Finite Group Databasebeta

← #27144908

Subgroups — #27144908

Order 12416.

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

normal subgroups only

This group has 520 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×#15690799C211yesmaximal, characteristic
2#20260642C211yesmaximal, characteristic, complemented
3#20260626C212yesmaximal, complemented
4#20260624C212yesmaximal, complemented
5#20260613C212yesmaximal
6#20260609C212yesmaximal, complemented
7#20260606C212yesmaximal, complemented
8#20260599C212yesmaximal, complemented
9#391×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C22×C4×D97C2212yes—
12#15690799C418yescomplemented
13C2×C4⋊Dic97C2211yescharacteristic
14C2×#13095490C2212yes—
15C2×#13095490C2212yes—
16C2×#13095488C2212yes—
17C2×#13095488C2212yes—
18C2×#13095488C2212yes—
19C2×#13095488C2212yes—
20C2×#13095488C2212yes—
21C2×C4×Dic97C2211yescharacteristic
22C2×C4×Dic97C2211yescharacteristic
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2212yes—
26#15690717C2212yes—
27C22×Q8×C97C2211yescharacteristic
28C2×C4⋊C4×C97C2211yescharacteristic
29C2×C42×C97C2211yescharacteristic
30#95×C97C2212yes—
31#95×C97C2212yes—
32C23×D97C2312yes—
33C2×D388—41no—
34C2×D388—41no—
35C2×D388C2×C414yes—
36C22×Dic97C2312yes—
37C22×Dic97C2312yes—
38C22×Dic97C2312yes—
39C2×C4×D97—42no—
40C2×C4×D97—42no—
41C2×C4×D97—42no—
42C4⋊Dic97—41no—
43#13095490—42no—
44#13095490—42no—
45#13095488—42no—
46#13095488—42no—
47#13095488—42no—
48#13095488C2×C418yescomplemented
49#13095488C2×C418yescomplemented
50C4×Dic97—41no—
51C4×Dic97C2×C414yescomplemented
52C2×Q8×C97—41no—
53C2×Q8×C97—41no—
54C2×Q8×C97C2×C414yescomplemented
55C22×C4×C97C2312yes—
56C22×C4×C97C2312yes—
57C22×C4×C97C2311yescharacteristic
58C22×C4×C97C2311yescharacteristic
59C22×C4×C97C2311yescharacteristic
60C4⋊C4×C97—41no—
61C42×C97—41no—
62C23×C97C2411yescharacteristic
63C22×D97—22no—
64C22×D97—22no—
65C22×D97—22no—
66C22×D97—22no—
67C22×D97—22no—
68C22×D97—22no—
69C22×D97C22×C414yes—
70Q8×C97—44no—
71C2×C4×C97—21no—
72C2×C4×C97—21no—
73C2×C4×C97—21no—
74C2×C4×C97—21no—
75C2×C4×C97—21no—
76C2×C4×C97—21no—
77C2×C4×C97—22no—
78C2×C4×C97—22no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97C22×C412yes—
84C2×C4×C97C22×C412yes—
85C2×C4×C97C22×C412yes—
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—22no—
89C2×Dic97—22no—
90C2×Dic97—22no—
91C2×Dic97—22no—
92C2×Dic97—22no—
93C2×Dic97—22no—
94C2×Dic97C22×C414yes—
95D388—44no—
96C4×D97—48no—
97C22×C97C23×C411yescharacteristic
98C22×C97H(5,𝔽2)11yescharacteristic
99C22×C97H(5,𝔽2)11yescharacteristic
100C22×C97H(5,𝔽2)12yes—
101C22×C972−1+412yes—
102C2×D97—24no—
103C2×D97—24no—
104C2×D97—24no—
105C2×D97—42no—
106C2×D97—42no—
107C2×D97—42no—
108C2×D97—42no—
109C4×C97—22no—
110C4×C97—22no—
111C4×C97—22no—
112C4×C97—42no—
113C4×C97—42no—
114Dic97—24no—
115Dic97—42no—
116Dic97—42no—
117C2×C97#51811yescharacteristic
118C2×C97#52012yes—
119C2×C97H(3,𝔽4)11yescharacteristic
120C2×C97#56212yes—
121C2×C97#56311yescharacteristic
122D97—44no—
123#2379—971nomaximal, Sylow
124C97#237911yescharacteristic, complemented, Sylow
125C2×#124—971no—
126#391—971no—
127#391D9711yescharacteristic, complemented
128#388—972no—
129#388—972no—
130#386—972no—
131#385—972no—
132#385—972no—
133#382—972no—
134C22×Q8C2×D9711yescharacteristic
135C22×D4—971no—
136C23×C4—972no—
137#124—978no—
138C2×C4⋊C4—972no—
139C2×C4⋊C4—972no—
140C2×C4⋊C4—971no—
141C2×C4⋊C4C2×D9711yescharacteristic
142C2×C22⋊C4—972no—
143C2×C22⋊C4—972no—
144C2×C22⋊C4—972no—
145C2×C22⋊C4—972no—
146C2×C22⋊C4—972no—
147C2×C42—971no—
148C2×C42—971no—
149C2×C42C2×D9711yescharacteristic
150#95—972no—
151#95—972no—
152#95—972no—
153#95—972no—
154#95C2×D9712yes—
155#95C2×D9712yes—
156C24—972no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C4×D9714yescomplemented
160C2×D4—974no—
161C2×D4—3881no—
162C2×D4—3881no—
163C22×C4—972no—
164C22×C4—972no—
165C22×C4—972no—
166C22×C4—3882no—
167C22×C4—3882no—
168C22×C4—3882no—
169C22×C4C22×D9712yes—
170C22×C4C22×D9712yes—
171C22×C4C22×D9711yescharacteristic
172C22×C4C22×D9711yescharacteristic
173C22×C4C22×D9711yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—3882no—
176C4⋊C4—3882no—
177C4⋊C4—3881no—
178C22⋊C4—978no—
179C22⋊C4—978no—
180C22⋊C4—3882no—
181C22⋊C4—3882no—
182C22⋊C4—3882no—
183C42—41no—
184C42—974no—
185C42—3881no—
186C23—974no—
187C23—1942no—
188C23—1942no—
189C23—1942no—
190C23—1942no—
191C23—1942no—
192C23—1942no—
193C23C23×D9711yescharacteristic
194Q8—44no—
195D4—3884no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—974no—
209C2×C4—3888no—
210C2×C4—1942no—
211C2×C4—1942no—
212C2×C4—1942no—
213C2×C4—1942no—
214C2×C4—1942no—
215C2×C4—1942no—
216C2×C4—1942no—
217C2×C4—1942no—
218C2×C4C2×C4×D9712yes—
219C2×C4C2×C4×D9712yes—
220C2×C4C2×C4×D9712yes—
221C22—3882no—
222C22—3882no—
223C22—3882no—
224C22—3882no—
225C22—1944no—
226C22—1944no—
227C22—1944no—
228C22C22×C4×D9711yescharacteristic
229C22#1569085511yescharacteristic
230C22#1569085511yescharacteristic
231C22#1569085612yes—
232C22#1569085812yes—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—3882no—
239C4—3882no—
240C4—1944no—
241C2—3884no—
242C2#2026120811yescharacteristic
243C2#2026121712yes—
244C2#2026147711yescharacteristic
245C2#2026148112yes—
246C2#2026148411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#391×C97C2
socle, cosocleC23×C97C24
prime radicalC22×D388C22
hypercenter#391D97
hypocenterC97#2379
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #20260642C2semidirect
3. #20260626C2semidirect
4. #20260624C2semidirect
6. #20260609C2semidirect
7. #20260606C2semidirect
8. #20260599C2semidirect
9. #391×C97C2semidirect
12. #15690799C4semidirect
48. #13095488C2×C4semidirect
49. #13095488C2×C4semidirect
51. C4×Dic97C2×C4semidirect
54. C2×Q8×C97C2×C4semidirect
124. C97#2379semidirect
127. #391D97semidirect
159. C2×Q8C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907992
#202606422
#202606262
#202606242
#202606132
#202606092
#202606062
#202605992
#391×C972
#237997