Finite Group Databasebeta

← C2×#20260738 #27150235

Subgroups — C2×#20260738 #27150235

Order 12416.

1008 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×D4×D97C211yesmaximal, characteristic
2C22×#13095494C211yesmaximal, characteristic, complemented
3C2×#15690768C211yesmaximal, characteristic, complemented
4#20260738C218yesmaximal, direct product factor
5C2×#15690732C211yesmaximal, characteristic, complemented
6C2×#15690731C211yesmaximal, characteristic, complemented
7C2×#15690730C211yesmaximal, characteristic, complemented
8C2×#102×C97C211yesmaximal, characteristic, complemented
9C24×D97—21no—
10C22×D388C2211yescharacteristic
11C22×C22⋊D97—21no—
12C2×D4×D97—22no—
13C2×D4×D97—24no—
14C2×D4×D97C418yescomplemented
15C2×D4×D97C2214yes—
16C22×C4×D97C2211yescharacteristic
17C2×#13095494—24no—
18C2×#13095494C2214yescomplemented
19C22×#11715895C2211yescharacteristic, complemented
20#15690768—22no—
21#15690768C2214yescomplemented
22C2×C4⋊Dic97C2211yescharacteristic, complemented
23C2×#13095488—21no—
24C2×C4×Dic97—21no—
25#15690732C2214yescomplemented
26#15690731C2214yescomplemented
27#15690730C2214yescomplemented
28C22×D4×C97C2211yescharacteristic
29C22×C8×C97C2211yescharacteristic, complemented
30C2×C4⋊C4×C97C2211yescharacteristic, complemented
31#102×C97C2214yescomplemented
32C23×D97—21no—
33C23×D97—28no—
34C23×D97—24no—
35C23×D97—44no—
36C23×D97—44no—
37C23×D97D411yescharacteristic
38C2×D388—21no—
39C2×D388—22no—
40C2×D388C2×C414yes—
41C2×D388C2312yes—
42C2×C22⋊D97—28no—
43C2×C22⋊D97—24no—
44D4×D97—216no—
45D4×D97—28no—
46D4×D97C2×C4116yescomplemented
47C22×Dic97—21no—
48C22×Dic97D411yescharacteristic
49C2×C4×D97C2×C414yes—
50C2×C4×D97D412yes—
51C2×C4×D97D414yes—
52C2×C4×D97C2312yes—
53#13095494—28no—
54C2×#11715895—22no—
55C2×#11715895C2312yescomplemented
56C4⋊Dic97—21no—
57C4⋊Dic97C2312yescomplemented
58#13095488—24no—
59C4×Dic97—24no—
60C24×C97—21no—
61C2×D4×C97—21no—
62C2×D4×C97—22no—
63C2×D4×C97C2×C414yes—
64C2×D4×C97C2312yes—
65C22×C4×C97—21no—
66C22×C4×C97C2311yescharacteristic
67C2×C8×C97—22no—
68C2×C8×C97C2312yescomplemented
69C4⋊C4×C97—21no—
70C4⋊C4×C97C2312yescomplemented
71C23×C97—24no—
72C23×C97—22no—
73C23×C97—42no—
74C23×C97—42no—
75C23×C97C2×D411yescharacteristic
76C22×D97—22no—
77C22×D97—22no—
78C22×D97—24no—
79C22×D97—22no—
80C22×D97—216no—
81C22×D97—48no—
82C22×D97—48no—
83C22×D97—432no—
84C22×D97—42no—
85C22×D97—42no—
86C22×D97C22⋊C414yes—
87C22×D97C2×D412yes—
88D4×C97—24no—
89D4×C97—22no—
90D4×C97C22×C414yes—
91C2×C4×C97—22no—
92C2×C4×C97—42no—
93C2×C4×C97C22×C412yes—
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2×D412yes—
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—216no—
98C2×Dic97—22no—
99C2×Dic97—42no—
100C2×Dic97C22⋊C414yes—
101C2×Dic97C2×D412yes—
102D388—24no—
103D388—22no—
104D388C22×C414yes—
105C4×D97C22⋊C418yes—
106C4×D97C22×C414yes—
107C4×D97C2×D414yes—
108C8×C97—22no—
109#11715895—22no—
110C22×C97—24no—
111C22×C97—42no—
112C22×C97—42no—
113C22×C97—48no—
114C22×C97C2×C22⋊C412yes—
115C22×C97AGL(1,ℤ/8)12yes—
116C22×C97AGL(1,ℤ/8)12yes—
117C22×C97C22×D411yescharacteristic
118C2×D97—22no—
119C2×D97—22no—
120C2×D97—28no—
121C2×D97—24no—
122C2×D97—432no—
123C2×D97—42no—
124C2×D97—42no—
125C2×D97—48no—
126C2×D97C2×C22⋊C414yes—
127C4×C97—42no—
128C4×C97C2×C22⋊C412yes—
129C4×C97C23×C411yescharacteristic
130C4×C97C22×D411yescharacteristic
131Dic97—42no—
132Dic97C2×C22⋊C414yes—
133C2×C97—44no—
134C2×C97#41814yescomplemented
135C2×C97C22×C22⋊C411yescharacteristic
136C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
138D97—24no—
139D97—44no—
140C2×#418—971nomaximal, Sylow
141C97C2×#41811yescharacteristic, complemented, Sylow
142C23×D4—971no—
143C22×M16—971no—
144C2×#117—971no—
145#418—978no—
146C2×#102—971no—
147C2×#102—971no—
148C2×#102—971no—
149C2×#102D9711yescharacteristic, complemented
150C25—1941no—
151C22×D4—978no—
152C22×D4—974no—
153C22×D4—971no—
154C22×D4—1942no—
155C22×D4—1944no—
156C22×D4—1941no—
157C22×D4C2×D9711yescharacteristic
158C23×C4—971no—
159C2×M16—974no—
160C2×M16—1944no—
161C22×C8—971no—
162C22×C8C2×D9711yescharacteristic, complemented
163#117—974no—
164#117—1942no—
165C2×C4⋊C4—971no—
166C2×C4⋊C4C2×D9711yescharacteristic, complemented
167C2×C22⋊C4—1941no—
168C2×C42—1941no—
169#102—974no—
170#102—974no—
171#102—974no—
172#102C2×D9714yescomplemented
173C24—21no—
174C24—971no—
175C24—3884no—
176C24—3884no—
177C24—1948no—
178C24—1944no—
179C24—1941no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—9716no—
183C2×D4—974no—
184C2×D4—972no—
185C2×D4—19416no—
186C2×D4—1948no—
187C2×D4—1948no—
188C2×D4—1944no—
189C2×D4—1941no—
190C2×D4—1942no—
191C2×D4C4×D9714yes—
192C2×D4C22×D9712yes—
193C22×C4—21no—
194C22×C4—971no—
195C22×C4—974no—
196C22×C4—972no—
197C22×C4—974no—
198C22×C4—972no—
199C22×C4—1941no—
200C22×C4C22×D9711yescharacteristic
201M16—1948no—
202C2×C8—22no—
203C2×C8—972no—
204C2×C8—1942no—
205C2×C8C22×D9712yescomplemented
206C4⋊C4—21no—
207C4⋊C4—972no—
208C4⋊C4—1941no—
209C4⋊C4C22×D9712yescomplemented
210C22⋊C4—1944no—
211C42—1944no—
212C23—24no—
213C23—22no—
214C23—42no—
215C23—42no—
216C23—974no—
217C23—972no—
218C23—38832no—
219C23—3888no—
220C23—3888no—
221C23—3882no—
222C23—3882no—
223C23—19416no—
224C23—1944no—
225C23—1942no—
226C23—1942no—
227C23—1942no—
228C23D4×D9711yescharacteristic
229D4—24no—
230D4—22no—
231D4—974no—
232D4—19416no—
233D4—1944no—
234D4—1942no—
235D4C2×C4×D9714yes—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—974no—
239C2×C4—972no—
240C2×C4—978no—
241C2×C4—974no—
242C2×C4—974no—
243C2×C4—3882no—
244C2×C4—1942no—
245C2×C4C2×C4×D9712yes—
246C2×C4D4×D9711yescharacteristic
247C2×C4D4×D9712yes—
248C2×C4C23×D9711yescharacteristic
249C8—22no—
250C8—1942no—
251C22—24no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22—974no—
256C22—38832no—
257C22—3888no—
258C22—3882no—
259C22—3882no—
260C22—1944no—
261C22—1948no—
262C22—1942no—
263C22—1942no—
264C22C22⋊C4×D9712yes—
265C22#1569083412yes—
266C22#1569083712yes—
267C22C2×D4×D9711yescharacteristic
268C4—42no—
269C4—974no—
270C4—3882no—
271C4C22⋊C4×D9712yes—
272C4C22×C4×D9711yescharacteristic
273C4C2×D4×D9711yescharacteristic
274C2—44no—
275C2—3884no—
276C2—1944no—
277C2#2026073814yesdirect product factor
278C2C2×C22⋊C4×D9711yescharacteristic
279C2C2×#1569083411yescharacteristic
280C2C2×#1569083711yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
FittingC2×#102×C97C2
socleC23×C97C2×D4
prime radicalC22×D4×D97C2
cosocleC2×C4×C97C24
hypercenterC2×#102D97
hypocenterC97C2×#418
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C22×#13095494C2semidirect
3. C2×#15690768C2semidirect
4. #20260738C2direct (N × H)
5. C2×#15690732C2semidirect
6. C2×#15690731C2semidirect
7. C2×#15690730C2semidirect
8. C2×#102×C97C2semidirect
14. C2×D4×D97C4semidirect
18. C2×#13095494C22semidirect
19. C22×#11715895C22semidirect
21. #15690768C22semidirect
22. C2×C4⋊Dic97C22semidirect
25. #15690732C22semidirect
26. #15690731C22semidirect
27. #15690730C22semidirect
29. C22×C8×C97C22semidirect
30. C2×C4⋊C4×C97C22semidirect
31. #102×C97C22semidirect
46. D4×D97C2×C4semidirect
55. C2×#11715895C23semidirect
57. C4⋊Dic97C23semidirect
68. C2×C8×C97C23semidirect
70. C4⋊C4×C97C23semidirect
134. C2×C97#418semidirect
141. C97C2×#418semidirect
149. C2×#102D97semidirect
162. C22×C8C2×D97semidirect
166. C2×C4⋊C4C2×D97semidirect
172. #102C2×D97semidirect
205. C2×C8C22×D97semidirect
209. C4⋊C4C22×D97semidirect
277. C2#20260738direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D972
C22×#130954942
C2×#156907682
#202607382
C2×#156907322
C2×#156907312
C2×#156907302
C2×#102×C972
C2×#41897