Finite Group Databasebeta

← D4×#15691346 #38754404

Subgroups — D4×#15691346 #38754404

Order 25216.

546 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#15691346C212yesmaximal, complemented
2D4×Dic394C211yesmaximal, characteristic, complemented
3#27161451C211yesmaximal, characteristic, complemented
4#27161306C212yesmaximal, complemented
5#27161284C211yesmaximal, characteristic, complemented
6#27161276C211yesmaximal, characteristic, complemented
7#27161230C212yesmaximal, complemented
8#27161217C212yesmaximal, complemented
9C4×#15691346C211yesmaximal, characteristic, complemented
10D4×#13096802C211yesmaximal, characteristic, complemented
11D42×C197C211yesmaximal, characteristic
12C22×Dic394C2212yescomplemented
13C2×#15691346—28no—
14C2×#15691346C2212yes—
15C2×#15691346C2211yescharacteristic
16C2×#15691346C2212yescomplemented
17C2×#15691346C2212yescomplemented
18C2×#15691346C2212yescomplemented
19C22×#13096802C2212yescomplemented
20#20263326C2211yescharacteristic, complemented
21#20263325—21no—
22#20263305C2212yescomplemented
23#20263303—22no—
24C4×Dic394C2211yescharacteristic, complemented
25D4×Dic197—21no—
26#20263257C2211yescharacteristic, complemented
27C4⋊#13096802C2211yescharacteristic, complemented
28#20263249C2211yescharacteristic, complemented
29#20263249C2212yescomplemented
30#20263247C2211yescharacteristic
31#20263247C2212yescomplemented
32C22⋊#13096802C2212yescomplemented
33C4×#13096802C2211yescharacteristic, complemented
34C22×D4×C197—21no—
35C22×D4×C197C2212yes—
36#127×C197C2211yescharacteristic
37#121×C197—21no—
38#121×C197C2212yes—
39#120×C197—22no—
40C4×D4×C197C2211yescharacteristic
41C4×D4×C197C2211yescharacteristic
42C2×Dic394—24no—
43C2×Dic394—22no—
44C2×Dic394—42no—
45C2×Dic394C2312yes—
46C2×Dic394C2311yescharacteristic
47C22×Dic197—22no—
48#15691346—24no—
49#15691346—24no—
50#15691346—24no—
51#15691346—42no—
52#15691346D414yesdirect product factor
53C2×#13096802—24no—
54C2×#13096802C2311yescharacteristic
55C2×#13096802C2312yes—
56C2×#13096802C2311yescharacteristic
57C4⋊Dic197—21no—
58C4⋊Dic197—21no—
59C4⋊Dic197C2311yescharacteristic
60#15691337—21no—
61#15691337—22no—
62C22⋊Dic197—22no—
63C4×Dic197—21no—
64C24×C197—22no—
65C2×D4×C197—24no—
66C2×D4×C197—22no—
67C2×D4×C197—22no—
68C2×D4×C197—21no—
69C2×D4×C197—21no—
70C2×D4×C197—42no—
71C2×D4×C197—44no—
72C2×D4×C197D411yescharacteristic
73C2×D4×C197C2312yes—
74C2×D4×C197C2311yescharacteristic
75C22×C4×C197—21no—
76C22×C4×C197C2312yes—
77C4⋊C4×C197D411yescharacteristic
78C4⋊C4×C197C2311yescharacteristic
79C22⋊C4×C197—21no—
80C22⋊C4×C197—22no—
81C22⋊C4×C197D412yes—
82C42×C197C2311yescharacteristic
83Dic394—21no—
84Dic394—22no—
85Dic394—22no—
86Dic394—44no—
87Dic394C2×D412yescomplemented
88C2×Dic197—21no—
89C2×Dic197—21no—
90C2×Dic197—22no—
91C2×Dic197—44no—
92#13096802—22no—
93#13096802—21no—
94#13096802C2×D412yescomplemented
95C23×C197—21no—
96C23×C197—22no—
97C23×C197—44no—
98C23×C197—44no—
99C23×C197—82no—
100C23×C197C2×D412yes—
101D4×C197—22no—
102D4×C197—21no—
103D4×C197—22no—
104D4×C197—44no—
105D4×C197—44no—
106D4×C197—42no—
107D4×C197—81no—
108D4×C197SD1614yescomplemented
109D4×C197C2×D412yes—
110C2×C4×C197—22no—
111C2×C4×C197—22no—
112C2×C4×C197—21no—
113C2×C4×C197—42no—
114C2×C4×C197C2×D411yescharacteristic
115C2×C4×C197C2×D412yes—
116C2×C4×C197C2×D411yescharacteristic
117C2×C4×C197C2411yescharacteristic
118C22×C197—22no—
119C22×C197—22no—
120C22×C197—22no—
121C22×C197—41no—
122C22×C197—41no—
123C22×C197—42no—
124C22×C197—42no—
125C22×C197—42no—
126C22×C197—88no—
127C22×C197C2×SD1614yescomplemented
128C22×C197C22×D411yescharacteristic
129C4×C197—22no—
130C4×C197—21no—
131C4×C197—41no—
132C4×C197C2×SD1612yescomplemented
133C4×C197C22×D411yescharacteristic
134C4×C197H(5,𝔽2)11yescharacteristic
135Dic197—22no—
136Dic197—42no—
137Dic197—41no—
138C2×C197—24no—
139C2×C197—42no—
140C2×C197—82no—
141C2×C197D4211yescharacteristic
142C2×C197C22×SD1611yescharacteristic
143C2×C197#57711yescharacteristic
144C197D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1971nomaximal, Sylow
146C22×SD16—1972no—
147D4×Q8—1971no—
148D42D19711yescharacteristic
149#492—1971no—
150#465—1972no—
151#461—1971no—
152#460—1971no—
153#450—1972no—
154#448—1972no—
155C4×SD16—1971no—
156C8×D4—1971no—
157C22×Q8—1972no—
158C22×D4—21no—
159C22×D4C2×D19712yes—
160C2×SD16—3948no—
161C2×SD16—1972no—
162C2×SD16—1972no—
163C2×SD16—1972no—
164C2×SD16—1972no—
165C2×SD16—1971no—
166C22×C8—1972no—
167C4⋊Q8—3941no—
168C4⋊Q8—1971no—
169#127C2×D19711yescharacteristic
170C22⋊Q8—3942no—
171C22⋊Q8—1972no—
172#121—21no—
173#121C2×D19712yes—
174#120—22no—
175C4×Q8—1971no—
176C4×D4—3941no—
177C4×D4C2×D19711yescharacteristic
178C4×D4C2×D19711yescharacteristic
179#106—1971no—
180C4⋊C8—1971no—
181#103—1971no—
182#103—1972no—
183#102—1971no—
184#102—1972no—
185C22⋊C8—1972no—
186C4×C8—1971no—
187C24—22no—
188C2×Q8—7882no—
189C2×Q8—3944no—
190C2×Q8—3942no—
191C2×Q8—1972no—
192C2×Q8—1971no—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—42no—
199C2×D4—44no—
200C2×D4C22⋊D19711yescharacteristic
201C2×D4C22×D19712yes—
202C2×D4C22×D19711yescharacteristic
203C22×C4—21no—
204C22×C4—3942no—
205C22×C4C22×D19712yes—
206SD16—7882no—
207SD16—3944no—
208SD16—3944no—
209SD16—3944no—
210SD16—1974no—
211C2×C8—3944no—
212C2×C8—1971no—
213C2×C8—1972no—
214C2×C8—1971no—
215C4⋊C4—3941no—
216C4⋊C4—3942no—
217C4⋊C4—3941no—
218C4⋊C4—3941no—
219C4⋊C4—1971no—
220C4⋊C4C22⋊D19711yescharacteristic
221C4⋊C4C22×D19711yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—22no—
224C22⋊C4—3942no—
225C22⋊C4C22⋊D19712yes—
226C42—3941no—
227C42C22×D19711yescharacteristic
228C23—21no—
229C23—22no—
230C23—44no—
231C23—44no—
232C23—82no—
233C23C2×C22⋊D19712yes—
234Q8—7884no—
235Q8—3942no—
236Q8—3942no—
237Q8—3941no—
238Q8—1972no—
239D4—22no—
240D4—21no—
241D4—22no—
242D4—44no—
243D4—44no—
244D4—42no—
245D4—81no—
246D4#1569134614yesdirect product factor
247D4D4×D19712yes—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—7884no—
253C2×C4—3941no—
254C2×C4—3942no—
255C2×C4—3941no—
256C2×C4C2×C22⋊D19711yescharacteristic
257C2×C4C2×C22⋊D19712yes—
258C2×C4C2×C22⋊D19711yescharacteristic
259C2×C4C23×D19711yescharacteristic
260C8—3942no—
261C8—3941no—
262C8—1972no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—88no—
272C22C2×#1569134614yescomplemented
273C22C22×C22⋊D19711yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—7881no—
278C4—7882no—
279C4—3942no—
280C4C2×#1569134612yescomplemented
281C4C2×D4×D19711yescharacteristic
282C4#2026337111yescharacteristic
283C2—24no—
284C2—42no—
285C2—82no—
286C2D4×C22⋊D19711yescharacteristic
287C2C22×#1569134611yescharacteristic
288C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting, prime radicalD42×C197C2
socleC22×C197C22×D4
hypercenterD42D197
hypocenterC197D4×SD16
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C22×#15691346C2semidirect
2. D4×Dic394C2semidirect
3. #27161451C2semidirect
4. #27161306C2semidirect
5. #27161284C2semidirect
6. #27161276C2semidirect
7. #27161230C2semidirect
8. #27161217C2semidirect
9. C4×#15691346C2semidirect
10. D4×#13096802C2semidirect
12. C22×Dic394C22semidirect
16. C2×#15691346C22semidirect
17. C2×#15691346C22semidirect
18. C2×#15691346C22semidirect
19. C22×#13096802C22semidirect
20. #20263326C22semidirect
22. #20263305C22semidirect
24. C4×Dic394C22semidirect
26. #20263257C22semidirect
27. C4⋊#13096802C22semidirect
28. #20263249C22semidirect
29. #20263249C22semidirect
31. #20263247C22semidirect
32. C22⋊#13096802C22semidirect
33. C4×#13096802C22semidirect
52. #15691346D4direct (N × H)
87. Dic394C2×D4semidirect
94. #13096802C2×D4semidirect
108. D4×C197SD16semidirect
127. C22×C197C2×SD16semidirect
132. C4×C197C2×SD16semidirect
144. C197D4×SD16semidirect
246. D4#15691346direct (N × H)
272. C22C2×#15691346semidirect
280. C4C2×#15691346semidirect

Maximal subgroups

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

subgroupindex
C22×#156913462
D4×Dic3942
#271614512
#271613062
#271612842
#271612762
#271612302
#271612172
C4×#156913462
D4×#130968022
D42×C1972
D4×SD16197