Finite Group Databasebeta

← C2×#20261056 #27151617

Subgroups — C2×#20261056 #27151617

Order 12416.

468 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#15690769C211yesmaximal, characteristic, complemented
2#20261056C218yesmaximal, direct product factor
3C2×#15690743C211yesmaximal, characteristic, complemented
4C2×#15690741C211yesmaximal, characteristic, complemented
5C2×#15690731C211yesmaximal, characteristic, complemented
6C2×#15690731C211yesmaximal, characteristic, complemented
7C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
8C2×#121×C97C211yesmaximal, characteristic
9C22×#11715895C2211yescharacteristic
10C22×#11715895C2211yescharacteristic
11#15690769—22no—
12#15690769C2214yescomplemented
13C2×C4⋊Dic97C2211yescharacteristic
14C2×C4⋊Dic97C2211yescharacteristic
15C2×C22⋊Dic97—21no—
16C2×C4×Dic97—21no—
17#15690743C2214yescomplemented
18#15690741C2214yescomplemented
19#15690731C2214yescomplemented
20#15690731C2214yescomplemented
21C22⋊#11715895C2214yescomplemented
22C22×D4×C97—21no—
23C22×D4×C97C2211yescharacteristic
24C23×C4×C97C2211yescharacteristic
25#121×C97—22no—
26#121×C97C2214yes—
27C2×C4⋊C4×C97C2211yescharacteristic
28C2×C22⋊C4×C97—21no—
29C22×Dic97—21no—
30C22×Dic97—21no—
31C2×#11715895—22no—
32C2×#11715895—22no—
33C2×#11715895C2312yes—
34C2×#11715895C2312yes—
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37C4⋊Dic97C2312yes—
38C4⋊Dic97C2312yes—
39C22⋊Dic97—24no—
40C4×Dic97—24no—
41C24×C97—21no—
42C24×C97D411yescharacteristic
43C2×D4×C97—24no—
44C2×D4×C97—21no—
45C2×D4×C97—22no—
46C2×D4×C97—22no—
47C2×D4×C97—44no—
48C2×D4×C97C2312yes—
49C22×C4×C97—22no—
50C22×C4×C97—22no—
51C22×C4×C97—21no—
52C22×C4×C97D411yescharacteristic
53C22×C4×C97D412yes—
54C22×C4×C97C2312yes—
55C22×C4×C97C2311yescharacteristic
56C4⋊C4×C97—21no—
57C4⋊C4×C97C2312yes—
58C22⋊C4×C97—24no—
59C23×C97—22no—
60C23×C97—22no—
61C23×C97—22no—
62C23×C97—22no—
63C23×C97—42no—
64C23×C97—42no—
65C23×C97—42no—
66C23×C97C2×D412yes—
67C23×C97C2×D411yescharacteristic
68D4×C97—22no—
69D4×C97—22no—
70D4×C97—48no—
71D4×C97—42no—
72C2×C4×C97—22no—
73C2×C4×C97—24no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—22no—
77C2×C4×C97—42no—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97#4112yes—
81C2×C4×C97#4112yes—
82C2×C4×C97C2411yescharacteristic
83C2×Dic97—22no—
84C2×Dic97—22no—
85C2×Dic97—42no—
86C2×Dic97—42no—
87#11715895—22no—
88#11715895—22no—
89C22×C97—22no—
90C22×C97—22no—
91C22×C97—22no—
92C22×C97—44no—
93C22×C97—42no—
94C22×C97—42no—
95C22×C97—42no—
96C22×C97—84no—
97C22×C97#12312yes—
98C22×C97#13512yes—
99C22×C97AGL(1,ℤ/8)12yes—
100C22×C97C22×D411yescharacteristic
101C4×C97—22no—
102C4×C97—42no—
103C4×C97#12312yes—
104C4×C97C2×#4111yescharacteristic
105C4×C97C2×#4111yescharacteristic
106Dic97—42no—
107Dic97—42no—
108C2×C97—42no—
109C2×C97—82no—
110C2×C97#48214yescomplemented
111C2×C97C2×#12311yescharacteristic
112C2×C97C2×#13511yescharacteristic
113C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
114C2×#482—971nomaximal, Sylow
115C97C2×#48211yescharacteristic, complemented, Sylow
116C2×#121D9711yescharacteristic
117C2×#117—971no—
118#482—978no—
119C2×#107—971no—
120C2×#106—971no—
121C2×#102—971no—
122C2×#102—971no—
123C2×C22⋊C8—971no—
124C22×D4—21no—
125C22×D4C2×D9711yescharacteristic
126C23×C4C2×D9711yescharacteristic
127C22×C8—971no—
128C22×C8—971no—
129#121—22no—
130#121C2×D9714yes—
131#117—974no—
132#117—1942no—
133C2×C4⋊C4—971no—
134C2×C4⋊C4—971no—
135C2×C4⋊C4C2×D9711yescharacteristic
136C2×C22⋊C4—21no—
137C2×C22⋊C4—1941no—
138C2×C42—1941no—
139#107—974no—
140#106—974no—
141#102—974no—
142#102—974no—
143C22⋊C8—974no—
144C24—21no—
145C24C22⋊D9711yescharacteristic
146C2×D4—24no—
147C2×D4—21no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—44no—
151C2×D4C22×D9712yes—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—1941no—
156C22×C4—1941no—
157C22×C4C22⋊D9712yes—
158C22×C4C22⋊D9711yescharacteristic
159C22×C4C22×D9712yes—
160C22×C4C22×D9711yescharacteristic
161C2×C8—972no—
162C2×C8—972no—
163C2×C8—1942no—
164C2×C8—1942no—
165C4⋊C4—21no—
166C4⋊C4—972no—
167C4⋊C4—972no—
168C4⋊C4—1941no—
169C4⋊C4—1941no—
170C4⋊C4C22×D9712yes—
171C22⋊C4—24no—
172C22⋊C4—1944no—
173C42—1944no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23C2×C22⋊D9712yes—
182C23C2×C22⋊D9711yescharacteristic
183D4—22no—
184D4—22no—
185D4—48no—
186D4—42no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—3882no—
194C2×C4—3882no—
195C2×C4—1942no—
196C2×C4—1942no—
197C2×C4C2×C22⋊D9712yes—
198C2×C4C2×C22⋊D9711yescharacteristic
199C2×C4#1309551012yes—
200C2×C4#1309551012yes—
201C2×C4C23×D9711yescharacteristic
202C8—1942no—
203C8—1942no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—44no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—84no—
212C22#1569079112yes—
213C22#1569083012yes—
214C22#1569083612yes—
215C22C22×C22⋊D9711yescharacteristic
216C4—22no—
217C4—42no—
218C4—3882no—
219C4—3882no—
220C4#1569079112yes—
221C4C2×#1309551011yescharacteristic
222C4C2×#1309551011yescharacteristic
223C2—42no—
224C2—82no—
225C2#2026105614yesdirect product factor
226C2C2×#1569079111yescharacteristic
227C2C2×#1569083011yescharacteristic
228C2C2×#1569083611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting, prime radicalC2×#121×C97C2
socleC23×C97C2×D4
hypercenterC2×#121D97
hypocenterC97C2×#482
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#15690769C2semidirect
2. #20261056C2direct (N × H)
3. C2×#15690743C2semidirect
4. C2×#15690741C2semidirect
5. C2×#15690731C2semidirect
6. C2×#15690731C2semidirect
7. C2×C22⋊#11715895C2semidirect
12. #15690769C22semidirect
17. #15690743C22semidirect
18. #15690741C22semidirect
19. #15690731C22semidirect
20. #15690731C22semidirect
21. C22⋊#11715895C22semidirect
110. C2×C97#482semidirect
115. C97C2×#482semidirect
225. C2#20261056direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156907692
#202610562
C2×#156907432
C2×#156907412
C2×#156907312
C2×C22⋊#117158952
C2×#121×C972
C2×#48297