Finite Group Databasebeta

← C2×#27161222 #38751911

Subgroups — C2×#27161222 #38751911

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×#15691358C211yesmaximal, characteristic, complemented
2C22×#15691346C211yesmaximal, characteristic, complemented
3C22×#15691343C211yesmaximal, characteristic, complemented
4#27161222C218yesmaximal, direct product factor
5C2×#20263249C211yesmaximal, characteristic, complemented
6C2×#20263246C211yesmaximal, characteristic, complemented
7C2×C22⋊#13096802C211yesmaximal, characteristic, complemented
8C2×#121×C197C211yesmaximal, characteristic, complemented
9C2×#15691358—24no—
10C2×#15691358—24no—
11C2×#15691358—22no—
12C2×#15691358C2214yescomplemented
13C22×Dic394C2211yescharacteristic
14C22×D788C2211yescharacteristic
15C22×C22⋊D197—21no—
16C22×C4×D197—21no—
17C2×#15691346—24no—
18C2×#15691346C2214yescomplemented
19C2×#15691343—24no—
20C2×#15691343C2214yescomplemented
21C22×#13096802C2211yescharacteristic
22C22×#13096802C2211yescharacteristic
23#20263249C2214yescomplemented
24#20263246C2214yescomplemented
25C22⋊#13096802C2214yescomplemented
26C22×D4×C197—21no—
27C22×D4×C197C2211yescharacteristic
28C23×C4×C197C2211yescharacteristic
29#121×C197—22no—
30#121×C197C2214yescomplemented
31C2×C4⋊C4×C197C2211yescharacteristic
32C2×C22⋊C4×C197—21no—
33C23×D197—21no—
34#15691358—216no—
35#15691358—28no—
36#15691358—28no—
37C2×Dic394—22no—
38C2×Dic394—21no—
39C2×Dic394D414yescomplemented
40C2×Dic394C2312yes—
41C2×D788—22no—
42C2×D788—21no—
43C2×D788D414yescomplemented
44C2×D788C2312yes—
45C2×C22⋊D197—28no—
46C2×C22⋊D197—24no—
47C22×Dic197—21no—
48C2×C4×D197—28no—
49C2×C4×D197—24no—
50#15691346—28no—
51#15691343—28no—
52C2×#13096802—22no—
53C2×#13096802—22no—
54C2×#13096802C2312yes—
55C2×#13096802C2312yes—
56C24×C197—21no—
57C24×C197D411yescharacteristic
58C2×D4×C197—24no—
59C2×D4×C197—22no—
60C2×D4×C197—22no—
61C2×D4×C197—21no—
62C2×D4×C197—44no—
63C2×D4×C197C2312yes—
64C22×C4×C197—22no—
65C22×C4×C197—22no—
66C22×C4×C197—21no—
67C22×C4×C197D411yescharacteristic
68C22×C4×C197D412yes—
69C22×C4×C197C2312yes—
70C22×C4×C197C2311yescharacteristic
71C4⋊C4×C197—21no—
72C4⋊C4×C197C2312yes—
73C22⋊C4×C197—24no—
74C22×D197—24no—
75C22×D197—22no—
76C22×D197—42no—
77C22×D197—42no—
78Dic394—24no—
79Dic394—22no—
80Dic394C2×D414yescomplemented
81D788—24no—
82D788—22no—
83D788C2×D414yescomplemented
84C22⋊D197—216no—
85C2×Dic197—24no—
86C2×Dic197—22no—
87C4×D197—216no—
88#13096802—22no—
89#13096802—22no—
90C23×C197—22no—
91C23×C197—22no—
92C23×C197—22no—
93C23×C197—22no—
94C23×C197—42no—
95C23×C197—42no—
96C23×C197—42no—
97C23×C197C2×D412yes—
98C23×C197C2×D411yescharacteristic
99D4×C197—22no—
100D4×C197—22no—
101D4×C197—48no—
102D4×C197—42no—
103C2×C4×C197—22no—
104C2×C4×C197—24no—
105C2×C4×C197—22no—
106C2×C4×C197—22no—
107C2×C4×C197—22no—
108C2×C4×C197—42no—
109C2×C4×C197C2×D412yes—
110C2×C4×C197C2×D412yes—
111C2×C4×C197C2×D412yes—
112C2×C4×C197C2×D411yescharacteristic
113C2×C4×C197C2411yescharacteristic
114C22×C197—22no—
115C22×C197—22no—
116C22×C197—22no—
117C22×C197—44no—
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—24no—
127C2×D197—42no—
128C2×D197—42no—
129C2×D197—48no—
130C4×C197—22no—
131C4×C197—42no—
132C4×C197#12012yes—
133C4×C197C22×D411yescharacteristic
134C4×C197C22×D411yescharacteristic
135Dic197—24no—
136C2×C197—42no—
137C2×C197—82no—
138C2×C197#44914yescomplemented
139C2×C197C2×#12011yescharacteristic
140C2×C197C2×#13511yescharacteristic
141C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
142D197—44no—
143C197C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1971nomaximal, Sylow
145C22×#41—1971no—
146C22×SD16—1971no—
147C22×D8—1971no—
148C2×#121D19711yescharacteristic, complemented
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—21no—
159C22×D4—3941no—
160C22×D4—1971no—
161C22×D4C2×D19711yescharacteristic
162C23×C4—3941no—
163C23×C4C2×D19711yescharacteristic
164C2×SD16—3944no—
165C2×SD16—1974no—
166C2×D8—3944no—
167C2×D8—1974no—
168C22×C8—1971no—
169C22×C8—1971no—
170#121—22no—
171#121C2×D19714yescomplemented
172C2×C4⋊C4C2×D19711yescharacteristic
173C2×C22⋊C4—21no—
174#103—1974no—
175#102—1974no—
176C22⋊C8—1974no—
177C24—21no—
178C24—3941no—
179C24C22⋊D19711yescharacteristic
180#41—39416no—
181#41—3948no—
182#41—3948no—
183C2×Q8—3942no—
184C2×Q8—3941no—
185C2×Q8—1974no—
186C2×Q8—1972no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—44no—
192C2×D4—3948no—
193C2×D4—3944no—
194C2×D4—3942no—
195C2×D4—3941no—
196C2×D4—1974no—
197C2×D4—1972no—
198C2×D4C22×D19712yes—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—3948no—
203C22×C4—3941no—
204C22×C4—3944no—
205C22×C4C22⋊D19712yes—
206C22×C4C22⋊D19711yescharacteristic
207C22×C4C22×D19712yes—
208C22×C4C22×D19711yescharacteristic
209SD16—3948no—
210D8—3948no—
211C2×C8—3942no—
212C2×C8—3942no—
213C2×C8—1972no—
214C2×C8—1972no—
215C4⋊C4—21no—
216C4⋊C4C22×D19712yes—
217C22⋊C4—24no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—7882no—
226C23—7882no—
227C23—3944no—
228C23—3942no—
229C23C2×C22⋊D19712yes—
230C23C2×C22⋊D19711yescharacteristic
231Q8—3944no—
232Q8—3942no—
233Q8—1974no—
234D4—22no—
235D4—22no—
236D4—48no—
237D4—42no—
238D4—39416no—
239D4—3944no—
240D4—3942no—
241D4—1974no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—39416no—
249C2×C4—3944no—
250C2×C4—3942no—
251C2×C4D4×D19712yes—
252C2×C4D4×D19712yes—
253C2×C4C2×C22⋊D19712yes—
254C2×C4C2×C22⋊D19711yescharacteristic
255C2×C4C23×D19711yescharacteristic
256C8—3942no—
257C8—3942no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—44no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—7882no—
267C22—7882no—
268C22—7888no—
269C22—3944no—
270C22#2026329412yes—
271C22#2026334612yes—
272C22#2026335212yes—
273C22C22×C22⋊D19711yescharacteristic
274C4—22no—
275C4—42no—
276C4—3944no—
277C4#2026329412yes—
278C4C2×D4×D19711yescharacteristic
279C4C2×D4×D19711yescharacteristic
280C2—42no—
281C2—82no—
282C2—7884no—
283C2#2716122214yesdirect product factor
284C2C2×#2026329411yescharacteristic
285C2C2×#2026334611yescharacteristic
286C2C2×#2026335211yescharacteristic
subgroupG/H
centerC23C2×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#121×C197C2
socleC23×C197C2×D4
hypercenterC2×#121D197
hypocenterC197C2×#449
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#15691358C2semidirect
2. C22×#15691346C2semidirect
3. C22×#15691343C2semidirect
4. #27161222C2direct (N × H)
5. C2×#20263249C2semidirect
6. C2×#20263246C2semidirect
7. C2×C22⋊#13096802C2semidirect
8. C2×#121×C197C2semidirect
12. C2×#15691358C22semidirect
18. C2×#15691346C22semidirect
20. C2×#15691343C22semidirect
23. #20263249C22semidirect
24. #20263246C22semidirect
25. C22⋊#13096802C22semidirect
30. #121×C197C22semidirect
39. C2×Dic394D4semidirect
43. C2×D788D4semidirect
80. Dic394C2×D4semidirect
83. D788C2×D4semidirect
138. C2×C197#449semidirect
143. C197C2×#449semidirect
148. C2×#121D197semidirect
171. #121C2×D197semidirect
283. C2#27161222direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156913582
C22×#156913462
C22×#156913432
#271612222
C2×#202632492
C2×#202632462
C2×C22⋊#130968022
C2×#121×C1972
C2×#449197