Finite Group Databasebeta

← C2×#20261106 #27152141

Subgroups — C2×#20261106 #27152141

Order 12416.

656 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13095500C211yesmaximal, characteristic, complemented
2C22×#13095499C211yesmaximal, characteristic, complemented
3C22×#13095498C211yesmaximal, characteristic, complemented
4C2×#15690808C211yesmaximal, characteristic, complemented
5C2×#15690806C211yesmaximal, characteristic
6#20261106C218yesmaximal, direct product factor
7C2×C8×Dic97C211yesmaximal, characteristic, complemented
8C22×SD16×C97C211yesmaximal, characteristic, complemented
9C22×Dic194C2211yescharacteristic
10C22×D388C2211yescharacteristic
11C22×C22⋊D97—22no—
12C2×#13095500—24no—
13C2×#13095500C2214yescomplemented
14C2×#13095499—24no—
15C2×#13095499C2214yescomplemented
16C2×#13095498—24no—
17C2×#13095498C2214yescomplemented
18C22×#11715895C2211yescharacteristic
19#15690808—22no—
20#15690808C2214yescomplemented
21#15690806—22no—
22#15690806C2214yes—
23C2×#13095490—22no—
24C2×C4×Dic97C2211yescharacteristic
25C8×Dic97C2214yescomplemented
26C22×Q8×C97C2211yescharacteristic
27C22×D4×C97C2211yescharacteristic
28C2×SD16×C97—24no—
29C2×SD16×C97C2214yescomplemented
30C22×C8×C97C2211yescharacteristic
31C23×D97—21no—
32C2×Dic194—22no—
33C2×Dic194—22no—
34C2×Dic194—21no—
35C2×Dic194C2312yes—
36C2×D388—22no—
37C2×D388—22no—
38C2×D388—21no—
39C2×D388C2312yes—
40C2×C22⋊D97—28no—
41C2×C22⋊D97—48no—
42C22×Dic97—21no—
43C22×Dic97D412yes—
44#13095500—28no—
45#13095499—28no—
46#13095498—28no—
47C2×#11715895D414yescomplemented
48C2×#11715895C2312yes—
49#13095490—28no—
50C4×Dic97D412yes—
51C4×Dic97C2312yes—
52C24×C97—21no—
53C2×Q8×C97—21no—
54C2×Q8×C97—22no—
55C2×Q8×C97—22no—
56C2×Q8×C97C2312yes—
57C2×D4×C97—22no—
58C2×D4×C97—22no—
59C2×D4×C97—21no—
60C2×D4×C97C2312yes—
61C22×C4×C97—21no—
62C22×C4×C97C2311yescharacteristic
63SD16×C97—28no—
64C2×C8×C97D414yescomplemented
65C2×C8×C97C2312yes—
66C23×C97—22no—
67C23×C97—42no—
68C23×C97—44no—
69C23×C97C2×D411yescharacteristic
70C22×D97—22no—
71C22×D97—42no—
72C22×D97—44no—
73Q8×C97—22no—
74Q8×C97—22no—
75Q8×C97—42no—
76D4×C97—22no—
77D4×C97—22no—
78D4×C97—42no—
79C2×C4×C97—22no—
80C2×C4×C97—42no—
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D412yes—
83C2×C4×C97C2×D412yes—
84C2×C4×C97C2411yescharacteristic
85C22⋊D97—416no—
86C2×Dic97—22no—
87C2×Dic97—42no—
88C2×Dic97SD1618yescomplemented
89C2×Dic97C2×D414yes—
90D388—22no—
91D388—22no—
92D388—42no—
93Dic194—22no—
94Dic194—22no—
95Dic194—42no—
96C8×C97C2×D414yescomplemented
97#11715895C2×D414yescomplemented
98C22×C97—42no—
99C22×C97—44no—
100C22×C97—84no—
101C22×C97#12712yes—
102C22×C97C2×SD1614yes—
103C22×C97C22×D411yescharacteristic
104C2×D97—42no—
105C2×D97—44no—
106C2×D97—84no—
107C4×C97—42no—
108C4×C97#12712yes—
109C4×C97C22×D411yescharacteristic
110C4×C97C22×D411yescharacteristic
111Dic97—42no—
112Dic97C2×SD1618yescomplemented
113C2×C97—82no—
114C2×C97#49214yescomplemented
115C2×C97C2×#12711yescharacteristic
116C2×C97C22×SD1612yes—
117D97—82no—
118C2×#492—971nomaximal, Sylow
119C97C2×#49211yescharacteristic, complemented, Sylow
120C22×SD16—971no—
121C22×SD16—971no—
122C22×SD16—971no—
123C22×SD16D9711yescharacteristic, complemented
124C2×C4⋊Q8—971no—
125C2×#127—971no—
126#492—978no—
127C2×C4×C8—971no—
128C22×Q8—971no—
129C22×Q8C2×D9711yescharacteristic
130C22×D4—971no—
131C22×D4—1942no—
132C22×D4C2×D9711yescharacteristic
133C2×SD16—24no—
134C2×SD16—974no—
135C2×SD16—974no—
136C2×SD16—974no—
137C2×SD16—1944no—
138C2×SD16—1944no—
139C2×SD16—1944no—
140C2×SD16C2×D9714yescomplemented
141C22×C8—971no—
142C22×C8C2×D9711yescharacteristic
143C4⋊Q8—974no—
144C4⋊Q8—1942no—
145#127—974no—
146#127—1942no—
147C2×C4⋊C4—1942no—
148C2×C42—971no—
149C4×C8—974no—
150C24—21no—
151C24—1941no—
152C2×Q8—21no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—972no—
156C2×Q8—1942no—
157C2×Q8—1942no—
158C2×Q8—1941no—
159C2×Q8C22×D9712yes—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—972no—
164C2×D4—3888no—
165C2×D4—1948no—
166C2×D4—1941no—
167C2×D4—1942no—
168C2×D4—1942no—
169C2×D4C22×D9712yes—
170C22×C4—21no—
171C22×C4—972no—
172C22×C4—1941no—
173C22×C4C22×D9711yescharacteristic
174SD16—28no—
175SD16—1948no—
176SD16—1948no—
177SD16—1948no—
178C2×C8—974no—
179C2×C8—972no—
180C2×C8C22⋊D9714yescomplemented
181C2×C8C22×D9712yes—
182C4⋊C4—1948no—
183C42—972no—
184C42—972no—
185C23—22no—
186C23—42no—
187C23—44no—
188C23—3882no—
189C23—3884no—
190C23—1942no—
191C23D4×D9711yescharacteristic
192Q8—22no—
193Q8—22no—
194Q8—42no—
195Q8—3882no—
196Q8—1942no—
197Q8—1942no—
198D4—22no—
199D4—22no—
200D4—42no—
201D4—38816no—
202D4—3882no—
203D4—1942no—
204D4—1942no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—978no—
208C2×C4—974no—
209C2×C4—3882no—
210C2×C4—1942no—
211C2×C4D4×D9711yescharacteristic
212C2×C4D4×D9712yes—
213C2×C4C2×C22⋊D9712yes—
214C2×C4C23×D9711yescharacteristic
215C8—974no—
216C8C2×C22⋊D9714yescomplemented
217C22—44no—
218C22—42no—
219C22—84no—
220C22—7764no—
221C22—3884no—
222C22—3882no—
223C22#1569080612yes—
224C22SD16×D9714yes—
225C22C2×D4×D9711yescharacteristic
226C4—42no—
227C4—978no—
228C4—3882no—
229C4#1569080612yes—
230C4C2×D4×D9711yescharacteristic
231C4C22×C22⋊D9711yescharacteristic
232C2—82no—
233C2—7762no—
234C2#2026110614yesdirect product factor
235C2C2×#1569080611yescharacteristic
236C2C2×SD16×D9712yes—
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×SD16×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690806C2
hypercenterC22×SD16D97
hypocenterC97C2×#492
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#13095500C2semidirect
2. C22×#13095499C2semidirect
3. C22×#13095498C2semidirect
4. C2×#15690808C2semidirect
6. #20261106C2direct (N × H)
7. C2×C8×Dic97C2semidirect
8. C22×SD16×C97C2semidirect
13. C2×#13095500C22semidirect
15. C2×#13095499C22semidirect
17. C2×#13095498C22semidirect
20. #15690808C22semidirect
25. C8×Dic97C22semidirect
29. C2×SD16×C97C22semidirect
47. C2×#11715895D4semidirect
64. C2×C8×C97D4semidirect
88. C2×Dic97SD16semidirect
96. C8×C97C2×D4semidirect
97. #11715895C2×D4semidirect
112. Dic97C2×SD16semidirect
114. C2×C97#492semidirect
119. C97C2×#492semidirect
123. C22×SD16D97semidirect
140. C2×SD16C2×D97semidirect
180. C2×C8C22⋊D97semidirect
216. C8C2×C22⋊D97semidirect
234. C2#20261106direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955002
C22×#130954992
C22×#130954982
C2×#156908082
C2×#156908062
#202611062
C2×C8×Dic972
C22×SD16×C972
C2×#49297