Finite Group Databasebeta

← #27157434

Subgroups — #27157434

Order 12416.

420 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×#15690889C211yesmaximal, characteristic
2C2×#15690889C211yesmaximal, characteristic
3#20261612C214yesmaximal
4#117×D97C211yesmaximal, characteristic
5#15690889—21no—
6#15690889—21no—
7#15690889—21no—
8#15690889—21no—
9#15690889—21no—
10#15690889—21no—
11#15690889C412yescomplemented
12#15690889C412yescomplemented
13C2×C4⋊#87604—21no—
14C2×C4⋊#87604C412yescomplemented
15C2×C22⋊#87604C2212yes—
16C2×C22⋊#87604C2212yes—
17C2×C4×#87604—21no—
18C2×C4×#87604C412yescomplemented
19C22×C4×D97C2211yescharacteristic
20#15690769C411yescharacteristic, complemented
21#15690768—21no—
22#15690767—21no—
23#15690766C412yescomplemented
24C4⋊C4×D97—21no—
25C22⋊C4×D97C2212yes—
26C42×D97—21no—
27#117×C97C411yescharacteristic, complemented
28C22×#87604—22no—
29C22×#87604C2×C414yes—
30C4⋊#87604—22no—
31C4⋊#87604—22no—
32C4⋊#87604—22no—
33C4⋊#87604—22no—
34C22⋊#87604—22no—
35C22⋊#87604—22no—
36C22⋊#87604—22no—
37C22⋊#87604C2×C414yes—
38C4×#87604—22no—
39C4×#87604—22no—
40C4×#87604—22no—
41C4×#87604—22no—
42C23×D97C2311yescharacteristic
43C22×Dic97C2×C411yescharacteristic
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×D97C22×C411yescharacteristic
70C22×D97C22×C411yescharacteristic
71C22×D97C2×D411yescharacteristic
72C22×D97C2×D411yescharacteristic
73C22×D97C2×D411yescharacteristic
74C22×D97C2×Q811yescharacteristic
75C2×#87604—24no—
76C2×#87604—24no—
77C2×#87604—44no—
78C2×#87604—44no—
79C2×C4×C97—22no—
80C2×C4×C97C4211yescharacteristic
81C2×C4×C97C4211yescharacteristic
82C2×C4×C97C22⋊C411yescharacteristic
83C2×C4×C97C22⋊C411yescharacteristic
84C2×C4×C97C4⋊C411yescharacteristic
85C2×C4×C97C4⋊C411yescharacteristic
86C2×Dic97—22no—
87C2×Dic97C4211yescharacteristic
88C2×Dic97C4211yescharacteristic
89C2×Dic97C22⋊C411yescharacteristic
90C2×Dic97C22⋊C411yescharacteristic
91C2×Dic97C4⋊C411yescharacteristic
92C2×Dic97C4⋊C411yescharacteristic
93C4×D97—44no—
94C4×D97C22⋊C411yescharacteristic
95C4×D97C22⋊C411yescharacteristic
96C4×D97C22⋊C411yescharacteristic
97C4×D97C22⋊C411yescharacteristic
98C4×D97C22⋊C411yescharacteristic
99C4×D97C22⋊C411yescharacteristic
100C4×D97C22⋊C411yescharacteristic
101C4×D97C22⋊C411yescharacteristic
102C4×D97C4⋊C411yescharacteristic
103C4×D97C4⋊C411yescharacteristic
104C4×D97C4⋊C411yescharacteristic
105C4×D97C4⋊C411yescharacteristic
106C4×D97C4⋊C411yescharacteristic
107C4×D97C4⋊C411yescharacteristic
108C4×D97C4⋊C411yescharacteristic
109C4×D97C4⋊C411yescharacteristic
110C22×C97—41no—
111C22×C97C2×C4211yescharacteristic
112C22×C97C2×C22⋊C411yescharacteristic
113C22×C97C2×C4⋊C411yescharacteristic
114C2×D97—22no—
115C2×D97—22no—
116C2×D97—22no—
117C2×D97—41no—
118C2×D97—41no—
119C2×D97—41no—
120C2×D97C2×C22⋊C411yescharacteristic
121C2×D97C2×C22⋊C411yescharacteristic
122C2×D97C2×C4⋊C411yescharacteristic
123C2×D97C2×C4⋊C411yescharacteristic
124#87604—44no—
125#87604—44no—
126C4×C97—42no—
127C4×C97#9511yescharacteristic
128C4×C97#9511yescharacteristic
129C4×C97#9511yescharacteristic
130C4×C97#9511yescharacteristic
131Dic97—42no—
132Dic97#9511yescharacteristic
133Dic97#9511yescharacteristic
134Dic97#9511yescharacteristic
135Dic97#9511yescharacteristic
136C2×C97—21no—
137C2×C97—21no—
138C2×C97—21no—
139C2×C97C2×#9511yescharacteristic
140D97—21no—
141D97—21no—
142D97—21no—
143D97#41012yes—
144#1740—971nomaximal, Sylow
145C97#174011yescharacteristic, complemented, Sylow
146C2×#117—971no—
147C2×#117—971no—
148C2×#117—971no—
149#342—974no—
150C23×C4—971no—
151#117—972no—
152#117—972no—
153#117—971no—
154#117—972no—
155#117—1941no—
156#117—1941no—
157#117—1941no—
158#117—1941no—
159#117—1941no—
160#117—1941no—
161#117—1941no—
162#117—1941no—
163#117#8760411yescharacteristic, complemented
164C2×C4⋊C4—972no—
165C2×C4⋊C4—1941no—
166C2×C4⋊C4—1941no—
167C2×C22⋊C4—972no—
168C2×C22⋊C4—972no—
169C2×C22⋊C4—972no—
170C2×C42—972no—
171C2×C42—1941no—
172C2×C42—1941no—
173C24—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—971no—
184C22×C4—971no—
185C22×C4—971no—
186C22×C4—971no—
187C22×C4—974no—
188C22×C4—1942no—
189C22×C4—1942no—
190C22×C4C2×#8760411yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—1942no—
193C4⋊C4—1942no—
194C4⋊C4—1942no—
195C4⋊C4—1942no—
196C4⋊C4—1942no—
197C4⋊C4—1941no—
198C22⋊C4—974no—
199C22⋊C4—972no—
200C22⋊C4—1942no—
201C22⋊C4—1942no—
202C22⋊C4—1942no—
203C22⋊C4—1942no—
204C22⋊C4C2×#8760412yes—
205C42—21no—
206C42—1942no—
207C42—1942no—
208C42—1942no—
209C42—1942no—
210C42—1942no—
211C42—1941no—
212C23—971no—
213C23—971no—
214C23—971no—
215C23—971no—
216C23—971no—
217C23—971no—
218C23—3882no—
219C23C22×#8760411yescharacteristic
220C2×C4—22no—
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—971no—
240C2×C4—971no—
241C2×C4—971no—
242C2×C4—971no—
243C2×C4—3884no—
244C2×C4—3884no—
245C2×C4—3884no—
246C2×C4—1944no—
247C2×C4—1944no—
248C2×C4—1942no—
249C2×C4C4×#8760411yescharacteristic
250C2×C4C4×#8760411yescharacteristic
251C2×C4C22⋊#8760411yescharacteristic
252C2×C4C22⋊#8760411yescharacteristic
253C2×C4C4⋊#8760411yescharacteristic
254C2×C4C4⋊#8760411yescharacteristic
255C22—41no—
256C22—971no—
257C22—971no—
258C22—971no—
259C22—971no—
260C22—3881no—
261C22—3881no—
262C22—3881no—
263C22—1942no—
264C22—1942no—
265C22—1942no—
266C22C2×C4×#8760411yescharacteristic
267C22C2×C22⋊#8760411yescharacteristic
268C22C2×C4⋊#8760411yescharacteristic
269C4—42no—
270C4—971no—
271C4—971no—
272C4—971no—
273C4—971no—
274C4—3882no—
275C4—3884no—
276C4—3884no—
277C4(C2×#87604)⋊C411yescharacteristic
278C4(C2×#87604)⋊C411yescharacteristic
279C4(C2×#87604)⋊C411yescharacteristic
280C4(C2×#87604)⋊C411yescharacteristic
281C2—21no—
282C2—21no—
283C2—21no—
284C2—972no—
285C2—1941no—
286C2—1941no—
287C2—1941no—
288C2C2×(C2×#87604)⋊C411yescharacteristic
subgroupG/H
centerC4(C2×#87604)⋊C4
commutatorC22×C97C2×C42
FrattiniC23C22×#87604
Fitting#117×C97C4
socleC2×C97C2×#95
prime radical, cosocleC23×D97C23
hypercenter#117#87604
hypocenterC97#1740
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
11. #15690889C4semidirect
12. #15690889C4semidirect
14. C2×C4⋊#87604C4semidirect
18. C2×C4×#87604C4semidirect
20. #15690769C4semidirect
23. #15690766C4semidirect
27. #117×C97C4semidirect
145. C97#1740semidirect
163. #117#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908892
#202616122
#117×D972
#174097