Finite Group Databasebeta

← #27150963

Subgroups — #27150963

Order 12416.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#15690836C212yesmaximal, complemented
2#20261246C211yesmaximal, characteristic, complemented
3#20260886C214yesmaximal, complemented
4#20260869C214yesmaximal, complemented
5#20260740C212yesmaximal, complemented
6#20260699C211yesmaximal, characteristic, complemented
7C23×D4×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic, complemented
9#15690836—28no—
10C2×#13095499C2214yescomplemented
11C2×#13095496C2214yescomplemented
12C2×#13095495C2212yescomplemented
13#15690793—21no—
14#15690789C2212yescomplemented
15#15690784C2212yescomplemented
16#15690779—21no—
17C4×C22⋊D97—22no—
18#15690769C2211yescharacteristic, complemented
19#15690731C2218yescomplemented
20C22⋊#11715895C2214yescomplemented
21C25×C97—21no—
22C22×D4×C97—28no—
23C22×D4×C97—22no—
24C22×D4×C97—21no—
25C22×D4×C97C2214yes—
26C22×D4×C97C2212yes—
27C23×C4×C97C2211yescharacteristic
28#13095511—24no—
29C2×Dic194C2311yescharacteristic, complemented
30C2×D388C2311yescharacteristic, complemented
31C2×C22⋊D97—22no—
32C2×C22⋊D97—21no—
33C2×C4×D97—21no—
34#13095499—28no—
35#13095496—28no—
36#13095495—24no—
37C2×#11715895C2314yescomplemented
38C4⋊Dic97C2312yescomplemented
39#13095490—22no—
40C22⋊Dic97—21no—
41C22⋊Dic97—22no—
42#13095488—22no—
43C4×Dic97—21no—
44C24×C97—28no—
45C24×C97—24no—
46C24×C97—22no—
47C24×C97—48no—
48C24×C97D411yescharacteristic
49C2×D4×C97—28no—
50C2×D4×C97—216no—
51C2×D4×C97—216no—
52C2×D4×C97—24no—
53C2×D4×C97—22no—
54C2×D4×C97—24no—
55C2×D4×C97D418yescomplemented
56C2×D4×C97C2314yes—
57C22×C4×C97—24no—
58C22×C4×C97D412yes—
59C22×C4×C97D411yescharacteristic
60C22×C4×C97C2312yes—
61C22×C4×C97C2311yescharacteristic
62C23×C97—22no—
63C23×C97—24no—
64C23×C97—216no—
65C23×C97—28no—
66C23×C97—24no—
67C23×C97—432no—
68C23×C97—48no—
69C23×C97—48no—
70C23×C97—48no—
71C23×C97C2×D411yescharacteristic
72C23×C97C2×D412yes—
73C22×D97—21no—
74D4×C97—216no—
75D4×C97—28no—
76D4×C97—24no—
77D4×C97C2×D418yes—
78C2×C4×C97—22no—
79C2×C4×C97—24no—
80C2×C4×C97—24no—
81C2×C4×C97C2×D412yes—
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D414yes—
84C2×C4×C97C2411yescharacteristic
85C22⋊D97—42no—
86C22⋊D97—44no—
87C2×Dic97—22no—
88C2×Dic97—21no—
89D388—22no—
90C4×D97—42no—
91Dic194—22no—
92#11715895—24no—
93C22×C97—28no—
94C22×C97—22no—
95C22×C97—22no—
96C22×C97—22no—
97C22×C97—28no—
98C22×C97—416no—
99C22×C97—432no—
100C22×C97—48no—
101C22×C97#12012yes—
102C22×C97AGL(1,ℤ/8)14yescomplemented
103C22×C97C22×D411yescharacteristic
104C2×D97—41no—
105C2×D97—42no—
106C4×C97—22no—
107C4×C97#12012yes—
108C4×C97C22×D412yes—
109Dic97—41no—
110Dic97—42no—
111C2×C97—22no—
112C2×C97—24no—
113C2×C97—48no—
114C2×C97C2×#12011yescharacteristic
115C2×C97C2×AGL(1,ℤ/8)12yes—
116D97—81no—
117#3003—971nomaximal, Sylow
118C97#300311yescharacteristic, complemented, Sylow
119C23×D4D9711yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—972no—
121#525—971no—
122#450—974no—
123#447—974no—
124#418—972no—
125#407—971no—
126C25—21no—
127C2×#41—971no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D9714yes—
132C22×D4C2×D9712yes—
133C23×C4C2×D9711yescharacteristic
134AGL(1,ℤ/8)—1948no—
135C2×SD16—974no—
136C2×D8—974no—
137C2×M16—972no—
138#123—1941no—
139C22⋊Q8—972no—
140#121—972no—
141#120—1941no—
142C4×D4—1942no—
143#117—971no—
144#102—978no—
145C22⋊C8—974no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D9711yescharacteristic
151#41—1944no—
152C2×Q8—971no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—971no—
160C2×D4—1942no—
161C2×D4—1941no—
162C2×D4C22⋊D9718yescomplemented
163C2×D4C22×D9714yes—
164C22×C4—24no—
165C22×C4—1941no—
166C22×C4C22⋊D9712yes—
167C22×C4C22⋊D9711yescharacteristic
168C22×C4C22×D9712yes—
169C22×C4C22×D9711yescharacteristic
170SD16—1948no—
171D8—1948no—
172M16—1944no—
173C2×C8—974no—
174C4⋊C4—972no—
175C4⋊C4—1942no—
176C22⋊C4—1941no—
177C22⋊C4—1942no—
178C22⋊C4—1942no—
179C42—1941no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—1941no—
190C23C2×C22⋊D9711yescharacteristic
191C23C2×C22⋊D9712yes—
192Q8—1942no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—3882no—
197D4—3884no—
198D4—1942no—
199D4C2×C22⋊D9718yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—3882no—
204C2×C4—1942no—
205C2×C4—1941no—
206C2×C4C2×C22⋊D9712yes—
207C2×C4C2×C22⋊D9711yescharacteristic
208C2×C4C2×C22⋊D9714yes—
209C2×C4C23×D9711yescharacteristic
210C8—1944no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—3881no—
220C22—3882no—
221C22#1569077912yes—
222C22#1569083614yescomplemented
223C22C22×C22⋊D9711yescharacteristic
224C4—22no—
225C4—3881no—
226C4—3882no—
227C4#1569077912yes—
228C4C22×C22⋊D9712yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—7761no—
233C2C2×#1569077911yescharacteristic
234C2C2×#1569083612yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC23×D4×C97C2
socleC22×C97C22×D4
hypercenterC23×D4D97
hypocenterC97#3003
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. #20261246C2semidirect
3. #20260886C2semidirect
4. #20260869C2semidirect
5. #20260740C2semidirect
6. #20260699C2semidirect
7. C23×D4×C97C2semidirect
8. C2×#13095511C22semidirect
10. C2×#13095499C22semidirect
11. C2×#13095496C22semidirect
12. C2×#13095495C22semidirect
14. #15690789C22semidirect
15. #15690784C22semidirect
18. #15690769C22semidirect
19. #15690731C22semidirect
20. C22⋊#11715895C22semidirect
29. C2×Dic194C23semidirect
30. C2×D388C23semidirect
37. C2×#11715895C23semidirect
38. C4⋊Dic97C23semidirect
55. C2×D4×C97D4semidirect
102. C22×C97AGL(1,ℤ/8)semidirect
118. C97#3003semidirect
119. C23×D4D97semidirect
162. C2×D4C22⋊D97semidirect
222. C22#15690836semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
#202612462
#202608862
#202608692
#202607402
#202606992
C23×D4×C972
#300397