Finite Group Databasebeta

← C2×#27161221 #38751910

Subgroups — C2×#27161221 #38751910

Order 25216.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15691357C211yesmaximal, characteristic, complemented
2C22×#15691346C211yesmaximal, characteristic, complemented
3C2×#20263297C211yesmaximal, characteristic, complemented
4#27161221C218yesmaximal, direct product factor
5C2×#20263251C211yesmaximal, characteristic, complemented
6C2×#20263247C211yesmaximal, characteristic, complemented
7C2×#20263238C211yesmaximal, characteristic, complemented
8C22×D8×C197C211yesmaximal, characteristic, complemented
9C2×#15691357—24no—
10C2×#15691357—22no—
11C2×#15691357—24no—
12C2×#15691357C2214yescomplemented
13C22×Dic394C2211yescharacteristic
14C22×C22⋊D197—21no—
15C22×C22⋊D197—21no—
16C23×Dic197—21no—
17C22×C4×D197C2211yescharacteristic
18C2×#15691346—24no—
19C2×#15691346C2214yescomplemented
20C22×#13096802C2211yescharacteristic
21#20263297—22no—
22#20263297C2214yescomplemented
23C2×C4⋊Dic197C2211yescharacteristic
24C2×C22⋊Dic197—21no—
25#20263251C2214yescomplemented
26#20263247C2214yescomplemented
27#20263238C2214yescomplemented
28C22×D4×C197C2211yescharacteristic
29C22×D4×C197C2211yescharacteristic
30C2×D8×C197—24no—
31C2×D8×C197C2214yescomplemented
32C22×C8×C197C2211yescharacteristic
33C23×D197D411yescharacteristic
34#15691357—216no—
35#15691357—28no—
36#15691357—28no—
37C2×Dic394—21no—
38C2×Dic394—22no—
39C2×Dic394D414yescomplemented
40C2×Dic394C2312yes—
41C2×C22⋊D197—28no—
42C2×C22⋊D197—24no—
43C2×C22⋊D197—24no—
44C2×C22⋊D197—44no—
45C22×Dic197—28no—
46C22×Dic197—24no—
47C22×Dic197—21no—
48C22×Dic197—21no—
49C22×Dic197D411yescharacteristic
50C2×C4×D197—22no—
51C2×C4×D197—22no—
52C2×C4×D197D412yes—
53C2×C4×D197C2312yes—
54#15691346—28no—
55C2×#13096802—22no—
56C2×#13096802C2312yes—
57C4⋊Dic197—21no—
58C4⋊Dic197C2312yes—
59C22⋊Dic197—24no—
60C24×C197—21no—
61C24×C197—21no—
62C2×D4×C197—21no—
63C2×D4×C197—22no—
64C2×D4×C197—22no—
65C2×D4×C197—22no—
66C2×D4×C197—21no—
67C2×D4×C197D414yescomplemented
68C2×D4×C197C2312yes—
69C2×D4×C197C2312yes—
70C22×C4×C197C2311yescharacteristic
71D8×C197—28no—
72C2×C8×C197—22no—
73C2×C8×C197C2312yes—
74C22×D197—22no—
75C22×D197—22no—
76C22×D197—22no—
77C22×D197C2×D412yes—
78Dic394—24no—
79Dic394—22no—
80Dic394C2×D414yescomplemented
81C22⋊D197—216no—
82C22⋊D197—48no—
83C2×Dic197—216no—
84C2×Dic197—22no—
85C2×Dic197—24no—
86C2×Dic197—22no—
87C2×Dic197—22no—
88C2×Dic197—42no—
89C2×Dic197C2×D412yes—
90C4×D197—24no—
91C4×D197—22no—
92C4×D197—22no—
93#13096802—22no—
94C23×C197—24no—
95C23×C197—22no—
96C23×C197—22no—
97C23×C197—42no—
98C23×C197—42no—
99C23×C197—42no—
100C23×C197—42no—
101C23×C197—42no—
102C23×C197C2×D411yescharacteristic
103D4×C197—24no—
104D4×C197—22no—
105D4×C197—22no—
106D4×C197—22no—
107D4×C197—42no—
108D4×C197C2×D414yescomplemented
109C2×C4×C197C2×D411yescharacteristic
110C2×C4×C197C2×D412yes—
111C2×C4×C197C2×D412yes—
112C2×C4×C197C2411yescharacteristic
113C8×C197—22no—
114C22×C197—24no—
115C22×C197—42no—
116C22×C197—42no—
117C22×C197—48no—
118C22×C197—42no—
119C22×C197—42no—
120C22×C197—42no—
121C22×C197—84no—
122C22×C197#12012yes—
123C22×C197#13512yes—
124C22×C197AGL(1,ℤ/8)12yes—
125C22×C197C22×D411yescharacteristic
126C2×D197—22no—
127C2×D197—22no—
128C2×D197—22no—
129C2×D197—44no—
130C4×C197#12012yes—
131C4×C197C22×D411yescharacteristic
132C4×C197C22×D411yescharacteristic
133Dic197—22no—
134Dic197—24no—
135Dic197—42no—
136C2×C197—44no—
137C2×C197—82no—
138C2×C197#44914yescomplemented
139C2×C197C2×#12011yescharacteristic
140C2×C197C2×#13511yescharacteristic
141C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
142D197—42no—
143C197C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1971nomaximal, Sylow
145C22×#41—1971no—
146C22×SD16—1971no—
147C22×D8D19711yescharacteristic, complemented
148C2×#121—1971no—
149#449—1978no—
150C2×#103—1971no—
151C2×#102—1971no—
152C2×C22⋊C8—1971no—
153C2×#41—3944no—
154C2×#41—3944no—
155C2×#41—3942no—
156C2×#41—1974no—
157C22×Q8—1971no—
158C22×D4—3941no—
159C22×D4—3941no—
160C22×D4C2×D19711yescharacteristic
161C22×D4C2×D19711yescharacteristic
162C23×C4—3941no—
163C23×C4—1971no—
164C2×SD16—3944no—
165C2×SD16—1974no—
166C2×D8—24no—
167C2×D8C2×D19714yescomplemented
168C22×C8—1971no—
169C22×C8C2×D19711yescharacteristic
170#121—3942no—
171#121—1974no—
172C2×C4⋊C4—1971no—
173C2×C22⋊C4—3941no—
174#103—1974no—
175#102—1974no—
176C22⋊C8—1974no—
177C24—21no—
178C24—21no—
179C24—1971no—
180#41—39416no—
181#41—3948no—
182#41—3948no—
183C2×Q8—3941no—
184C2×Q8—3942no—
185C2×Q8—1974no—
186C2×Q8—1972no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—7884no—
193C2×D4—3944no—
194C2×D4—3948no—
195C2×D4—3944no—
196C2×D4C22⋊D19714yescomplemented
197C2×D4C22×D19712yes—
198C2×D4C22×D19712yes—
199C22×C4—3941no—
200C22×C4—3948no—
201C22×C4—3941no—
202C22×C4—3944no—
203C22×C4—3942no—
204C22×C4—3942no—
205C22×C4—1971no—
206C22×C4—1972no—
207C22×C4—1972no—
208C22×C4C22×D19711yescharacteristic
209SD16—3948no—
210D8—28no—
211C2×C8—22no—
212C2×C8—3942no—
213C2×C8—1972no—
214C2×C8C22×D19712yes—
215C4⋊C4—3941no—
216C4⋊C4—1972no—
217C22⋊C4—3944no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—3942no—
227C23—3942no—
228C23—3942no—
229C23—1972no—
230C23D4×D19711yescharacteristic
231Q8—3944no—
232Q8—3942no—
233Q8—1974no—
234D4—24no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—42no—
239D4—7888no—
240D4—39416no—
241D4C2×C22⋊D19714yescomplemented
242C2×C4—7882no—
243C2×C4—3942no—
244C2×C4—39416no—
245C2×C4—3942no—
246C2×C4—3944no—
247C2×C4—3942no—
248C2×C4—3944no—
249C2×C4—3942no—
250C2×C4—3942no—
251C2×C4—1972no—
252C2×C4D4×D19711yescharacteristic
253C2×C4D4×D19712yes—
254C2×C4C2×C22⋊D19712yes—
255C2×C4C23×D19711yescharacteristic
256C8—22no—
257C8—3942no—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—48no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—7884no—
267C22—3942no—
268C22—3942no—
269C22—3942no—
270C22#2026329412yes—
271C22#2026334512yes—
272C22#2026335012yes—
273C22C2×D4×D19711yescharacteristic
274C4—7882no—
275C4—3942no—
276C4—3944no—
277C4#2026329412yes—
278C4C2×D4×D19711yescharacteristic
279C4C22×C22⋊D19711yescharacteristic
280C2—44no—
281C2—82no—
282C2—7882no—
283C2#2716122114yesdirect product factor
284C2C2×#2026329411yescharacteristic
285C2C2×#2026334511yescharacteristic
286C2C2×#2026335011yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×D8×C197C2
socleC23×C197C2×D4
hypercenterC22×D8D197
hypocenterC197C2×#449
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#15691357C2semidirect
2. C22×#15691346C2semidirect
3. C2×#20263297C2semidirect
4. #27161221C2direct (N × H)
5. C2×#20263251C2semidirect
6. C2×#20263247C2semidirect
7. C2×#20263238C2semidirect
8. C22×D8×C197C2semidirect
12. C2×#15691357C22semidirect
19. C2×#15691346C22semidirect
22. #20263297C22semidirect
25. #20263251C22semidirect
26. #20263247C22semidirect
27. #20263238C22semidirect
31. C2×D8×C197C22semidirect
39. C2×Dic394D4semidirect
67. C2×D4×C197D4semidirect
80. Dic394C2×D4semidirect
108. D4×C197C2×D4semidirect
138. C2×C197#449semidirect
143. C197C2×#449semidirect
147. C22×D8D197semidirect
167. C2×D8C2×D197semidirect
196. C2×D4C22⋊D197semidirect
241. D4C2×C22⋊D197semidirect
283. C2#27161221direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156913572
C22×#156913462
C2×#202632972
#271612212
C2×#202632512
C2×#202632472
C2×#202632382
C22×D8×C1972
C2×#449197