Finite Group Databasebeta

← C2×#20260964 #27151296

Subgroups — C2×#20260964 #27151296

Order 12416.

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

normal subgroups only

This group has 564 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
2C2×#15690789C211yesmaximal, characteristic, complemented
3C2×#15690784C211yesmaximal, characteristic
4#20260964C218yesmaximal, direct product factor
5C2×#15690742C211yesmaximal, characteristic, complemented
6C2×#15690735C211yesmaximal, characteristic, complemented
7C2×#15690732C211yesmaximal, characteristic, complemented
8C23×C8×C97C211yesmaximal, characteristic, complemented
9C22×Dic194C2211yescharacteristic
10C22×D388C2211yescharacteristic
11C22×C22⋊D97—21no—
12C2×#13095500—24no—
13C2×#13095500C2214yescomplemented
14#15690789—22no—
15#15690789C2214yescomplemented
16#15690784—22no—
17#15690784C2214yes—
18C2×C4⋊Dic97C2211yescharacteristic
19C2×C4⋊Dic97C2211yescharacteristic
20C2×#13095490—21no—
21C2×C22⋊Dic97—21no—
22C2×#13095488—21no—
23#15690742C2214yescomplemented
24#15690735C2214yescomplemented
25#15690732C2214yescomplemented
26C23×C4×C97C2211yescharacteristic
27C22×C8×C97—24no—
28C22×C8×C97C2211yescharacteristic
29C22×C8×C97C2211yescharacteristic
30C22×C8×C97C2214yescomplemented
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—24no—
41C2×C22⋊D97—44no—
42C22×Dic97—21no—
43C22×Dic97—21no—
44C22×Dic97—21no—
45#13095500—28no—
46C4⋊Dic97—21no—
47C4⋊Dic97—21no—
48C4⋊Dic97C2312yes—
49C4⋊Dic97C2312yes—
50#13095490—24no—
51C22⋊Dic97—24no—
52#13095488—24no—
53C24×C97D411yescharacteristic
54C22×C4×C97—22no—
55C22×C4×C97—22no—
56C22×C4×C97D411yescharacteristic
57C22×C4×C97D412yes—
58C22×C4×C97C2312yes—
59C22×C4×C97C2311yescharacteristic
60C2×C8×C97—22no—
61C2×C8×C97—28no—
62C2×C8×C97D414yescomplemented
63C2×C8×C97C2312yes—
64C2×C8×C97C2312yes—
65C23×C97—22no—
66C23×C97—22no—
67C23×C97SD1614yescomplemented
68C23×C97C2×D412yes—
69C23×C97C2×D411yescharacteristic
70C22×D97—22no—
71C22×D97—42no—
72C22×D97—42no—
73C22×D97—42no—
74C2×C4×C97—22no—
75C2×C4×C97—24no—
76C2×C4×C97—22no—
77C2×C4×C97—22no—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×D412yes—
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97#4112yes—
82C2×C4×C97C2411yescharacteristic
83C22⋊D97—48no—
84C2×Dic97—22no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—42no—
88C2×Dic97—42no—
89C2×Dic97—42no—
90D388—22no—
91D388—22no—
92D388—42no—
93Dic194—22no—
94Dic194—22no—
95Dic194—42no—
96C8×C97—22no—
97C8×C97C2×D414yescomplemented
98C22×C97—22no—
99C22×C97—22no—
100C22×C97—28no—
101C22×C97#12112yes—
102C22×C97C2×SD1612yes—
103C22×C97C2×SD1614yescomplemented
104C22×C97#13512yes—
105C22×C97C22×D411yescharacteristic
106C2×D97—42no—
107C2×D97—42no—
108C2×D97—42no—
109C2×D97—84no—
110C4×C97—22no—
111C4×C97#12112yes—
112C4×C97C22×D411yescharacteristic
113C4×C97C2×#4111yescharacteristic
114Dic97—42no—
115Dic97—42no—
116Dic97—42no—
117C2×C97—24no—
118C2×C97#46514yescomplemented
119C2×C97C2×#12111yescharacteristic
120C2×C97C22×SD1611yescharacteristic
121C2×C97C2×#13511yescharacteristic
122D97—82no—
123C2×#465—971nomaximal, Sylow
124C97C2×#46511yescharacteristic, complemented, Sylow
125C22×SD16—971no—
126C23×C8D9711yescharacteristic, complemented
127C2×C22⋊Q8—971no—
128C2×#121—971no—
129#465—978no—
130C2×#106—971no—
131C2×#103—971no—
132C2×#102—971no—
133C22×Q8—971no—
134C22×D4—971no—
135C22×D4—1941no—
136C23×C4C2×D9711yescharacteristic
137C2×SD16—974no—
138C2×SD16—1944no—
139C22×C8—24no—
140C22×C8C2×D9711yescharacteristic
141C22×C8C2×D9711yescharacteristic
142C22×C8C2×D9714yescomplemented
143C22⋊Q8—974no—
144C22⋊Q8—1942no—
145#121—974no—
146#121—1942no—
147C2×C4⋊C4—971no—
148C2×C4⋊C4—971no—
149C2×C4⋊C4—1941no—
150C2×C22⋊C4—1941no—
151C2×C22⋊C4—1941no—
152#106—974no—
153#103—974no—
154#102—974no—
155C24—1941no—
156C24D38811yescharacteristic
157C2×Q8—972no—
158C2×Q8—1942no—
159C2×Q8—1942no—
160C2×Q8—1941no—
161C2×D4—972no—
162C2×D4—3884no—
163C2×D4—1944no—
164C2×D4—1942no—
165C2×D4—1941no—
166C2×D4—1942no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—1941no—
170C22×C4—1941no—
171C22×C4—1941no—
172C22×C4D38812yes—
173C22×C4D38811yescharacteristic
174C22×C4C22×D9712yes—
175C22×C4C22×D9711yescharacteristic
176SD16—1948no—
177C2×C8—22no—
178C2×C8—28no—
179C2×C8C22⋊D9714yescomplemented
180C2×C8C22×D9712yes—
181C2×C8C22×D9712yes—
182C4⋊C4—972no—
183C4⋊C4—972no—
184C4⋊C4—1944no—
185C4⋊C4—1941no—
186C4⋊C4—1941no—
187C22⋊C4—1944no—
188C22⋊C4—1944no—
189C23—22no—
190C23—22no—
191C23—3882no—
192C23—3882no—
193C23—3882no—
194C23—1942no—
195C23#1309550014yescomplemented
196C23C2×D38812yes—
197C23C2×D38811yescharacteristic
198Q8—3882no—
199Q8—1942no—
200Q8—1942no—
201D4—3888no—
202D4—3882no—
203D4—1942no—
204D4—1942no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—3882no—
210C2×C4—3882no—
211C2×C4—3882no—
212C2×C4—1942no—
213C2×C4—1942no—
214C2×C4—1942no—
215C2×C4C2×C22⋊D9712yes—
216C2×C4C2×D38812yes—
217C2×C4C2×D38811yescharacteristic
218C2×C4#1309551112yes—
219C2×C4C23×D9711yescharacteristic
220C8—22no—
221C8C2×C22⋊D9714yescomplemented
222C22—28no—
223C22—22no—
224C22—22no—
225C22—7764no—
226C22—3882no—
227C22—3882no—
228C22—3882no—
229C22#1569078412yes—
230C22C2×#1309550012yes—
231C22C2×#1309550014yescomplemented
232C22#1569083212yes—
233C22C22×D38811yescharacteristic
234C4—22no—
235C4—3882no—
236C4—3882no—
237C4—3882no—
238C4#1569078412yes—
239C4C22×C22⋊D9711yescharacteristic
240C4C2×#1309551111yescharacteristic
241C2—24no—
242C2—7762no—
243C2#2026096414yesdirect product factor
244C2C2×#1569078411yescharacteristic
245C2C22×#1309550011yescharacteristic
246C2C2×#1569083211yescharacteristic
subgroupG/H
centerC23C2×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC23×C8×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690784C2
hypercenterC23×C8D97
hypocenterC97C2×#465
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. C22×#13095500C2semidirect
2. C2×#15690789C2semidirect
4. #20260964C2direct (N × H)
5. C2×#15690742C2semidirect
6. C2×#15690735C2semidirect
7. C2×#15690732C2semidirect
8. C23×C8×C97C2semidirect
13. C2×#13095500C22semidirect
15. #15690789C22semidirect
23. #15690742C22semidirect
24. #15690735C22semidirect
25. #15690732C22semidirect
30. C22×C8×C97C22semidirect
62. C2×C8×C97D4semidirect
67. C23×C97SD16semidirect
97. C8×C97C2×D4semidirect
103. C22×C97C2×SD16semidirect
118. C2×C97#465semidirect
124. C97C2×#465semidirect
126. C23×C8D97semidirect
142. C22×C8C2×D97semidirect
179. C2×C8C22⋊D97semidirect
195. C23#13095500semidirect
221. C8C2×C22⋊D97semidirect
231. C22C2×#13095500semidirect
243. C2#20260964direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955002
C2×#156907892
C2×#156907842
#202609642
C2×#156907422
C2×#156907352
C2×#156907322
C23×C8×C972
C2×#46597