Finite Group Databasebeta

← C2×#20260982 #27151325

Subgroups — C2×#20260982 #27151325

Order 12416.

544 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13095494C211yesmaximal, characteristic, complemented
3C2×#15690790C211yesmaximal, characteristic, complemented
4C2×#15690785C211yesmaximal, characteristic
5#20260982C218yesmaximal, direct product factor
6C2×#15690733C211yesmaximal, characteristic, complemented
7C2×#15690730C211yesmaximal, characteristic, complemented
8C2×#107×C97C211yesmaximal, characteristic, complemented
9C22×Dic194C2211yescharacteristic
10C22×D388—21no—
11C22×D388C2211yescharacteristic
12C22×C4×D97C2211yescharacteristic
13C2×#13095500—24no—
14C2×#13095500C2214yescomplemented
15C2×#13095494—24no—
16C2×#13095494C2214yescomplemented
17C22×#11715895C2211yescharacteristic
18#15690790—22no—
19#15690790C2214yescomplemented
20#15690785—22no—
21#15690785C2214yes—
22C2×C4⋊Dic97—21no—
23C2×#13095488—21no—
24C2×#13095488—21no—
25#15690733C2214yescomplemented
26#15690730C2214yescomplemented
27C22×C8×C97C2211yescharacteristic
28C2×C4⋊C4×C97C2211yescharacteristic
29C2×C4⋊C4×C97C2211yescharacteristic
30#107×C97C2214yescomplemented
31C23×D97—21no—
32C23×D97D411yescharacteristic
33C2×Dic194—22no—
34C2×Dic194—22no—
35C2×Dic194—21no—
36C2×Dic194C2312yes—
37C2×D388—22no—
38C2×D388—22no—
39C2×D388—24no—
40C2×D388—21no—
41C2×D388—44no—
42C2×D388C2312yes—
43C22×Dic97—21no—
44C22×Dic97D411yescharacteristic
45C2×C4×D97—22no—
46C2×C4×D97—22no—
47C2×C4×D97D412yes—
48C2×C4×D97C2312yes—
49#13095500—28no—
50#13095494—28no—
51C2×#11715895—22no—
52C2×#11715895C2312yes—
53C4⋊Dic97—24no—
54#13095488—24no—
55#13095488—24no—
56C22×C4×C97—21no—
57C22×C4×C97—21no—
58C22×C4×C97C2311yescharacteristic
59C2×C8×C97D414yescomplemented
60C2×C8×C97C2312yes—
61C4⋊C4×C97—21no—
62C4⋊C4×C97—21no—
63C4⋊C4×C97C2312yes—
64C4⋊C4×C97C2312yes—
65C23×C97C2×D411yescharacteristic
66C22×D97—22no—
67C22×D97—22no—
68C22×D97—22no—
69C22×D97—22no—
70C22×D97—42no—
71C22×D97—42no—
72C22×D97—42no—
73C22×D97C2×D412yes—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—42no—
77C2×C4×C97—42no—
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D412yes—
80C2×C4×C97#4112yes—
81C2×C4×C97C2411yescharacteristic
82C2×Dic97—22no—
83C2×Dic97—22no—
84C2×Dic97—42no—
85C2×Dic97C2×D412yes—
86D388—22no—
87D388—22no—
88D388—48no—
89D388—42no—
90C4×D97—24no—
91C4×D97—22no—
92C4×D97—22no—
93Dic194—22no—
94Dic194—22no—
95Dic194—42no—
96C8×C97C2×D414yescomplemented
97#11715895—22no—
98C22×C97#12112yes—
99C22×C97AGL(1,ℤ/8)12yes—
100C22×C97#13712yes—
101C22×C97C22×D411yescharacteristic
102C2×D97—22no—
103C2×D97—22no—
104C2×D97—22no—
105C2×D97—44no—
106C2×D97—42no—
107C2×D97—42no—
108C2×D97—42no—
109C2×D97—84no—
110C4×C97—42no—
111C4×C97—42no—
112C4×C97#12112yes—
113C4×C97C22×D411yescharacteristic
114C4×C97C2×#4111yescharacteristic
115Dic97—22no—
116Dic97—42no—
117C2×C97#46814yescomplemented
118C2×C97C2×#12111yescharacteristic
119C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C97C2×#13711yescharacteristic
121D97—42no—
122D97—82no—
123C2×#468—971nomaximal, Sylow
124C97C2×#46811yescharacteristic, complemented, Sylow
125C22×SD16—971no—
126C22×M16—971no—
127C2×C22⋊Q8—971no—
128C2×#121—971no—
129#468—978no—
130C2×#107D9711yescharacteristic, complemented
131C2×#103—971no—
132C2×#102—971no—
133C22×Q8—971no—
134C22×D4—971no—
135C22×D4—1941no—
136C23×C4—971no—
137C2×SD16—974no—
138C2×SD16—1944no—
139C2×M16—974no—
140C2×M16—1944no—
141C22×C8—971no—
142C22×C8C2×D9711yescharacteristic
143C22⋊Q8—974no—
144C22⋊Q8—1942no—
145#121—974no—
146#121—1942no—
147C2×C4⋊C4—1941no—
148C2×C4⋊C4C2×D9711yescharacteristic
149C2×C4⋊C4C2×D9711yescharacteristic
150C2×C22⋊C4—1941no—
151C2×C22⋊C4—1941no—
152#107C2×D9714yescomplemented
153#103—974no—
154#102—974no—
155C24—971no—
156C24—1941no—
157C2×Q8—972no—
158C2×Q8—1942no—
159C2×Q8—1942no—
160C2×Q8—1941no—
161C2×D4—972no—
162C2×D4—3884no—
163C2×D4—1944no—
164C2×D4—1941no—
165C2×D4—1942no—
166C2×D4—1942no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—971no—
170C22×C4—972no—
171C22×C4—972no—
172C22×C4—1941no—
173C22×C4—1942no—
174C22×C4—1942no—
175C22×C4C22×D9711yescharacteristic
176SD16—1948no—
177M16—1948no—
178C2×C8—972no—
179C2×C8—1942no—
180C2×C8D38814yescomplemented
181C2×C8C22×D9712yes—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—1944no—
185C4⋊C4C22×D9712yes—
186C4⋊C4C22×D9712yes—
187C22⋊C4—1944no—
188C22⋊C4—1944no—
189C23—972no—
190C23—3882no—
191C23—3882no—
192C23—3882no—
193C23—1942no—
194C23—1942no—
195C23—1942no—
196C23—1942no—
197C23D4×D9711yescharacteristic
198Q8—3882no—
199Q8—1942no—
200Q8—1942no—
201D4—3888no—
202D4—3882no—
203D4—1942no—
204D4—1942no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—972no—
210C2×C4—3882no—
211C2×C4—1942no—
212C2×C4—1942no—
213C2×C4—1944no—
214C2×C4—1942no—
215C2×C4—1942no—
216C2×C4D4×D9711yescharacteristic
217C2×C4C2×D38812yes—
218C2×C4#1309551212yes—
219C2×C4C23×D9711yescharacteristic
220C8—1942no—
221C8C2×D38814yescomplemented
222C22—7764no—
223C22—3882no—
224C22—3882no—
225C22—3882no—
226C22—3884no—
227C22—1942no—
228C22—1942no—
229C22—1942no—
230C22#1569078512yes—
231C22#1569083412yes—
232C22#1569084212yes—
233C22C2×D4×D9711yescharacteristic
234C4—42no—
235C4—42no—
236C4—3882no—
237C4—1942no—
238C4#1569078512yes—
239C4C22×D38811yescharacteristic
240C4C2×#1309551211yescharacteristic
241C2—7762no—
242C2—3882no—
243C2#2026098214yesdirect product factor
244C2C2×#1569078511yescharacteristic
245C2C2×#1569083411yescharacteristic
246C2C2×#1569084211yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#107×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690785C2
hypercenterC2×#107D97
hypocenterC97C2×#468
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
1. C22×#13095500C2semidirect
2. C22×#13095494C2semidirect
3. C2×#15690790C2semidirect
5. #20260982C2direct (N × H)
6. C2×#15690733C2semidirect
7. C2×#15690730C2semidirect
8. C2×#107×C97C2semidirect
14. C2×#13095500C22semidirect
16. C2×#13095494C22semidirect
19. #15690790C22semidirect
25. #15690733C22semidirect
26. #15690730C22semidirect
30. #107×C97C22semidirect
59. C2×C8×C97D4semidirect
96. C8×C97C2×D4semidirect
117. C2×C97#468semidirect
124. C97C2×#468semidirect
130. C2×#107D97semidirect
152. #107C2×D97semidirect
180. C2×C8D388semidirect
221. C8C2×D388semidirect
243. C2#20260982direct (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×#130954942
C2×#156907902
C2×#156907852
#202609822
C2×#156907332
C2×#156907302
C2×#107×C972
C2×#46897