Finite Group Databasebeta

← C2×#20261061 #27151622

Subgroups — C2×#20261061 #27151622

Order 12416.

468 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#15690785C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic, complemented
3#20261061C218yesmaximal, direct product factor
4C2×#15690743C211yesmaximal, characteristic, complemented
5C2×#15690732C211yesmaximal, characteristic, complemented
6C2×#15690730C211yesmaximal, characteristic, complemented
7C2×#15690722C211yesmaximal, characteristic, complemented
8C2×#106×C97C211yesmaximal, characteristic, complemented
9C22×D388—21no—
10C22×D388C2211yescharacteristic
11C22×C4×D97C2211yescharacteristic
12C22×#11715895C2211yescharacteristic
13#15690785—22no—
14#15690785C2214yes—
15#15690768—22no—
16#15690768C2214yescomplemented
17C2×C4⋊Dic97C2211yescharacteristic
18C2×#13095488—21no—
19C2×#13095488—21no—
20C2×C4×Dic97—21no—
21#15690743C2214yescomplemented
22#15690732C2214yescomplemented
23#15690730C2214yescomplemented
24#15690722C2214yescomplemented
25C22×C8×C97C2211yescharacteristic
26C2×C4⋊C4×C97C2211yescharacteristic
27C2×C4⋊C4×C97C2211yescharacteristic
28#106×C97C2214yescomplemented
29C23×D97—21no—
30C23×D97D411yescharacteristic
31C2×D388—22no—
32C2×D388—22no—
33C2×D388—24no—
34C2×D388—21no—
35C2×D388—44no—
36C2×D388C2312yes—
37C22×Dic97—21no—
38C22×Dic97D411yescharacteristic
39C2×C4×D97—22no—
40C2×C4×D97—22no—
41C2×C4×D97D412yes—
42C2×C4×D97C2312yes—
43C2×#11715895—22no—
44C2×#11715895C2312yes—
45C4⋊Dic97—21no—
46C4⋊Dic97C2312yes—
47#13095488—24no—
48#13095488—24no—
49C4×Dic97—24no—
50C22×C4×C97—21no—
51C22×C4×C97—21no—
52C22×C4×C97C2311yescharacteristic
53C2×C8×C97—22no—
54C2×C8×C97C2312yes—
55C4⋊C4×C97—21no—
56C4⋊C4×C97—21no—
57C4⋊C4×C97C2312yes—
58C4⋊C4×C97C2312yes—
59C23×C97C2×D411yescharacteristic
60C22×D97—22no—
61C22×D97—22no—
62C22×D97—22no—
63C22×D97—22no—
64C22×D97—42no—
65C22×D97—42no—
66C22×D97—42no—
67C22×D97C2×D412yes—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—42no—
71C2×C4×C97—42no—
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97#4112yes—
74C2×C4×C97#4112yes—
75C2×C4×C97C2411yescharacteristic
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—42no—
79C2×Dic97C2×D412yes—
80D388—22no—
81D388—22no—
82D388—48no—
83D388—42no—
84C4×D97—24no—
85C4×D97—22no—
86C4×D97—22no—
87C8×C97—22no—
88#11715895—22no—
89C22×C97#12312yes—
90C22×C97#13512yes—
91C22×C97AGL(1,ℤ/8)12yes—
92C22×C97C22×D411yescharacteristic
93C2×D97—22no—
94C2×D97—22no—
95C2×D97—22no—
96C2×D97—44no—
97C2×D97—42no—
98C2×D97—42no—
99C2×D97—42no—
100C2×D97—84no—
101C4×C97—42no—
102C4×C97—42no—
103C4×C97#12312yes—
104C4×C97C2×#4111yescharacteristic
105C4×C97C2×#4111yescharacteristic
106Dic97—22no—
107Dic97—42no—
108C2×C97#48214yescomplemented
109C2×C97C2×#12311yescharacteristic
110C2×C97C2×#13511yescharacteristic
111C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
112D97—42no—
113D97—82no—
114C2×#482—971nomaximal, Sylow
115C97C2×#48211yescharacteristic, complemented, Sylow
116C2×#121—971no—
117C2×#117—971no—
118#482—978no—
119C2×#107—971no—
120C2×#106D9711yescharacteristic, complemented
121C2×#102—971no—
122C2×#102—971no—
123C2×C22⋊C8—971no—
124C22×D4—971no—
125C22×D4—1941no—
126C23×C4—971no—
127C22×C8—971no—
128C22×C8C2×D9711yescharacteristic
129#121—974no—
130#121—1942no—
131#117—974no—
132#117—1942no—
133C2×C4⋊C4—971no—
134C2×C4⋊C4C2×D9711yescharacteristic
135C2×C4⋊C4C2×D9711yescharacteristic
136C2×C22⋊C4—1941no—
137C2×C22⋊C4—1941no—
138C2×C42—1941no—
139#107—974no—
140#106C2×D9714yescomplemented
141#102—974no—
142#102—974no—
143C22⋊C8—974no—
144C24—971no—
145C24—1941no—
146C2×D4—972no—
147C2×D4—3884no—
148C2×D4—1944no—
149C2×D4—1941no—
150C2×D4—1942no—
151C2×D4—1942no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—971no—
155C22×C4—972no—
156C22×C4—972no—
157C22×C4—1941no—
158C22×C4—1942no—
159C22×C4—1942no—
160C22×C4C22×D9711yescharacteristic
161C2×C8—22no—
162C2×C8—972no—
163C2×C8—1942no—
164C2×C8C22×D9712yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—972no—
168C4⋊C4—1941no—
169C4⋊C4C22×D9712yes—
170C4⋊C4C22×D9712yes—
171C22⋊C4—1944no—
172C22⋊C4—1944no—
173C42—1944no—
174C23—972no—
175C23—3882no—
176C23—3882no—
177C23—3882no—
178C23—1942no—
179C23—1942no—
180C23—1942no—
181C23—1942no—
182C23D4×D9711yescharacteristic
183D4—3888no—
184D4—3882no—
185D4—1942no—
186D4—1942no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—972no—
192C2×C4—3882no—
193C2×C4—1942no—
194C2×C4—1942no—
195C2×C4—1944no—
196C2×C4—1942no—
197C2×C4—1942no—
198C2×C4D4×D9711yescharacteristic
199C2×C4#1309551112yes—
200C2×C4#1309551212yes—
201C2×C4C23×D9711yescharacteristic
202C8—22no—
203C8—1942no—
204C22—7764no—
205C22—3882no—
206C22—3882no—
207C22—3882no—
208C22—3884no—
209C22—1942no—
210C22—1942no—
211C22—1942no—
212C22#1569079512yes—
213C22#1569083112yes—
214C22#1569083712yes—
215C22C2×D4×D9711yescharacteristic
216C4—42no—
217C4—42no—
218C4—3882no—
219C4—1942no—
220C4#1569079512yes—
221C4C2×#1309551111yescharacteristic
222C4C2×#1309551211yescharacteristic
223C2—7762no—
224C2—3882no—
225C2#2026106114yesdirect product factor
226C2C2×#1569079511yescharacteristic
227C2C2×#1569083111yescharacteristic
228C2C2×#1569083711yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#106×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690785C2
hypercenterC2×#106D97
hypocenterC97C2×#482
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
2. C2×#15690768C2semidirect
3. #20261061C2direct (N × H)
4. C2×#15690743C2semidirect
5. C2×#15690732C2semidirect
6. C2×#15690730C2semidirect
7. C2×#15690722C2semidirect
8. C2×#106×C97C2semidirect
16. #15690768C22semidirect
21. #15690743C22semidirect
22. #15690732C22semidirect
23. #15690730C22semidirect
24. #15690722C22semidirect
28. #106×C97C22semidirect
108. C2×C97#482semidirect
115. C97C2×#482semidirect
120. C2×#106D97semidirect
140. #106C2×D97semidirect
225. C2#20261061direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156907852
C2×#156907682
#202610612
C2×#156907432
C2×#156907322
C2×#156907302
C2×#156907222
C2×#106×C972
C2×#48297