Finite Group Databasebeta

← #27150962

Subgroups — #27150962

Order 12416.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D388C211yesmaximal, characteristic, complemented
2C2×#15690835C212yesmaximal, complemented
3#20260885C214yesmaximal, complemented
4#20260868C214yesmaximal, complemented
5#20260739C212yesmaximal, complemented
6#20260699C211yesmaximal, characteristic, complemented
7#525×C97C211yesmaximal, characteristic, complemented
8C24×D97—21no—
9C22×D388—28no—
10C22×D388—22no—
11C22×D388—21no—
12C22×D388C2214yes—
13C22×D388C2212yes—
14#15690835—28no—
15C2×#13095498C2214yescomplemented
16C2×#13095496C2214yescomplemented
17C2×#13095495C2212yescomplemented
18#15690730C2218yescomplemented
19C22⋊#11715895C2214yescomplemented
20C2×#41×C97C2211yescharacteristic, complemented
21C23×C4×C97C2211yescharacteristic
22#123×C97—21no—
23C22⋊Q8×C97C2212yescomplemented
24#121×C97C2212yescomplemented
25#120×C97—21no—
26C4×D4×C97—22no—
27#117×C97C2211yescharacteristic, complemented
28C23×D97—28no—
29C23×D97—24no—
30C23×D97—22no—
31C23×D97—48no—
32C2×D388—28no—
33C2×D388—216no—
34C2×D388—24no—
35C2×D388—24no—
36C2×D388—22no—
37C2×D388—216no—
38C2×D388D418yescomplemented
39C2×D388C2314yes—
40#13095498—28no—
41#13095496—28no—
42#13095495—24no—
43C2×#11715895C2314yescomplemented
44C24×C97D411yescharacteristic
45#41×C97—24no—
46C2×Q8×C97C2311yescharacteristic, complemented
47C2×D4×C97—22no—
48C2×D4×C97—21no—
49C2×D4×C97C2311yescharacteristic, complemented
50C22×C4×C97—24no—
51C22×C4×C97—21no—
52C22×C4×C97D412yes—
53C22×C4×C97D411yescharacteristic
54C22×C4×C97C2312yes—
55C22×C4×C97C2311yescharacteristic
56C4⋊C4×C97—22no—
57C4⋊C4×C97C2312yescomplemented
58C22⋊C4×C97—21no—
59C22⋊C4×C97—22no—
60C22⋊C4×C97—22no—
61C42×C97—21no—
62C23×C97—22no—
63C23×C97—24no—
64C23×C97—21no—
65C23×C97C2×D411yescharacteristic
66C23×C97C2×D412yes—
67C22×D97—216no—
68C22×D97—28no—
69C22×D97—24no—
70C22×D97—432no—
71C22×D97—48no—
72C22×D97—48no—
73C22×D97—48no—
74Q8×C97—22no—
75D4×C97—22no—
76D4×C97—42no—
77D4×C97—44no—
78C2×C4×C97—22no—
79C2×C4×C97—24no—
80C2×C4×C97—24no—
81C2×C4×C97—22no—
82C2×C4×C97—21no—
83C2×C4×C97—42no—
84C2×C4×C97C2×D414yes—
85C2×C4×C97C2×D412yes—
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97C2411yescharacteristic
88D388—216no—
89D388—28no—
90D388—24no—
91D388C2×D418yes—
92#11715895—24no—
93C22×C97—28no—
94C22×C97—22no—
95C22×C97—22no—
96C22×C97—22no—
97C22×C97—41no—
98C22×C97—42no—
99C22×C97#12012yes—
100C22×C97AGL(1,ℤ/8)14yescomplemented
101C22×C97C22×D411yescharacteristic
102C2×D97—28no—
103C2×D97—416no—
104C2×D97—432no—
105C2×D97—48no—
106C4×C97—22no—
107C4×C97—41no—
108C4×C97—42no—
109C4×C97#12012yes—
110C4×C97C22×D412yes—
111C2×C97—22no—
112C2×C97—24no—
113C2×C97—81no—
114C2×C97C2×#12011yescharacteristic
115C2×C97C2×AGL(1,ℤ/8)12yes—
116D97—48no—
117#3003—971nomaximal, Sylow
118C97#300311yescharacteristic, complemented, Sylow
119C23×D4—971no—
120C2×AGL(1,ℤ/8)—972no—
121#525D9711yescharacteristic, complemented
122#450—974no—
123#447—974no—
124#418—972no—
125#407—971no—
126C25—1941no—
127C2×#41C2×D9711yescharacteristic, complemented
128C22×D4—972no—
129C22×D4—974no—
130C22×D4—1948no—
131C22×D4—1942no—
132C22×D4—1941no—
133C23×C4C2×D9711yescharacteristic
134AGL(1,ℤ/8)—1948no—
135C2×SD16—974no—
136C2×D8—974no—
137C2×M16—972no—
138#123—21no—
139C22⋊Q8C2×D9712yescomplemented
140#121C2×D9712yescomplemented
141#120—21no—
142C4×D4—22no—
143#117C2×D9711yescharacteristic, complemented
144#102—978no—
145C22⋊C8—974no—
146C24—3888no—
147C24—1948no—
148C24—1942no—
149C24—1944no—
150C24C22⋊D9711yescharacteristic
151#41—24no—
152C2×Q8C22×D9711yescharacteristic, complemented
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—978no—
156C2×D4—974no—
157C2×D4—19416no—
158C2×D4—1948no—
159C2×D4—19416no—
160C2×D4—1944no—
161C2×D4—1944no—
162C2×D4—1942no—
163C2×D4C22×D9711yescharacteristic, complemented
164C22×C4—24no—
165C22×C4—21no—
166C22×C4C22⋊D9712yes—
167C22×C4C22⋊D9711yescharacteristic
168C22×C4C22×D9712yes—
169C22×C4C22×D9711yescharacteristic
170SD16—1948no—
171D8—1948no—
172M16—1944no—
173C2×C8—974no—
174C4⋊C4—22no—
175C4⋊C4C22×D9712yescomplemented
176C22⋊C4—21no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C42—21no—
180C23—22no—
181C23—24no—
182C23—21no—
183C23—38832no—
184C23—3888no—
185C23—3888no—
186C23—3888no—
187C23—19416no—
188C23—1948no—
189C23—1944no—
190C23C2×C22⋊D9711yescharacteristic
191C23C2×C22⋊D9712yes—
192Q8—22no—
193D4—22no—
194D4—42no—
195D4—44no—
196D4—978no—
197D4—19416no—
198D4—1948no—
199D4—1944no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4D4×D9714yes—
207C2×C4C2×C22⋊D9712yes—
208C2×C4C2×C22⋊D9711yescharacteristic
209C2×C4C23×D9711yescharacteristic
210C8—1944no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—41no—
216C22—42no—
217C22—38816no—
218C22—38832no—
219C22—3888no—
220C22—1948no—
221C22#1569077812yes—
222C22#1569083514yescomplemented
223C22C22×C22⋊D9711yescharacteristic
224C4—22no—
225C4—41no—
226C4—42no—
227C4#1569077812yes—
228C4C2×D4×D9712yes—
229C2—22no—
230C2—24no—
231C2—81no—
232C2—3888no—
233C2C2×#1569077811yescharacteristic
234C2C2×#1569083512yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#525×C97C2
socleC22×C97C22×D4
hypercenter#525D97
hypocenterC97#3003
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C23×D388C2semidirect
2. C2×#15690835C2semidirect
3. #20260885C2semidirect
4. #20260868C2semidirect
5. #20260739C2semidirect
6. #20260699C2semidirect
7. #525×C97C2semidirect
15. C2×#13095498C22semidirect
16. C2×#13095496C22semidirect
17. C2×#13095495C22semidirect
18. #15690730C22semidirect
19. C22⋊#11715895C22semidirect
20. C2×#41×C97C22semidirect
23. C22⋊Q8×C97C22semidirect
24. #121×C97C22semidirect
27. #117×C97C22semidirect
38. C2×D388D4semidirect
43. C2×#11715895C23semidirect
46. C2×Q8×C97C23semidirect
49. C2×D4×C97C23semidirect
57. C4⋊C4×C97C23semidirect
100. C22×C97AGL(1,ℤ/8)semidirect
118. C97#3003semidirect
121. #525D97semidirect
127. C2×#41C2×D97semidirect
139. C22⋊Q8C2×D97semidirect
140. #121C2×D97semidirect
143. #117C2×D97semidirect
152. C2×Q8C22×D97semidirect
163. C2×D4C22×D97semidirect
175. C4⋊C4C22×D97semidirect
222. C22#15690835semidirect

Maximal subgroups

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

subgroupindex
C23×D3882
C2×#156908352
#202608852
#202608682
#202607392
#202606992
#525×C972
#300397