Finite Group Databasebeta

← #27140240

Subgroups — #27140240

Order 12416.

334 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#20260699C211yesmaximal, characteristic, complemented
2C2×#15690717C211yesmaximal, characteristic, complemented
3#20260472C212yesmaximal, complemented
4#20260469C212yesmaximal, complemented
5#525×C97C211yesmaximal, characteristic
6C23×Dic97—21no—
7C2×#13095495—21no—
8C22⋊#11715895—21no—
9C22⋊#11715895C2212yes—
10#15690717—21no—
11#15690717C2212yes—
12C2×#41×C97C411yescharacteristic, complemented
13C23×C4×C97C2211yescharacteristic
14#123×C97—21no—
15C22⋊Q8×C97C411yescharacteristic, complemented
16C22⋊Q8×C97C2211yescharacteristic
17#121×C97C411yescharacteristic, complemented
18#121×C97C2211yescharacteristic
19#120×C97—21no—
20C4×D4×C97—21no—
21C4×D4×C97—21no—
22#117×C97C411yescharacteristic, complemented
23C22×Dic97—24no—
24C22×Dic97—22no—
25C22×Dic97—42no—
26C22×Dic97—42no—
27#13095495—42no—
28C2×#11715895—22no—
29C24×C97D411yescharacteristic
30#41×C97—42no—
31C2×Q8×C97C2×C411yescharacteristic, complemented
32C2×D4×C97—21no—
33C2×D4×C97—21no—
34C2×D4×C97—21no—
35C2×D4×C97C2×C411yescharacteristic, complemented
36C22×C4×C97—21no—
37C22×C4×C97—42no—
38C22×C4×C97C2×C411yescharacteristic
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97D411yescharacteristic
41C22×C4×C97D411yescharacteristic
42C22×C4×C97D411yescharacteristic
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic, complemented
47C4⋊C4×C97C2×C411yescharacteristic, complemented
48C22⋊C4×C97—21no—
49C22⋊C4×C97—21no—
50C22⋊C4×C97—21no—
51C22⋊C4×C97—21no—
52C22⋊C4×C97—21no—
53C42×C97—21no—
54C23×C97—22no—
55C23×C97—22no—
56C23×C97—21no—
57C23×C97—21no—
58C23×C97—21no—
59C23×C97C22⋊C411yescharacteristic
60C23×C97C22⋊C411yescharacteristic
61C23×C97C2×D411yescharacteristic
62Q8×C97—41no—
63D4×C97—41no—
64D4×C97—42no—
65D4×C97—42no—
66D4×C97—41no—
67D4×C97—41no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—21no—
71C2×C4×C97—21no—
72C2×C4×C97—21no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97—42no—
78C2×C4×C97C22⋊C411yescharacteristic
79C2×C4×C97C22⋊C411yescharacteristic
80C2×C4×C97C22×C411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×Dic97—24no—
83C2×Dic97—48no—
84C2×Dic97—42no—
85C2×Dic97—42no—
86#11715895—42no—
87C22×C97—24no—
88C22×C97—24no—
89C22×C97—21no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97—41no—
96C22×C97—41no—
97C22×C97—41no—
98C22×C97#9912yes—
99C22×C97#10412yes—
100C22×C97C2×C22⋊C411yescharacteristic
101C22×C97AGL(1,ℤ/8)11yescharacteristic
102C22×C97#13711yescharacteristic
103C4×C97—21no—
104C4×C97—21no—
105C4×C97—21no—
106C4×C97—21no—
107C4×C97—41no—
108C4×C97—41no—
109C4×C97—41no—
110Dic97—44no—
111C2×C97—21no—
112C2×C97—22no—
113C2×C97—22no—
114C2×C97—21no—
115C2×C97—81no—
116C2×C97C2×#9911yescharacteristic
117C2×C97#41711yescharacteristic
118C2×C97C2×#10411yescharacteristic
119#1504—971nomaximal, Sylow
120C97#150411yescharacteristic, complemented, Sylow
121#525D9711yescharacteristic
122#407—971no—
123C2×#95—971no—
124#328—972no—
125#327—972no—
126C2×#41Dic9711yescharacteristic, complemented
127C23×C4—1941no—
128C23×C4C2×D9711yescharacteristic
129C2×M16—1941no—
130#123—21no—
131C22⋊Q8Dic9711yescharacteristic, complemented
132C22⋊Q8C2×D9711yescharacteristic
133#121Dic9711yescharacteristic, complemented
134#121C2×D9711yescharacteristic
135#120—21no—
136C4×D4—21no—
137C4×D4—21no—
138#117Dic9711yescharacteristic, complemented
139C22⋊C8—972no—
140C22⋊C8—1941no—
141#95—972no—
142#95—1941no—
143C24C22⋊D9711yescharacteristic
144#41—42no—
145C2×Q8C2×Dic9711yescharacteristic, complemented
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C2×Dic9711yescharacteristic, complemented
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—3882no—
153C22×C4—3882no—
154C22×C4—1944no—
155C22×C4—1942no—
156C22×C4C2×Dic9711yescharacteristic
157C22×C4C2×Dic9711yescharacteristic
158C22×C4C22⋊D9711yescharacteristic
159C22×C4C22⋊D9711yescharacteristic
160C22×C4C22⋊D9711yescharacteristic
161C22×C4C22×D9711yescharacteristic
162M16—3882no—
163C2×C8—1942no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4C2×Dic9711yescharacteristic, complemented
167C4⋊C4C2×Dic9711yescharacteristic, complemented
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C42—21no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23—21no—
179C23C22⋊Dic9711yescharacteristic
180C23C22⋊Dic9711yescharacteristic
181C23C2×C22⋊D9711yescharacteristic
182Q8—41no—
183D4—41no—
184D4—41no—
185D4—41no—
186D4—42no—
187D4—42no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—3888no—
199C2×C4—3882no—
200C2×C4—3882no—
201C2×C4—1944no—
202C2×C4C22⋊Dic9711yescharacteristic
203C2×C4C22⋊Dic9711yescharacteristic
204C2×C4C22×Dic9711yescharacteristic
205C2×C4C2×C22⋊D9711yescharacteristic
206C8—3882no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22#1569072512yes—
219C22#1569073712yes—
220C22C2×C22⋊Dic9711yescharacteristic
221C22#1569083511yescharacteristic
222C22#1569084011yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—3884no—
231C2—21no—
232C2—21no—
233C2—22no—
234C2—22no—
235C2—81no—
236C2C2×#1569072511yescharacteristic
237C2#2026073611yescharacteristic
238C2C2×#1569073711yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radical#525×C97C2
socleC22×C97C2×C22⋊C4
cosocleC22×C4×C97C23
hypercenter#525D97
hypocenterC97#1504
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #20260699C2semidirect
2. C2×#15690717C2semidirect
3. #20260472C2semidirect
4. #20260469C2semidirect
12. C2×#41×C97C4semidirect
15. C22⋊Q8×C97C4semidirect
17. #121×C97C4semidirect
22. #117×C97C4semidirect
31. C2×Q8×C97C2×C4semidirect
35. C2×D4×C97C2×C4semidirect
46. C4⋊C4×C97C2×C4semidirect
47. C4⋊C4×C97C2×C4semidirect
120. C97#1504semidirect
126. C2×#41Dic97semidirect
131. C22⋊Q8Dic97semidirect
133. #121Dic97semidirect
138. #117Dic97semidirect
145. C2×Q8C2×Dic97semidirect
149. C2×D4C2×Dic97semidirect
166. C4⋊C4C2×Dic97semidirect
167. C4⋊C4C2×Dic97semidirect

Maximal subgroups

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

subgroupindex
#202606992
C2×#156907172
#202604722
#202604692
#525×C972
#150497