Finite Group Databasebeta

← C2×#20261128 #27152175

Subgroups — C2×#20261128 #27152175

Order 12416.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13095501C211yesmaximal, characteristic, complemented
2C22×#13095500C211yesmaximal, characteristic
3C22×#13095498C211yesmaximal, characteristic
4C2×#15690808C211yesmaximal, characteristic, complemented
5C2×#15690799C211yesmaximal, characteristic
6#20261128C218yesmaximal, direct product factor
7C2×#15690720C211yesmaximal, characteristic, complemented
8C22×Q16×C97C211yesmaximal, characteristic, complemented
9C22×Dic194C2211yescharacteristic
10C22×D388C2211yescharacteristic
11C2×#13095501—24no—
12C2×#13095501C2214yescomplemented
13C2×#13095500—24no—
14C2×#13095500C2214yes—
15C2×#13095498—24no—
16C2×#13095498C2214yes—
17C22×#11715895C2211yescharacteristic
18#15690808—22no—
19#15690808C2214yescomplemented
20#15690799—22no—
21#15690799C2214yes—
22C2×#13095490—22no—
23C2×#13095488—22no—
24C2×C4×Dic97C2211yescharacteristic
25#15690720C2214yescomplemented
26C22×Q8×C97C2211yescharacteristic
27C22×Q8×C97C2211yescharacteristic
28C2×Q16×C97—24no—
29C2×Q16×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—
40C22×Dic97—21no—
41C22×Dic97D412yes—
42#13095501—28no—
43#13095500—28no—
44#13095498—28no—
45C2×#11715895D414yes—
46C2×#11715895C2312yes—
47#13095490—28no—
48#13095488—28no—
49C4×Dic97D412yes—
50C4×Dic97C2312yes—
51C2×Q8×C97—21no—
52C2×Q8×C97—21no—
53C2×Q8×C97—22no—
54C2×Q8×C97—22no—
55C2×Q8×C97—22no—
56C2×Q8×C97—22no—
57C2×Q8×C97C2312yes—
58C2×Q8×C97C2312yes—
59C22×C4×C97—21no—
60C22×C4×C97—21no—
61C22×C4×C97C2311yescharacteristic
62Q16×C97—28no—
63C2×C8×C97D414yes—
64C2×C8×C97C2312yes—
65C23×C97C2×D411yescharacteristic
66C22×D97—22no—
67C22×D97—42no—
68C22×D97—44no—
69Q8×C97—22no—
70Q8×C97—22no—
71Q8×C97—22no—
72Q8×C97—22no—
73Q8×C97—42no—
74Q8×C97—42no—
75C2×C4×C97—22no—
76C2×C4×C97—22no—
77C2×C4×C97—42no—
78C2×C4×C97—42no—
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D412yes—
81C2×C4×C97C2×D412yes—
82C2×C4×C97C2411yescharacteristic
83C2×Dic97—24no—
84C2×Dic97—22no—
85C2×Dic97—42no—
86C2×Dic97C2×D414yes—
87D388—22no—
88D388—22no—
89D388—42no—
90Dic194—22no—
91Dic194—22no—
92Dic194—42no—
93C8×C97C2×D414yes—
94#11715895C2×D414yes—
95C22×C97#12712yes—
96C22×C97#13714yes—
97C22×C97C22×D411yescharacteristic
98C2×D97—42no—
99C2×D97—44no—
100C2×D97—84no—
101C4×C97—42no—
102C4×C97—42no—
103C4×C97#12712yes—
104C4×C97C22×D411yescharacteristic
105C4×C97C22×D411yescharacteristic
106Dic97—24no—
107Dic97—42no—
108C2×C97#49714yescomplemented
109C2×C97C2×#12711yescharacteristic
110C2×C97C2×#13712yes—
111D97—82no—
112C2×#497—971nomaximal, Sylow
113C97C2×#49711yescharacteristic, complemented, Sylow
114C22×Q16—971no—
115C22×Q16D9711yescharacteristic, complemented
116C22×SD16—971no—
117C22×SD16—971no—
118C2×C4⋊Q8—971no—
119C2×#124—971no—
120#497—978no—
121C2×#97—971no—
122C22×Q8—971no—
123C22×Q8C2×D9711yescharacteristic
124C22×Q8C2×D9711yescharacteristic
125C22×D4—971no—
126C2×Q16—24no—
127C2×Q16—974no—
128C2×Q16—1944no—
129C2×Q16C2×D9714yescomplemented
130C2×SD16—974no—
131C2×SD16—974no—
132C2×SD16—1944no—
133C2×SD16—1944no—
134C22×C8—971no—
135C22×C8C2×D9711yescharacteristic
136C4⋊Q8—974no—
137C4⋊Q8—1942no—
138#124—974no—
139#124—1942no—
140C2×C4⋊C4—1942no—
141C2×C22⋊C4—1942no—
142C2×C42—971no—
143#97—974no—
144C24—1941no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—972no—
152C2×Q8—1942no—
153C2×Q8—1942no—
154C2×Q8—1941no—
155C2×Q8C22×D9712yes—
156C2×Q8C22×D9712yes—
157C2×D4—972no—
158C2×D4—1941no—
159C2×D4—1942no—
160C2×D4—1942no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—972no—
164C22×C4—1941no—
165C22×C4C22×D9711yescharacteristic
166Q16—28no—
167Q16—1948no—
168SD16—1948no—
169SD16—1948no—
170C2×C8—974no—
171C2×C8—972no—
172C2×C8C22⋊D9714yes—
173C2×C8C22×D9712yes—
174C4⋊C4—1948no—
175C22⋊C4—1948no—
176C42—972no—
177C42—972no—
178C23—3882no—
179C23—3884no—
180C23—1942no—
181C23D4×D9711yescharacteristic
182Q8—22no—
183Q8—22no—
184Q8—22no—
185Q8—22no—
186Q8—42no—
187Q8—42no—
188Q8—3882no—
189Q8—1942no—
190Q8—1942no—
191D4—3882no—
192D4—1942no—
193D4—1942no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—974no—
199C2×C4—3882no—
200C2×C4—1942no—
201C2×C4—1944no—
202C2×C4D4×D9711yescharacteristic
203C2×C4D4×D9712yes—
204C2×C4C2×C22⋊D9712yes—
205C2×C4C23×D9711yescharacteristic
206C8—974no—
207C8C2×C22⋊D9714yes—
208C22—7764no—
209C22—3884no—
210C22—3882no—
211C22#1569080612yes—
212C22#1569084214yes—
213C22C2×D4×D9711yescharacteristic
214C4—42no—
215C4—42no—
216C4—3882no—
217C4—1944no—
218C4#1569080612yes—
219C4C2×D4×D9711yescharacteristic
220C4C22×C22⋊D9711yescharacteristic
221C2—7762no—
222C2#2026112814yesdirect product factor
223C2C2×#1569080611yescharacteristic
224C2C2×#1569084212yes—
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×Q16×C97C2
socleC23×C97C2×D4
prime radicalC22×D388C22
hypercenterC22×Q16D97
hypocenterC97C2×#497
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C22×#13095501C2semidirect
4. C2×#15690808C2semidirect
6. #20261128C2direct (N × H)
7. C2×#15690720C2semidirect
8. C22×Q16×C97C2semidirect
12. C2×#13095501C22semidirect
19. #15690808C22semidirect
25. #15690720C22semidirect
29. C2×Q16×C97C22semidirect
108. C2×C97#497semidirect
113. C97C2×#497semidirect
115. C22×Q16D97semidirect
129. C2×Q16C2×D97semidirect
222. C2#20261128direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955012
C22×#130955002
C22×#130954982
C2×#156908082
C2×#156907992
#202611282
C2×#156907202
C22×Q16×C972
C2×#49797