Finite Group Databasebeta

← #27144546

Subgroups — #27144546

Order 12416.

540 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 177 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×#15690798C211yesmaximal, characteristic
2#20260639C212yesmaximal, complemented
3#20260627C212yesmaximal, complemented
4#20260624C212yesmaximal, complemented
5#20260613C212yesmaximal
6#20260609C212yesmaximal, complemented
7C4×C4⋊Dic97C211yesmaximal, characteristic, complemented
8#20260582C212yesmaximal, complemented
9#383×C97C211yesmaximal, characteristic, complemented
10C22×Dic194C2211yescharacteristic
11C22×D388C2211yescharacteristic
12C22×C4×D97C2212yes—
13#15690798C418yescomplemented
14C2×C4⋊Dic97C2212yes—
15C2×#13095490C2212yes—
16C2×#13095488C2212yes—
17C2×#13095488C2212yes—
18C2×#13095488C2212yes—
19C2×#13095488C2212yes—
20C2×#13095488C2212yes—
21C2×C4×Dic97C2211yescharacteristic
22C2×C4×Dic97C2212yes—
23C2×C4×Dic97C2211yescharacteristic
24#15690717C2212yes—
25#15690717C2212yes—
26#15690717C2212yes—
27C2×C42×C97C2211yescharacteristic
28#95×C97C2211yescharacteristic
29#95×C97C2212yes—
30#95×C97C2212yes—
31#95×C97C2211yescharacteristic
32C23×D97C2312yes—
33C2×Dic194—42no—
34C2×Dic194C2×C414yescomplemented
35C2×D388—42no—
36C2×D388C2×C414yes—
37C22×Dic97C2312yes—
38C22×Dic97C2312yes—
39C22×Dic97C2312yes—
40C2×C4×D97—24no—
41C2×C4×D97—42no—
42C2×C4×D97—42no—
43C4⋊Dic97—24no—
44#13095490—42no—
45#13095488—24no—
46#13095488—42no—
47#13095488—42no—
48#13095488C2×C418yescomplemented
49#13095488C2×C418yescomplemented
50C4×Dic97—41no—
51C4×Dic97—42no—
52C4×Dic97—41no—
53C22×C4×C97C2312yes—
54C22×C4×C97C2312yes—
55C22×C4×C97C2311yescharacteristic
56C22×C4×C97C2311yescharacteristic
57C22×C4×C97C2311yescharacteristic
58C42×C97C2×C414yescomplemented
59C23×C97C2411yescharacteristic
60C22×D97—22no—
61C22×D97—22no—
62C22×D97—22no—
63C22×D97—22no—
64C22×D97—22no—
65C22×D97—22no—
66C22×D97C22×C414yes—
67C2×C4×C97—22no—
68C2×C4×C97—21no—
69C2×C4×C97—22no—
70C2×C4×C97—21no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—22no—
77C2×C4×C97C22×C412yes—
78C2×C4×C97C22×C412yes—
79C2×C4×C97C22×C412yes—
80C2×Dic97—22no—
81C2×Dic97—22no—
82C2×Dic97—22no—
83C2×Dic97—22no—
84C2×Dic97—22no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97C22×C414yes—
88C2×Dic97#4114yes—
89D388—44no—
90C4×D97—48no—
91Dic194—44no—
92C22×C97C23×C411yescharacteristic
93C22×C97C2×#4112yes—
94C22×C97H(5,𝔽2)12yes—
95C22×C97H(5,𝔽2)11yescharacteristic
96C22×C972−1+411yescharacteristic
97C2×D97—24no—
98C2×D97—24no—
99C2×D97—24no—
100C2×D97—42no—
101C2×D97—42no—
102C2×D97—42no—
103C2×D97—42no—
104C4×C97—22no—
105C4×C97—22no—
106C4×C97—22no—
107C4×C97—42no—
108C4×C97—42no—
109Dic97—24no—
110Dic97—24no—
111Dic97—42no—
112C2×C97C4×#4111yescharacteristic
113C2×C97#51811yescharacteristic
114C2×C97#52011yescharacteristic
115C2×C97#53812yes—
116C2×C97#54212yes—
117D97—44no—
118#2337—971nomaximal, Sylow
119C97#233711yescharacteristic, complemented, Sylow
120C2×#124—971no—
121#391—972no—
122#388—972no—
123#388—972no—
124#386—972no—
125#385—972no—
126#383D9711yescharacteristic, complemented
127C4×C4⋊C4—971no—
128C4×C22⋊C4—972no—
129C22×Q8—971no—
130C22×D4—971no—
131C23×C4—972no—
132#124—978no—
133C2×C4⋊C4—972no—
134C2×C4⋊C4—972no—
135C2×C22⋊C4—972no—
136C2×C22⋊C4—972no—
137C2×C22⋊C4—972no—
138C2×C22⋊C4—972no—
139C2×C22⋊C4—972no—
140C2×C42—971no—
141C2×C42—972no—
142C2×C42—971no—
143C2×C42C2×D9711yescharacteristic
144#95—972no—
145#95—972no—
146#95—972no—
147#95C2×D9712yes—
148#95C2×D9711yescharacteristic
149#95C2×D9712yes—
150#95C2×D9711yescharacteristic
151C24—972no—
152C2×Q8—974no—
153C2×Q8—3882no—
154C2×D4—974no—
155C2×D4—3882no—
156C22×C4—972no—
157C22×C4—972no—
158C22×C4—972no—
159C22×C4—3882no—
160C22×C4—3882no—
161C22×C4—1944no—
162C22×C4C22×D9712yes—
163C22×C4C22×D9712yes—
164C22×C4C22×D9711yescharacteristic
165C22×C4C22×D9711yescharacteristic
166C22×C4C22×D9711yescharacteristic
167C4⋊C4—3882no—
168C4⋊C4—1944no—
169C22⋊C4—978no—
170C22⋊C4—978no—
171C22⋊C4—3882no—
172C22⋊C4—3882no—
173C22⋊C4—1944no—
174C42—3881no—
175C42—3881no—
176C42—3882no—
177C42C4×D9714yescomplemented
178C23—974no—
179C23—1942no—
180C23—1942no—
181C23—1942no—
182C23—1942no—
183C23—1942no—
184C23—1942no—
185C23C23×D9711yescharacteristic
186Q8—3884no—
187D4—3884no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—974no—
199C2×C4—974no—
200C2×C4—3888no—
201C2×C4—1942no—
202C2×C4—1942no—
203C2×C4—1942no—
204C2×C4—1942no—
205C2×C4—1942no—
206C2×C4—1942no—
207C2×C4—1942no—
208C2×C4C2×C4×D9712yes—
209C2×C4C2×C4×D9712yes—
210C2×C4C2×C4×D9712yes—
211C22—3882no—
212C22—3882no—
213C22—3882no—
214C22—3882no—
215C22—1944no—
216C22—1944no—
217C22—1944no—
218C22C22×C4×D9711yescharacteristic
219C22#41×D9712yes—
220C22#1569085612yes—
221C22#1569085611yescharacteristic
222C22#1569085711yescharacteristic
223C4—22no—
224C4—22no—
225C4—22no—
226C4—42no—
227C4—42no—
228C4—3882no—
229C4—1944no—
230C4—1944no—
231C2—3884no—
232C2#2026120311yescharacteristic
233C2#2026120911yescharacteristic
234C2#2026121511yescharacteristic
235C2#2026132412yes—
236C2#2026135112yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#383×C97C2
socle, cosocleC23×C97C24
prime radicalC22×D388C22
hypercenter#383D97
hypocenterC97#2337
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
2. #20260639C2semidirect
3. #20260627C2semidirect
4. #20260624C2semidirect
6. #20260609C2semidirect
7. C4×C4⋊Dic97C2semidirect
8. #20260582C2semidirect
9. #383×C97C2semidirect
13. #15690798C4semidirect
34. C2×Dic194C2×C4semidirect
48. #13095488C2×C4semidirect
49. #13095488C2×C4semidirect
58. C42×C97C2×C4semidirect
119. C97#2337semidirect
126. #383D97semidirect
177. C42C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907982
#202606392
#202606272
#202606242
#202606132
#202606092
C4×C4⋊Dic972
#202605822
#383×C972
#233797