Finite Group Databasebeta

← #38751970

Subgroups — #38751970

Order 25216.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
2C2×#20263357C212yesmaximal, complemented
3#27161239C214yesmaximal, complemented
4#27161234C212yesmaximal, complemented
5#27161229C212yesmaximal, complemented
6#27161078C212yesmaximal, complemented
7#27161043C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C197C211yesmaximal, characteristic, complemented
9C2×#15691358—28no—
10C2×#15691358—21no—
11C2×#15691358—21no—
12C2×#15691358C2212yes—
13C2×#15691358C2212yes—
14C22×Dic394C2211yescharacteristic
15C22×D788C2211yescharacteristic
16C22×C22⋊D197—21no—
17C22×C4×D197—21no—
18#20263357—28no—
19C2×#15691348C2214yescomplemented
20C2×#15691345C2214yescomplemented
21C2×#15691342C2212yescomplemented
22#20263249C2214yescomplemented
23#20263246C2214yescomplemented
24C22⋊#13096802C2212yes—
25C22⋊#13096802C2212yescomplemented
26C22×Q8×C197C2211yescharacteristic, complemented
27C23×C4×C197C2211yescharacteristic
28C22⋊Q8×C197—21no—
29C22⋊Q8×C197—21no—
30C22⋊Q8×C197C2212yes—
31C22⋊Q8×C197C2212yescomplemented
32C2×C4⋊C4×C197—21no—
33C2×C4⋊C4×C197C2211yescharacteristic, complemented
34C2×C22⋊C4×C197—21no—
35C23×D197—21no—
36#15691358—28no—
37#15691358—28no—
38#15691358—24no—
39#15691358—24no—
40#15691358—24no—
41#15691358D418yes—
42C2×Dic394—24no—
43C2×Dic394—21no—
44C2×Dic394C2312yes—
45C2×D788—24no—
46C2×D788—21no—
47C2×D788C2312yes—
48C2×C22⋊D197—28no—
49C2×C22⋊D197—22no—
50C2×C22⋊D197—22no—
51C22×Dic197—21no—
52C2×C4×D197—28no—
53C2×C4×D197—22no—
54C2×C4×D197—22no—
55#15691348—28no—
56#15691345—28no—
57#15691342—24no—
58C2×#13096802C2314yes—
59C24×C197D411yescharacteristic
60C2×Q8×C197—21no—
61C2×Q8×C197—24no—
62C2×Q8×C197C2312yes—
63C22×C4×C197—24no—
64C22×C4×C197—21no—
65C22×C4×C197—21no—
66C22×C4×C197D411yescharacteristic
67C22×C4×C197D411yescharacteristic
68C22×C4×C197D411yescharacteristic
69C22×C4×C197C2311yescharacteristic
70C22×C4×C197C2311yescharacteristic
71C22×C4×C197C2311yescharacteristic
72C4⋊C4×C197—22no—
73C4⋊C4×C197—22no—
74C4⋊C4×C197—21no—
75C4⋊C4×C197C2312yes—
76C22⋊C4×C197—22no—
77C22⋊C4×C197—22no—
78C22×D197—24no—
79C22×D197—22no—
80C22×D197—42no—
81C22×D197—42no—
82Dic394—24no—
83Dic394—22no—
84Dic394C2×D414yes—
85D788—24no—
86D788—22no—
87D788C2×D414yes—
88C22⋊D197—28no—
89C22⋊D197—28no—
90C2×Dic197—24no—
91C2×Dic197—22no—
92C4×D197—28no—
93C4×D197—28no—
94#13096802—24no—
95C23×C197—22no—
96C23×C197—22no—
97C23×C197—41no—
98C23×C197C2×D411yescharacteristic
99C23×C197C2×D411yescharacteristic
100C23×C197C2×D411yescharacteristic
101Q8×C197—24no—
102Q8×C197—42no—
103C2×C4×C197—22no—
104C2×C4×C197—22no—
105C2×C4×C197—22no—
106C2×C4×C197—22no—
107C2×C4×C197—22no—
108C2×C4×C197—22no—
109C2×C4×C197—22no—
110C2×C4×C197—42no—
111C2×C4×C197—42no—
112C2×C4×C197C2×D414yes—
113C2×C4×C197C2×D411yescharacteristic
114C2×C4×C197C2×D411yescharacteristic
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2411yescharacteristic
117C22×C197—21no—
118C22×C197—21no—
119C22×C197—21no—
120C22×C197—21no—
121C22×C197—21no—
122C22×C197—21no—
123C22×C197—21no—
124C22×C197—44no—
125C22×C197#12012yes—
126C22×C197#13712yescomplemented
127C22×C197C22×D411yescharacteristic
128C2×D197—24no—
129C2×D197—48no—
130C2×D197—42no—
131C2×D197—42no—
132C4×C197—21no—
133C4×C197—21no—
134C4×C197—42no—
135C4×C197—42no—
136C4×C197#12012yes—
137C4×C197C22×D412yes—
138Dic197—24no—
139C2×C197—22no—
140C2×C197—22no—
141C2×C197—41no—
142C2×C197C2×#12011yescharacteristic
143C2×C197C2×#13711yescharacteristic
144C2×C197#57511yescharacteristic
145D197—44no—
146C197#300711yescharacteristic, complemented, Sylow
147#3007—1971nomaximal, Sylow
148C22×#41—1971no—
149C2×#137—1972no—
150C2×C22⋊Q8D19711yescharacteristic, complemented
151#452—1974no—
152#451—1972no—
153#450—1972no—
154#417—1972no—
155#407—1971no—
156C2×#41—3948no—
157C2×#41—3941no—
158C2×#41—3941no—
159C2×#41—1972no—
160C2×#41—1972no—
161C22×Q8—1971no—
162C22×Q8C2×D19711yescharacteristic, complemented
163C22×D4—3941no—
164C22×D4—1971no—
165C23×C4—3941no—
166C23×C4C2×D19711yescharacteristic
167#137—3948no—
168C2×Q16—1974no—
169C2×SD16—1974no—
170C2×M16—1972no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D19712yes—
174C22⋊Q8C2×D19712yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D19711yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—1974no—
179#102—1974no—
180C22⋊C8—1972no—
181C22⋊C8—1972no—
182C24—3941no—
183C24C22⋊D19711yescharacteristic
184#41—3948no—
185#41—3948no—
186#41—3944no—
187#41—3944no—
188#41—3944no—
189#41—1978no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—3944no—
193C2×Q8—3941no—
194C2×Q8—1972no—
195C2×Q8C22×D19712yes—
196C2×D4—3944no—
197C2×D4—3948no—
198C2×D4—3942no—
199C2×D4—3942no—
200C2×D4—3941no—
201C2×D4—1972no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—3948no—
206C22×C4—3941no—
207C22×C4—3942no—
208C22×C4—3942no—
209C22×C4C22⋊D19711yescharacteristic
210C22×C4C22⋊D19711yescharacteristic
211C22×C4C22⋊D19711yescharacteristic
212C22×C4C22×D19711yescharacteristic
213C22×C4C22×D19711yescharacteristic
214C22×C4C22×D19711yescharacteristic
215Q16—3948no—
216SD16—3948no—
217M16—3944no—
218C2×C8—1974no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D19712yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—7882no—
229C23—7882no—
230C23—3944no—
231C23—3942no—
232C23C2×C22⋊D19711yescharacteristic
233C23C2×C22⋊D19711yescharacteristic
234C23C2×C22⋊D19711yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—3944no—
238Q8—3942no—
239Q8—1974no—
240D4—3944no—
241D4—3948no—
242D4—3948no—
243D4—3942no—
244D4—1974no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—3948no—
255C2×C4—3944no—
256C2×C4—3942no—
257C2×C4—3948no—
258C2×C4D4×D19714yes—
259C2×C4C2×C22⋊D19711yescharacteristic
260C2×C4C2×C22⋊D19711yescharacteristic
261C2×C4C2×C22⋊D19711yescharacteristic
262C2×C4C23×D19711yescharacteristic
263C8—3944no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—7888no—
273C22—7882no—
274C22—7882no—
275C22—3944no—
276C22#2026329412yes—
277C22#2026335712yescomplemented
278C22C22×C22⋊D19711yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—3944no—
284C4#2026329412yes—
285C4C2×D4×D19712yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—7884no—
290C2C2×#2026329411yescharacteristic
291C2C2×#2026335711yescharacteristic
292C2#2716187711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×C22⋊Q8×C197C2
socleC22×C197C22×D4
prime radicalC22×#15691358C2
hypercenterC2×C22⋊Q8D197
hypocenterC197#3007
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#20263357C2semidirect
3. #27161239C2semidirect
4. #27161234C2semidirect
5. #27161229C2semidirect
6. #27161078C2semidirect
7. #27161043C2semidirect
8. C2×C22⋊Q8×C197C2semidirect
19. C2×#15691348C22semidirect
20. C2×#15691345C22semidirect
21. C2×#15691342C22semidirect
22. #20263249C22semidirect
23. #20263246C22semidirect
25. C22⋊#13096802C22semidirect
26. C22×Q8×C197C22semidirect
31. C22⋊Q8×C197C22semidirect
33. C2×C4⋊C4×C197C22semidirect
126. C22×C197#137semidirect
146. C197#3007semidirect
150. C2×C22⋊Q8D197semidirect
162. C22×Q8C2×D197semidirect
174. C22⋊Q8C2×D197semidirect
176. C2×C4⋊C4C2×D197semidirect
277. C22#20263357semidirect

Maximal subgroups

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

subgroupindex
C22×#156913582
C2×#202633572
#271612392
#271612342
#271612292
#271610782
#271610432
C2×C22⋊Q8×C1972
#3007197