Finite Group Databasebeta

← #27157444

Subgroups — #27157444

Order 12416.

388 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D97⋊M16C211yesmaximal, characteristic
2C2×D97⋊M16C211yesmaximal, characteristic
3#20261616C214yesmaximal, complemented
4#117×D97C211yesmaximal, characteristic
5C2×#13095521—21no—
6C2×#13095521C412yescomplemented
7D97⋊M16—21no—
8D97⋊M16—21no—
9D97⋊M16—21no—
10D97⋊M16—21no—
11D97⋊M16—21no—
12D97⋊M16—21no—
13D97⋊M16C412yescomplemented
14D97⋊M16C412yescomplemented
15C2×#13095520C2212yes—
16C2×#13095520C2212yes—
17C2×#13095519—21no—
18C2×#13095519C412yescomplemented
19C22×C4×D97C2211yescharacteristic
20#15690769C411yescharacteristic
21#15690768—21no—
22#15690767—21no—
23#15690766C2212yes—
24C4⋊C4×D97—21no—
25C22⋊C4×D97C412yes—
26C42×D97—21no—
27#117×C97C411yescharacteristic
28#13095521—22no—
29#13095521—22no—
30#13095521—22no—
31#13095521—22no—
32#13095520—22no—
33#13095520—22no—
34#13095520—22no—
35#13095520C2×C414yescomplemented
36#13095519—22no—
37#13095519—22no—
38#13095519—22no—
39#13095519—22no—
40C2×#11715897—22no—
41C2×#11715897C2×C414yescomplemented
42C23×D97C2×C411yescharacteristic
43C22×Dic97C2311yescharacteristic
44C2×C4×D97—22no—
45C2×C4×D97C2×C411yescharacteristic
46C2×C4×D97C2×C411yescharacteristic
47C2×C4×D97C2×C411yescharacteristic
48C2×C4×D97C2×C411yescharacteristic
49C2×C4×D97D411yescharacteristic
50C2×C4×D97D411yescharacteristic
51C2×C4×D97D411yescharacteristic
52C2×C4×D97D411yescharacteristic
53C2×C4×D97D411yescharacteristic
54C2×C4×D97D411yescharacteristic
55C2×C4×D97Q811yescharacteristic
56C2×C4×D97Q811yescharacteristic
57C4⋊Dic97—21no—
58#13095490—22no—
59C22⋊Dic97C2×C412yes—
60#13095488—22no—
61C4×Dic97—21no—
62C4×Dic97—22no—
63C22×C4×C97C2×C411yescharacteristic
64C4⋊C4×C97—21no—
65C22⋊C4×C97C2×C412yes—
66C42×C97—21no—
67C23×C97C22×C411yescharacteristic
68C22×D97—42no—
69C22×D97C4211yescharacteristic
70C22×D97C4211yescharacteristic
71C22×D97C22⋊C411yescharacteristic
72C22×D97C22⋊C411yescharacteristic
73C22×D97C4⋊C411yescharacteristic
74C22×D97C4⋊C411yescharacteristic
75C2×C4×C97—22no—
76C2×C4×C97C4211yescharacteristic
77C2×C4×C97C4211yescharacteristic
78C2×C4×C97C22⋊C411yescharacteristic
79C2×C4×C97C22⋊C411yescharacteristic
80C2×C4×C97C4⋊C411yescharacteristic
81C2×C4×C97C4⋊C411yescharacteristic
82C2×Dic97—22no—
83C2×Dic97C22×C411yescharacteristic
84C2×Dic97C22×C411yescharacteristic
85C2×Dic97C2×D411yescharacteristic
86C2×Dic97C2×D411yescharacteristic
87C2×Dic97C2×D411yescharacteristic
88C2×Dic97C2×Q811yescharacteristic
89C4×D97—44no—
90C4×D97C22⋊C411yescharacteristic
91C4×D97C22⋊C411yescharacteristic
92C4×D97C22⋊C411yescharacteristic
93C4×D97C22⋊C411yescharacteristic
94C4×D97C22⋊C411yescharacteristic
95C4×D97C22⋊C411yescharacteristic
96C4×D97C22⋊C411yescharacteristic
97C4×D97C22⋊C411yescharacteristic
98C4×D97C4⋊C411yescharacteristic
99C4×D97C4⋊C411yescharacteristic
100C4×D97C4⋊C411yescharacteristic
101C4×D97C4⋊C411yescharacteristic
102C4×D97C4⋊C411yescharacteristic
103C4×D97C4⋊C411yescharacteristic
104C4×D97C4⋊C411yescharacteristic
105C4×D97C4⋊C411yescharacteristic
106#11715897—24no—
107#11715897—24no—
108C22×C97—41no—
109C22×C97C2×C4211yescharacteristic
110C22×C97C2×C22⋊C411yescharacteristic
111C22×C97C2×C4⋊C411yescharacteristic
112C2×D97—22no—
113C2×D97—22no—
114C2×D97—22no—
115C2×D97—41no—
116C2×D97—41no—
117C2×D97—41no—
118C2×D97#9511yescharacteristic
119C2×D97#9511yescharacteristic
120C2×D97#9511yescharacteristic
121C2×D97#9511yescharacteristic
122C4×C97—42no—
123C4×C97#9511yescharacteristic
124C4×C97#9511yescharacteristic
125C4×C97#9511yescharacteristic
126C4×C97#9511yescharacteristic
127Dic97—42no—
128Dic97C2×C22⋊C411yescharacteristic
129Dic97C2×C22⋊C411yescharacteristic
130Dic97C2×C4⋊C411yescharacteristic
131Dic97C2×C4⋊C411yescharacteristic
132C2×C97—21no—
133C2×C97—21no—
134C2×C97—21no—
135C2×C97C2×#9511yescharacteristic
136D97—21no—
137D97—21no—
138D97—21no—
139D97#34412yescomplemented
140C2×#344—971nomaximal, Sylow
141C97C2×#34411yescharacteristic, complemented, Sylow
142C22×M16—971no—
143C22×M16—971no—
144C2×#117—971no—
145#344—974no—
146C23×C4—971no—
147C2×M16—972no—
148C2×M16—972no—
149C2×M16—972no—
150C2×M16—972no—
151C2×M16—972no—
152C2×M16—1941no—
153C2×M16—1941no—
154C2×M16—1941no—
155C2×M16—1941no—
156C2×M16—1941no—
157C2×M16—1941no—
158C2×M16—1941no—
159C22×C8—972no—
160C22×C8—1941no—
161#117—971no—
162#117—972no—
163#117—1941no—
164#117—1941no—
165#117#8760411yescharacteristic
166C2×C4⋊C4—1941no—
167C2×C22⋊C4—972no—
168C2×C42—1941no—
169C24—971no—
170C22×C4—971no—
171C22×C4—971no—
172C22×C4—971no—
173C22×C4—971no—
174C22×C4—971no—
175C22×C4—971no—
176C22×C4—971no—
177C22×C4—971no—
178C22×C4—971no—
179C22×C4—971no—
180C22×C4—971no—
181C22×C4—971no—
182C22×C4—971no—
183C22×C4—1942no—
184C22×C4C2×#8760411yescharacteristic
185M16—974no—
186M16—1942no—
187M16—1942no—
188M16—1942no—
189M16—1942no—
190M16—1942no—
191M16—1942no—
192M16—1942no—
193C2×C8—974no—
194C2×C8—1942no—
195C2×C8—1942no—
196C2×C8—1942no—
197C2×C8—1942no—
198C2×C8—1942no—
199C4⋊C4—21no—
200C4⋊C4—1942no—
201C4⋊C4—1941no—
202C22⋊C4—972no—
203C22⋊C4—1942no—
204C22⋊C4C2×#8760412yes—
205C42—21no—
206C42—1942no—
207C42—1941no—
208C23—971no—
209C23—971no—
210C23—971no—
211C23—971no—
212C23—971no—
213C23—971no—
214C23—3882no—
215C23C22×#8760411yescharacteristic
216C2×C4—22no—
217C2×C4—971no—
218C2×C4—971no—
219C2×C4—971no—
220C2×C4—971no—
221C2×C4—971no—
222C2×C4—971no—
223C2×C4—971no—
224C2×C4—971no—
225C2×C4—971no—
226C2×C4—971no—
227C2×C4—971no—
228C2×C4—971no—
229C2×C4—971no—
230C2×C4—971no—
231C2×C4—971no—
232C2×C4—971no—
233C2×C4—971no—
234C2×C4—971no—
235C2×C4—971no—
236C2×C4—971no—
237C2×C4—971no—
238C2×C4—971no—
239C2×C4—3884no—
240C2×C4—1942no—
241C2×C4C4×#8760411yescharacteristic
242C2×C4C4×#8760411yescharacteristic
243C2×C4C22⋊#8760411yescharacteristic
244C2×C4C22⋊#8760411yescharacteristic
245C2×C4C4⋊#8760411yescharacteristic
246C2×C4C4⋊#8760411yescharacteristic
247C8—1944no—
248C8—1944no—
249C22—41no—
250C22—971no—
251C22—971no—
252C22—971no—
253C22—971no—
254C22—3881no—
255C22—3881no—
256C22—3881no—
257C22—1942no—
258C22—1942no—
259C22—1942no—
260C22C2×C4×#8760411yescharacteristic
261C22C2×C22⋊#8760411yescharacteristic
262C22C2×C4⋊#8760411yescharacteristic
263C4—42no—
264C4—971no—
265C4—971no—
266C4—971no—
267C4—971no—
268C4—3882no—
269C4(C2×#87604)⋊C411yescharacteristic
270C4(C2×#87604)⋊C411yescharacteristic
271C4(C2×#87604)⋊C411yescharacteristic
272C4(C2×#87604)⋊C411yescharacteristic
273C2—21no—
274C2—21no—
275C2—21no—
276C2—972no—
277C2—1941no—
278C2—1941no—
279C2—1941no—
280C2C2×(C2×#87604)⋊C411yescharacteristic
subgroupG/H
centerC4(C2×#87604)⋊C4
commutatorC22×C97C2×C42
FrattiniC23C22×#87604
Fitting#117×C97C4
socleC2×C97C2×#95
prime radicalC23×D97C2×C4
cosocleC22×Dic97C23
hypercenter#117#87604
hypocenterC97C2×#344
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #20261616C2semidirect
6. C2×#13095521C4semidirect
13. D97⋊M16C4semidirect
14. D97⋊M16C4semidirect
18. C2×#13095519C4semidirect
35. #13095520C2×C4semidirect
41. C2×#11715897C2×C4semidirect
139. D97#344semidirect
141. C97C2×#344semidirect

Maximal subgroups

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

subgroupindex
C2×D97⋊M162
#202616162
#117×D972
C2×#34497