611yescharacteristic
138C2×C197C22×D811yescharacteristic
139C2×C197#57711yescharacteristic
140D197—24no—
141D197—82no—
142C197#329411yescharacteristic, complemented, Sylow
143#3294—1971nomaximal, Sylow
144#547—1971no—
145D42—1971no—
146#486—1971no—
147#480—1972no—
148#474—1971no—
149#466—1972no—
150#459—1971no—
151#447—1972no—
152C4×D8D19711yescharacteristic, complemented
153C8×D4—1971no—
154C2×#102—1972no—
155C2×#41—3941no—
156C22×D4—3941no—
157C22×D4—1972no—
158C2×D8—1972no—
159C2×D8C2×D19711yescharacteristic
160C22×C8—1972no—
161C4⋊Q8—1971no—
162#127—1971no—
163#123—3942no—
164C22⋊Q8—3941no—
165#121—3941no—
166#121—1972no—
167#121—1972no—
168#120—3942no—
169C4×D4—1971no—
170C4×D4C2×D19711yescharacteristic
171C4×D4C2×D19711yescharacteristic, complemented
172C2×C4⋊C4—1972no—
173#107—1972no—
174#107C2×D19711yescharacteristic, complemented
175C4⋊C8—1971no—
176#102—1972no—
177#102—1972no—
178#102—1972no—
179#102—1972no—
180#102C2×D19711yescharacteristic
181#102C2×D19711yescharacteristic, complemented
182C22⋊C8—1972no—
183C4×C8C2×D19711yescharacteristic, complemented
184C24—3942no—
185#41—7884no—
186C2×Q8—3941no—
187C2×D4—7884no—
188C2×D4—7882no—
189C2×D4—3942no—
190C2×D4—3942no—
191C2×D4—3944no—
192C2×D4—3942no—
193C2×D4—3941no—
194C2×D4—3941no—
195C2×D4—1971no—
196C2×D4—1972no—
197C2×D4C22×D19711yescharacteristic
198C2×D4C22×D19711yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—3942no—
202C22×C4—1972no—
203D8—22no—
204D8—7882no—
205C2×C8—7882no—
206C2×C8—1972no—
207C2×C8C22×D19711yescharacteristic
208C2×C8C22×D19711yescharacteristic
209C4⋊C4—3942no—
210C4⋊C4—3941no—
211C4⋊C4—1972no—
212C4⋊C4—1971no—
213C4⋊C4C22×D19711yescharacteristic
214C4⋊C4C22×D19711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—3942no—
218C22⋊C4—3942no—
219C22⋊C4—1972no—
220C42C22×D19711yescharacteristic
221C23—21no—
222C23—21no—
223C23—7884no—
224C23—7884no—
225C23—15762no—
226C23—3942no—
227C23—1972no—
228Q8—7882no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—7884no—
233D4—7884no—
234D4—7884no—
235D4—7882no—
236D4—15761no—
237D4—3942no—
238D4—3942no—
239D4—1974no—
240D4#1569135812yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—7884no—
246C2×C4—3942no—
247C2×C4—3942no—
248C2×C4—3942no—
249C2×C4—1972no—
250C2×C4D4×D19711yescharacteristic
251C2×C4D4×D19711yescharacteristic
252C2×C4C23×D19711yescharacteristic
253C8—21no—
254C8—21no—
255C8—3942no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—7882no—
263C22—7882no—
264C22—7882no—
265C22—15768no—
266C22—3942no—
267C22—3942no—
268C22—1974no—
269C22C2×D4×D19711yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—7882no—
274C4—3942no—
275C4D8×D19712yes—
276C4C2×#1569135811yescharacteristic
277C4#2026337211yescharacteristic
278C2—42no—
279C2—81no—
280C2—15762no—
281C2—3944no—
282C2#2716172511yescharacteristic
283C2C2×D8×D19711yescharacteristic
284C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC4×D8×C197C2
socleC22×C197C22×D4
hypercenterC4×D8D197
hypocenterC197#3294
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #27161734C2semidirect
2. D4×D788C2semidirect
3. #27161426C2semidirect
4. #27161394C2semidirect
5. #27161359C2semidirect
6. #27161314C2semidirect
7. #27161274C2semidirect
8. #27161211C2semidirect
9. C8×D788C2semidirect
10. #102×D197C2semidirect
11. C4×D8×C197C2semidirect
15. C2×#15691343C22semidirect
16. C2×C8×D197C22semidirect
17. #20263326C22semidirect
23. #20263297C22semidirect
26. C4⋊C4×D197C22semidirect
27. #20263259C22semidirect
28. C4⋊#13096802C22semidirect
29. #20263248C22semidirect
30. #20263248C22semidirect
32. #20263246C22semidirect
33. #20263238C22semidirect
36. C4×D4×C197C22semidirect
37. #107×C197C22semidirect
39. #102×C197C22semidirect
40. C4×C8×C197C22semidirect
89. D788D8semidirect
103. D4×C197#41semidirect
142. C197#3294semidirect
152. C4×D8D197semidirect
171. C4×D4C2×D197semidirect
174. #107C2×D197semidirect
181. #102C2×D197semidirect
183. C4×C8C2×D197semidirect
240. D4#15691358semidirect

Maximal subgroups

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

subgroupindex
#271617342
D4×D7882
#271614262
#271613942
#271613592
#271613142
#271612742
#271612112
C8×D7882
#102×D1972
C4×D8×C1972
#3294197