Finite Group Databasebeta

← C2×#20260952 #27151162

Subgroups — C2×#20260952 #27151162

Order 12416.

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

normal subgroups only

This group has 564 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×#13095496C211yesmaximal, characteristic
2C2×#15690799C211yesmaximal, characteristic, complemented
3C2×D4×Dic97C211yesmaximal, characteristic, complemented
4#20260952C218yesmaximal, direct product factor
5C2×#15690739C211yesmaximal, characteristic, complemented
6C2×#15690734C211yesmaximal, characteristic, complemented
7C2×#15690732C211yesmaximal, characteristic, complemented
8C22×SD16×C97C211yesmaximal, characteristic, complemented
9C22×D388C2211yescharacteristic
10C23×Dic97—21no—
11C2×#13095496—24no—
12C2×#13095496C2214yes—
13C22×#11715895C2211yescharacteristic
14#15690799—22no—
15#15690799C2214yescomplemented
16D4×Dic97—22no—
17D4×Dic97C2214yescomplemented
18C2×C4⋊Dic97C2211yescharacteristic
19C2×C22⋊Dic97—21no—
20C2×#13095488—21no—
21C2×#13095488—21no—
22C2×C4×Dic97C2211yescharacteristic
23#15690739C2214yescomplemented
24#15690734C2214yescomplemented
25#15690732C2214yescomplemented
26C22×Q8×C97C2211yescharacteristic
27C22×D4×C97C2211yescharacteristic
28C2×SD16×C97—24no—
29C2×SD16×C97C2214yescomplemented
30C22×C8×C97C2211yescharacteristic
31C23×D97—21no—
32C2×D388—22no—
33C2×D388—22no—
34C2×D388—21no—
35C2×D388C2312yes—
36C22×Dic97—21no—
37C22×Dic97—24no—
38C22×Dic97—44no—
39C22×Dic97D411yescharacteristic
40C22×Dic97D411yescharacteristic
41#13095496—28no—
42C2×#11715895—22no—
43C2×#11715895C2312yes—
44C4⋊Dic97—21no—
45C4⋊Dic97C2312yes—
46C22⋊Dic97—24no—
47#13095488—24no—
48#13095488—24no—
49C4×Dic97D412yes—
50C4×Dic97C2312yes—
51C24×C97—21no—
52C2×Q8×C97—21no—
53C2×Q8×C97—22no—
54C2×Q8×C97—22no—
55C2×Q8×C97C2312yes—
56C2×D4×C97—21no—
57C2×D4×C97—22no—
58C2×D4×C97D414yes—
59C2×D4×C97C2312yes—
60C22×C4×C97—21no—
61C22×C4×C97C2311yescharacteristic
62SD16×C97—28no—
63C2×C8×C97—22no—
64C2×C8×C97C2312yes—
65C23×C97—24no—
66C23×C97—22no—
67C23×C97—42no—
68C23×C97—42no—
69C23×C97C2×D411yescharacteristic
70C22×D97—22no—
71C22×D97—42no—
72C22×D97—42no—
73C22×D97—42no—
74Q8×C97—22no—
75Q8×C97—22no—
76Q8×C97—42no—
77D4×C97—24no—
78D4×C97—22no—
79D4×C97C2×D414yes—
80C2×C4×C97—22no—
81C2×C4×C97—42no—
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D412yes—
84C2×C4×C97#4112yes—
85C2×C4×C97C2411yescharacteristic
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—22no—
89C2×Dic97—48no—
90C2×Dic97—42no—
91C2×Dic97C2×D412yes—
92C2×Dic97C2×D412yes—
93D388—22no—
94D388—22no—
95D388—42no—
96C8×C97—22no—
97#11715895—22no—
98C22×C97—24no—
99C22×C97—42no—
100C22×C97—42no—
101C22×C97—48no—
102C22×C97#12112yes—
103C22×C97#13512yes—
104C22×C97AGL(1,ℤ/8)12yes—
105C22×C97C22×D411yescharacteristic
106C2×D97—42no—
107C2×D97—42no—
108C2×D97—42no—
109C2×D97—84no—
110C4×C97—42no—
111C4×C97#12112yes—
112C4×C97C22×D411yescharacteristic
113C4×C97C2×#4111yescharacteristic
114Dic97—22no—
115Dic97—22no—
116Dic97—42no—
117C2×C97—44no—
118C2×C97#46314yescomplemented
119C2×C97C2×#12111yescharacteristic
120C2×C97C2×#13511yescharacteristic
121C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
122D97—82no—
123C2×#463—971nomaximal, Sylow
124C97C2×#46311yescharacteristic, complemented, Sylow
125C22×SD16D9711yescharacteristic, complemented
126C22×D8—971no—
127C2×#124—971no—
128C2×C4×D4—971no—
129#463—978no—
130C2×C4⋊C8—971no—
131C2×#103—971no—
132C2×#102—971no—
133C22×Q8C2×D9711yescharacteristic
134C22×D4—971no—
135C22×D4C2×D9711yescharacteristic
136C23×C4—1941no—
137C2×SD16—24no—
138C2×SD16C2×D9714yescomplemented
139C2×D8—974no—
140C2×D8—1944no—
141C22×C8—971no—
142C22×C8C2×D9711yescharacteristic
143#124—974no—
144#124—1942no—
145C4×D4—974no—
146C4×D4—1942no—
147C2×C4⋊C4—971no—
148C2×C22⋊C4—1941no—
149C2×C22⋊C4—1941no—
150C2×C22⋊C4—1941no—
151C2×C42—971no—
152C4⋊C8—974no—
153#103—974no—
154#102—974no—
155C24—21no—
156C24—1941no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D9712yes—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—972no—
164C2×D4—1941no—
165C2×D4—1942no—
166C2×D4—1942no—
167C2×D4C22⋊D9714yes—
168C2×D4C22×D9712yes—
169C22×C4—21no—
170C22×C4—971no—
171C22×C4—971no—
172C22×C4—3884no—
173C22×C4—1944no—
174C22×C4—1941no—
175C22×C4C22×D9711yescharacteristic
176SD16—28no—
177D8—1948no—
178C2×C8—22no—
179C2×C8—972no—
180C2×C8—1942no—
181C2×C8C22×D9712yes—
182C4⋊C4—972no—
183C4⋊C4—1941no—
184C22⋊C4—1944no—
185C22⋊C4—1944no—
186C22⋊C4—1944no—
187C42—972no—
188C42—972no—
189C23—24no—
190C23—22no—
191C23—42no—
192C23—42no—
193C23—3882no—
194C23—3882no—
195C23—3882no—
196C23—1942no—
197C23D4×D9711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—24no—
202D4—22no—
203D4—3882no—
204D4—1942no—
205D4—1942no—
206D4C2×C22⋊D9714yes—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—972no—
210C2×C4—972no—
211C2×C4—3882no—
212C2×C4—3888no—
213C2×C4—1942no—
214C2×C4—1942no—
215C2×C4—1942no—
216C2×C4D4×D9711yescharacteristic
217C2×C4C2×C22⋊D9712yes—
218C2×C4#1309551012yes—
219C2×C4C23×D9711yescharacteristic
220C8—22no—
221C8—1942no—
222C22—24no—
223C22—48no—
224C22—42no—
225C22—42no—
226C22—7764no—
227C22—3882no—
228C22—3882no—
229C22—3882no—
230C22#1569078212yes—
231C22#1569083112yes—
232C22#1569083712yes—
233C22C2×D4×D9711yescharacteristic
234C4—42no—
235C4—3882no—
236C4—1942no—
237C4—1942no—
238C4#1569078212yes—
239C4C22×C22⋊D9711yescharacteristic
240C4C2×#1309551011yescharacteristic
241C2—44no—
242C2—7762no—
243C2#2026095214yesdirect product factor
244C2C2×#1569078211yescharacteristic
245C2C2×#1569083111yescharacteristic
246C2C2×#1569083711yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×SD16×C97C2
socleC23×C97C2×D4
prime radicalC22×#13095496C2
hypercenterC22×SD16D97
hypocenterC97C2×#463
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
2. C2×#15690799C2semidirect
3. C2×D4×Dic97C2semidirect
4. #20260952C2direct (N × H)
5. C2×#15690739C2semidirect
6. C2×#15690734C2semidirect
7. C2×#15690732C2semidirect
8. C22×SD16×C97C2semidirect
15. #15690799C22semidirect
17. D4×Dic97C22semidirect
23. #15690739C22semidirect
24. #15690734C22semidirect
25. #15690732C22semidirect
29. C2×SD16×C97C22semidirect
118. C2×C97#463semidirect
124. C97C2×#463semidirect
125. C22×SD16D97semidirect
138. C2×SD16C2×D97semidirect
243. C2#20260952direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130954962
C2×#156907992
C2×D4×Dic972
#202609522
C2×#156907392
C2×#156907342
C2×#156907322
C22×SD16×C972
C2×#46397