Finite Group Databasebeta

← #27140239

Subgroups — #27140239

Order 12416.

334 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261248C211yesmaximal, characteristic
2#20260697C211yesmaximal, characteristic, complemented
3#95×D97C211yesmaximal, characteristic, complemented
4#20260472C211yesmaximal, characteristic, complemented
5#20260470C211yesmaximal, characteristic, complemented
6#20260468C211yesmaximal, characteristic, complemented
7#328×C97C211yesmaximal, characteristic, complemented
8C2×#13095512C411yescharacteristic, complemented
9C22×C4×D97—21no—
10C22×C4×D97C2211yescharacteristic
11C2×#13095494—21no—
12#15690795—21no—
13#15690786C411yescharacteristic, complemented
14#15690785C411yescharacteristic, complemented
15#15690784C2211yescharacteristic
16#15690780—21no—
17#15690772—21no—
18#15690770—21no—
19#15690768C411yescharacteristic, complemented
20C22⋊#11715895C2211yescharacteristic
21#15690722—21no—
22#15690717—21no—
23#15690717C2211yescharacteristic
24C22⋊Q8×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26#95×C97C2211yescharacteristic
27C23×D97D411yescharacteristic
28#13095512—42no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388C2×C411yescharacteristic, complemented
32C2×C22⋊D97—21no—
33C22×Dic97—21no—
34C22×Dic97D411yescharacteristic
35C2×C4×D97—24no—
36C2×C4×D97—21no—
37C2×C4×D97—42no—
38C2×C4×D97—42no—
39C2×C4×D97—42no—
40C2×C4×D97C2×C411yescharacteristic
41C2×C4×D97C2×C411yescharacteristic
42C2×C4×D97D411yescharacteristic
43C2×C4×D97D411yescharacteristic
44#13095494—42no—
45C2×#11715895—21no—
46C4⋊Dic97C2×C411yescharacteristic, complemented
47#13095490—21no—
48#13095488—21no—
49#13095488—21no—
50#13095488—21no—
51#13095488—21no—
52C4×Dic97—21no—
53C2×Q8×C97C2×C411yescharacteristic, complemented
54C22×C4×C97—21no—
55C22×C4×C97C2311yescharacteristic
56C2×C8×C97—21no—
57C4⋊C4×C97—21no—
58C4⋊C4×C97C2×C411yescharacteristic, complemented
59C22⋊C4×C97—21no—
60C23×C97C2×D411yescharacteristic
61C22×D97—21no—
62C22×D97—21no—
63C22×D97—22no—
64C22×D97—22no—
65C22×D97—21no—
66C22×D97C22⋊C411yescharacteristic
67C22×D97C22⋊C411yescharacteristic
68Q8×C97—41no—
69C2×C4×C97—22no—
70C2×C4×C97—21no—
71C2×C4×C97—21no—
72C2×C4×C97—41no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97C22×C411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C22⋊D97—41no—
78C22⋊D97—41no—
79C2×Dic97—22no—
80C2×Dic97—21no—
81C2×Dic97—41no—
82C2×Dic97—41no—
83C2×Dic97—41no—
84C2×Dic97C22⋊C411yescharacteristic
85C2×Dic97C22⋊C411yescharacteristic
86D388—41no—
87D388—42no—
88D388—42no—
89C4×D97—22no—
90C4×D97—22no—
91C4×D97—41no—
92C4×D97—41no—
93C4×D97—42no—
94C4×D97—48no—
95C8×C97—41no—
96#11715895—41no—
97C22×C97—21no—
98C22×C97—21no—
99C22×C97C2×C22⋊C411yescharacteristic
100C22×C97AGL(1,ℤ/8)11yescharacteristic
101C22×C97#13711yescharacteristic
102C2×D97—24no—
103C2×D97—24no—
104C2×D97—21no—
105C2×D97—21no—
106C2×D97—21no—
107C2×D97—21no—
108C2×D97—41no—
109C2×D97—41no—
110C2×D97—41no—
111C2×D97#9912yes—
112C2×D97#10412yes—
113C4×C97—21no—
114C4×C97—21no—
115C4×C97—42no—
116C4×C97—41no—
117C4×C97—41no—
118Dic97—21no—
119Dic97—21no—
120Dic97—42no—
121Dic97—41no—
122C2×C97—21no—
123C2×C97—21no—
124C2×C97C2×#9911yescharacteristic
125C2×C97#41711yescharacteristic
126C2×C97C2×#10411yescharacteristic
127D97—22no—
128D97—22no—
129D97—81no—
130#1504—971nomaximal, Sylow
131C97#150411yescharacteristic, complemented, Sylow
132#525—971no—
133#407—971no—
134C2×#95—971no—
135#328—971no—
136#328D9711yescharacteristic, complemented
137#327—971no—
138#327—971no—
139C2×#41—971no—
140C23×C4—971no—
141C23×C4—1941no—
142C2×M16—1941no—
143#123—1941no—
144C22⋊Q8—971no—
145C22⋊Q8C2×D9711yescharacteristic
146#121—971no—
147#121—971no—
148#120—1941no—
149C4×D4—1941no—
150C4×D4—1941no—
151#117—971no—
152C22⋊C8—971no—
153C22⋊C8—1941no—
154C22⋊C8C2×D9711yescharacteristic
155#95—971no—
156#95—1941no—
157#95C2×D9711yescharacteristic
158C24—971no—
159#41—3882no—
160C2×Q8C4×D9711yescharacteristic, complemented
161C2×D4—971no—
162C2×D4—1941no—
163C2×D4—1941no—
164C2×D4—1941no—
165C22×C4—21no—
166C22×C4—971no—
167C22×C4—971no—
168C22×C4—971no—
169C22×C4—971no—
170C22×C4—971no—
171C22×C4—3882no—
172C22×C4—3882no—
173C22×C4—3882no—
174C22×C4—1941no—
175C22×C4—1944no—
176C22×C4—1941no—
177C22×C4C22×D9711yescharacteristic
178M16—3882no—
179C2×C8—21no—
180C2×C8—1941no—
181C4⋊C4—21no—
182C4⋊C4—971no—
183C4⋊C4—1941no—
184C4⋊C4C4×D9711yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—1941no—
187C22⋊C4—1941no—
188C22⋊C4—1941no—
189C22⋊C4—1941no—
190C42—1941no—
191C23—971no—
192C23—971no—
193C23—1942no—
194C23—1942no—
195C23—1941no—
196C23—1941no—
197C23—1941no—
198C23D4×D9711yescharacteristic
199Q8—41no—
200D4—3881no—
201D4—3882no—
202D4—3882no—
203D4—3881no—
204D4—3881no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—971no—
212C2×C4—971no—
213C2×C4—3881no—
214C2×C4—3881no—
215C2×C4—3888no—
216C2×C4—3882no—
217C2×C4—3881no—
218C2×C4—3881no—
219C2×C4—3881no—
220C2×C4—1942no—
221C2×C4—1942no—
222C2×C4—1942no—
223C2×C4—1941no—
224C2×C4C2×C4×D9711yescharacteristic
225C2×C4D4×D9711yescharacteristic
226C8—41no—
227C8—3881no—
228C22—21no—
229C22—21no—
230C22—972no—
231C22—972no—
232C22—3881no—
233C22—3881no—
234C22—3881no—
235C22—1944no—
236C22—1944no—
237C22—1941no—
238C22—1941no—
239C22—1941no—
240C22—1941no—
241C22C22⋊C4×D9711yescharacteristic
242C22#1569083711yescharacteristic
243C22#1569084211yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—41no—
249C4—3882no—
250C4—3881no—
251C4—1941no—
252C4—1941no—
253C2—21no—
254C2—21no—
255C2—7761no—
256C2—1942no—
257C2—1942no—
258C2#99×D9711yescharacteristic
259C2#2026073511yescharacteristic
260C2#104×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#328×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261248C2
cosocleC22×C4×C97C23
hypercenter#328D97
hypocenterC97#1504
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #20260697C2semidirect
3. #95×D97C2semidirect
4. #20260472C2semidirect
5. #20260470C2semidirect
6. #20260468C2semidirect
7. #328×C97C2semidirect
8. C2×#13095512C4semidirect
13. #15690786C4semidirect
14. #15690785C4semidirect
19. #15690768C4semidirect
31. C2×D388C2×C4semidirect
46. C4⋊Dic97C2×C4semidirect
53. C2×Q8×C97C2×C4semidirect
58. C4⋊C4×C97C2×C4semidirect
131. C97#1504semidirect
136. #328D97semidirect
160. C2×Q8C4×D97semidirect
184. C4⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
#202612482
#202606972
#95×D972
#202604722
#202604702
#202604682
#328×C972
#150497