Finite Group Databasebeta

← #27144405

Subgroups — #27144405

Order 12416.

632 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690793C211yesmaximal, characteristic
2C2×C4×C22⋊D97C211yesmaximal, characteristic
3#20260630C212yesmaximal, complemented
4#20260625C212yesmaximal, complemented
5#20260616C212yesmaximal, complemented
6#20260614C211yesmaximal, characteristic
7#20260608C211yesmaximal, characteristic, complemented
8#20260606C212yesmaximal, complemented
9#20260598C211yesmaximal, characteristic, complemented
10C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
11C22×C4⋊C4×C97C211yesmaximal, characteristic, complemented
12C22×C22⋊D97C2211yescharacteristic
13C22×C4×D97C2211yescharacteristic
14#15690793C418yescomplemented
15C4×C22⋊D97—24no—
16C2×#13095490C2211yescharacteristic
17C2×#13095490C2212yes—
18C2×#13095490C2212yes—
19C2×C22⋊Dic97C2211yescharacteristic
20C2×C22⋊Dic97C2211yescharacteristic
21C2×C22⋊Dic97C2211yescharacteristic
22C2×#13095488C2211yescharacteristic
23C2×#13095488C2212yes—
24C2×#13095488C2212yes—
25C2×C4×Dic97C2211yescharacteristic
26C2×C4×Dic97C2212yes—
27#15690717C2212yes—
28#15690717C2212yes—
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2212yes—
32C23×C4×C97C2211yescharacteristic
33C23×C4×C97C2211yescharacteristic
34C23×C4×C97C2211yescharacteristic
35C2×C4⋊C4×C97—24no—
36C2×C4⋊C4×C97C2212yes—
37C2×C4⋊C4×C97C2212yes—
38C23×D97C2311yescharacteristic
39C2×C22⋊D97—22no—
40C2×C22⋊D97—22no—
41C2×C22⋊D97C2×C414yes—
42C22×Dic97C2311yescharacteristic
43C22×Dic97C2312yes—
44C22×Dic97C2311yescharacteristic
45C22×Dic97C2312yes—
46C22×Dic97C2311yescharacteristic
47C2×C4×D97—22no—
48C2×C4×D97—42no—
49#13095490—22no—
50#13095490—42no—
51#13095490C2×C418yescomplemented
52C22⋊Dic97—22no—
53C22⋊Dic97—22no—
54C22⋊Dic97C2×C414yescomplemented
55#13095488—22no—
56#13095488—42no—
57#13095488C2×C418yescomplemented
58C4×Dic97—22no—
59C4×Dic97—42no—
60C24×C97C2311yescharacteristic
61C22×C4×C97—22no—
62C22×C4×C97—22no—
63C22×C4×C97—22no—
64C22×C4×C97—42no—
65C22×C4×C97—42no—
66C22×C4×C97C2×C414yescomplemented
67C22×C4×C97D414yes—
68C22×C4×C97C2312yes—
69C22×C4×C97C2312yes—
70C22×C4×C97C2311yescharacteristic
71C22×C4×C97C2311yescharacteristic
72C4⋊C4×C97—24no—
73C4⋊C4×C97—24no—
74C23×C97—22no—
75C23×C97—22no—
76C23×C97C22×C412yes—
77C23×C97C2×D412yes—
78C23×C97#4112yes—
79C23×C97C2411yescharacteristic
80C22×D97—21no—
81C22×D97—21no—
82C22×D97—22no—
83C22×D97—22no—
84C22×D97C22×C412yes—
85C2×C4×C97—28no—
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97—22no—
89C2×C4×C97—22no—
90C2×C4×C97—22no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97—44no—
94C2×C4×C97—44no—
95C2×C4×C97C22×C414yes—
96C2×C4×C97C2×D412yes—
97C2×C4×C97C2×D412yes—
98C22⋊D97—28no—
99C2×Dic97—21no—
100C2×Dic97—21no—
101C2×Dic97—22no—
102C2×Dic97—22no—
103C2×Dic97—21no—
104C2×Dic97—21no—
105C2×Dic97—22no—
106C2×Dic97—22no—
107C2×Dic97—22no—
108C2×Dic97—22no—
109C2×Dic97—22no—
110C2×Dic97C22×C414yes—
111C2×Dic97C22×C412yes—
112C4×D97—44no—
113C22×C97—24no—
114C22×C97—22no—
115C22×C97—22no—
116C22×C97—24no—
117C22×C97C4×D414yes—
118C22×C97C23×C411yescharacteristic
119C22×C97C22×D411yescharacteristic
120C22×C97C2×#4111yescharacteristic
121C22×C97H(5,𝔽2)12yes—
122C22×C972−1+412yes—
123C2×D97—22no—
124C2×D97—24no—
125C2×D97—41no—
126C2×D97—41no—
127C2×D97—42no—
128C4×C97—24no—
129C4×C97—22no—
130C4×C97—22no—
131C4×C97—42no—
132Dic97—22no—
133Dic97—24no—
134Dic97—41no—
135Dic97—41no—
136Dic97—42no—
137C2×C97—24no—
138C2×C97C2×C4×D411yescharacteristic
139C2×C97#52012yes—
140C2×C97#53411yescharacteristic
141C2×C97#53611yescharacteristic
142C2×C97#53912yes—
143D97—42no—
144#2322—971nomaximal, Sylow
145C97#232211yescharacteristic, complemented, Sylow
146C2×#123—971no—
147C2×C4×D4—971no—
148C22×C4⋊C4D9711yescharacteristic, complemented
149#389—972no—
150#388—972no—
151#387—972no—
152#386—971no—
153#385—971no—
154#385—972no—
155#381—971no—
156C4×C22⋊C4—971no—
157C22×D4—971no—
158C23×C4—971no—
159C23×C4C2×D9711yescharacteristic
160C23×C4C2×D9711yescharacteristic
161C23×C4C2×D9711yescharacteristic
162#123—978no—
163C4×D4—1944no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—972no—
166C2×C4⋊C4—972no—
167C2×C4⋊C4—971no—
168C2×C4⋊C4C2×D9712yes—
169C2×C4⋊C4C2×D9712yes—
170C2×C22⋊C4—971no—
171C2×C22⋊C4—972no—
172C2×C22⋊C4—972no—
173C2×C22⋊C4—971no—
174C2×C22⋊C4—971no—
175C2×C22⋊C4—971no—
176C2×C42—972no—
177C2×C42—971no—
178#95—972no—
179#95—972no—
180#95—972no—
181#95—972no—
182#95—972no—
183C24—971no—
184C24C22×D9711yescharacteristic
185C2×D4—974no—
186C2×D4—1942no—
187C2×D4—1942no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—972no—
194C22×C4—971no—
195C22×C4—971no—
196C22×C4—972no—
197C22×C4—971no—
198C22×C4—3882no—
199C22×C4—1942no—
200C22×C4C4×D9714yescomplemented
201C22×C4C22⋊D9714yes—
202C22×C4C22×D9712yes—
203C22×C4C22×D9712yes—
204C22×C4C22×D9711yescharacteristic
205C22×C4C22×D9711yescharacteristic
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—978no—
209C4⋊C4—3882no—
210C4⋊C4—1942no—
211C22⋊C4—974no—
212C22⋊C4—978no—
213C22⋊C4—3882no—
214C22⋊C4—1942no—
215C22⋊C4—1942no—
216C22⋊C4—1942no—
217C42—3882no—
218C42—1942no—
219C23—22no—
220C23—22no—
221C23—972no—
222C23—1941no—
223C23—1941no—
224C23—1942no—
225C23—1942no—
226C23C2×C4×D9712yes—
227C23C2×C22⋊D9712yes—
228C23#1309551112yes—
229C23C23×D9711yescharacteristic
230D4—1948no—
231C2×C4—28no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—974no—
242C2×C4—972no—
243C2×C4—3884no—
244C2×C4—1941no—
245C2×C4—1941no—
246C2×C4—1942no—
247C2×C4—1942no—
248C2×C4—1942no—
249C2×C4—1942no—
250C2×C4—1942no—
251C2×C4—1941no—
252C2×C4—1941no—
253C2×C4—1942no—
254C2×C4—1942no—
255C2×C4C2×C4×D9714yes—
256C2×C4C2×C22⋊D9712yes—
257C2×C4C2×C22⋊D9712yes—
258C22—24no—
259C22—22no—
260C22—22no—
261C22—24no—
262C22—3881no—
263C22—3881no—
264C22—3882no—
265C22—1942no—
266C22—1944no—
267C22C4×C22⋊D9714yes—
268C22C22×C4×D9711yescharacteristic
269C22C22×C22⋊D9711yescharacteristic
270C22C2×#1309551111yescharacteristic
271C22#1569085512yes—
272C22#1569085812yes—
273C4—24no—
274C4—22no—
275C4—22no—
276C4—42no—
277C4—3881no—
278C4—3881no—
279C4—3882no—
280C4—1942no—
281C4—1944no—
282C2—24no—
283C2—3882no—
284C2C2×C4×C22⋊D9711yescharacteristic
285C2#2026121612yes—
286C2#2026129411yescharacteristic
287C2#2026130911yescharacteristic
288C2#2026133212yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC22×C4⋊C4×C97C2
socle, cosocleC23×C97C24
prime radicalC22×C22⋊D97C22
hypercenterC22×C4⋊C4D97
hypocenterC97#2322
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20260630C2semidirect
4. #20260625C2semidirect
5. #20260616C2semidirect
7. #20260608C2semidirect
8. #20260606C2semidirect
9. #20260598C2semidirect
10. C4×C22⋊Dic97C2semidirect
11. C22×C4⋊C4×C97C2semidirect
14. #15690793C4semidirect
51. #13095490C2×C4semidirect
54. C22⋊Dic97C2×C4semidirect
57. #13095488C2×C4semidirect
66. C22×C4×C97C2×C4semidirect
145. C97#2322semidirect
148. C22×C4⋊C4D97semidirect
200. C22×C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907932
C2×C4×C22⋊D972
#202606302
#202606252
#202606162
#202606142
#202606082
#202606062
#202605982
C4×C22⋊Dic972
C22×C4⋊C4×C972
#232297