Finite Group Databasebeta

← C2×#20260912 #27151065

Subgroups — C2×#20260912 #27151065

Order 12416.

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

normal subgroups only

This group has 760 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×#15690855C211yesmaximal, characteristic
2C2×#15690841C212yesmaximal, complemented
3#20260912C218yesmaximal, direct product factor
4C2×#15690738C212yesmaximal, complemented
5C2×#15690727C211yesmaximal, characteristic, complemented
6C2×#124×C97C211yesmaximal, characteristic, complemented
7#15690855—24no—
8#15690855—22no—
9#15690855C2214yes—
10C2×#13095511C2212yes—
11C2×#13095510—22no—
12C22×C22⋊D97—22no—
13C2×D4×D97—22no—
14#15690841—24no—
15#15690841C2218yescomplemented
16C2×#13095501—22no—
17C2×#13095498—22no—
18C2×#13095495C2212yescomplemented
19#15690738C2218yescomplemented
20#15690727C2214yescomplemented
21C22×Q8×C97C2211yescharacteristic, complemented
22C22×D4×C97C2211yescharacteristic
23#124×C97—22no—
24#124×C97C2214yescomplemented
25C2×C22⋊C4×C97—22no—
26C2×C42×C97C2211yescharacteristic, complemented
27C23×D97—22no—
28#13095511—22no—
29#13095511—22no—
30#13095511D414yes—
31#13095511C2314yes—
32#13095510—28no—
33#13095510—28no—
34C2×Dic194D412yes—
35C2×D388D412yes—
36C2×C22⋊D97—24no—
37C2×C22⋊D97—28no—
38C2×C22⋊D97—22no—
39C2×C22⋊D97—28no—
40D4×D97—28no—
41D4×D97—28no—
42C22×Dic97—22no—
43C2×C4×D97—22no—
44#13095501—28no—
45#13095498—28no—
46#13095495—22no—
47#13095495C2314yescomplemented
48C2×#11715895—22no—
49C24×C97D412yes—
50C2×Q8×C97—21no—
51C2×Q8×C97—22no—
52C2×Q8×C97—22no—
53C2×Q8×C97C2312yescomplemented
54C2×D4×C97—22no—
55C2×D4×C97—22no—
56C2×D4×C97D412yes—
57C2×D4×C97C2312yes—
58C22×C4×C97—21no—
59C22×C4×C97—21no—
60C22×C4×C97C2311yescharacteristic
61C22⋊C4×C97—28no—
62C42×C97—21no—
63C42×C97C2312yescomplemented
64C23×C97—22no—
65C23×C97—22no—
66C23×C97—44no—
67C23×C97C2×D411yescharacteristic
68C23×C97C2×D414yes—
69C22×D97—22no—
70C22×D97—28no—
71C22×D97—84no—
72Q8×C97—24no—
73Q8×C97—42no—
74D4×C97—24no—
75D4×C97—24no—
76C2×C4×C97—22no—
77C2×C4×C97—22no—
78C2×C4×C97—42no—
79C2×C4×C97—41no—
80C2×C4×C97—41no—
81C2×C4×C97C2×D412yes—
82C2×C4×C97C2×D412yes—
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97C2411yescharacteristic
85C22⋊D97—216no—
86C22⋊D97—24no—
87C22⋊D97—24no—
88C2×Dic97—28no—
89C2×Dic97—22no—
90D388—22no—
91D388C2×D414yes—
92C4×D97—24no—
93C4×D97—24no—
94Dic194—22no—
95Dic194C2×D414yes—
96#11715895—24no—
97C22×C97—22no—
98C22×C97—24no—
99C22×C97—44no—
100C22×C97—44no—
101C22×C97—82no—
102C22×C97#12011yescharacteristic
103C22×C97#12011yescharacteristic
104C22×C97C22×D411yescharacteristic
105C2×D97—24no—
106C2×D97—44no—
107C2×D97—88no—
108C4×C97—42no—
109C4×C97—42no—
110C4×C97#12012yes—
111C4×C97C22×D412yes—
112Dic97—24no—
113C2×C97—21no—
114C2×C97—21no—
115C2×C97—44no—
116C2×C97#45512yescomplemented
117C2×C97C2×#12011yescharacteristic
118D97—44no—
119C2×#455—971nomaximal, Sylow
120C97C2×#45511yescharacteristic, complemented, Sylow
121C2×H(5,𝔽2)—971no—
122C2×#137—972no—
123C2×#124D9711yescharacteristic, complemented
124#455—978no—
125C2×#104—972no—
126C2×#100—971no—
127H(5,𝔽2)—974no—
128H(5,𝔽2)—1944no—
129H(5,𝔽2)—1942no—
130C2×#41—972no—
131C2×#41—1942no—
132C22×Q8C2×D9711yescharacteristic, complemented
133C22×D4—1942no—
134C22×D4—1942no—
135C22×D4C2×D9711yescharacteristic
136#137—978no—
137#137—1944no—
138C2×Q16—1942no—
139C2×SD16—1942no—
140C2×M16—972no—
141#124—22no—
142#124C2×D9714yescomplemented
143C2×C22⋊C4—22no—
144C2×C42C2×D9711yescharacteristic, complemented
145#104—978no—
146#100—974no—
147C24—1942no—
148C24C22⋊D9712yes—
149#41—974no—
150#41—974no—
151#41—1948no—
152#41—1948no—
153#41—1942no—
154#41—1942no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—972no—
159C2×Q8C22×D9712yescomplemented
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—972no—
163C2×D4—1948no—
164C2×D4—1948no—
165C2×D4—1944no—
166C2×D4—1948no—
167C2×D4—1942no—
168C2×D4—1948no—
169C2×D4C22⋊D9712yes—
170C2×D4C22×D9712yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—1942no—
174C22×C4—1942no—
175C22×C4C22×D9711yescharacteristic
176Q16—1948no—
177SD16—1948no—
178M16—974no—
179M16—1942no—
180C2×C8—1942no—
181C22⋊C4—28no—
182C42—21no—
183C42C22×D9712yescomplemented
184C23—22no—
185C23—22no—
186C23—44no—
187C23—7764no—
188C23—1948no—
189C23—1942no—
190C23C2×C22⋊D9711yescharacteristic
191C23C2×C22⋊D9714yes—
192Q8—24no—
193Q8—42no—
194Q8—974no—
195Q8—1942no—
196D4—24no—
197D4—24no—
198D4—974no—
199D4—19416no—
200D4—1944no—
201D4—1944no—
202D4—1942no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—1948no—
209C2×C4—1942no—
210C2×C4—1944no—
211C2×C4—1944no—
212C2×C4D4×D9712yes—
213C2×C4D4×D9712yes—
214C2×C4C2×C22⋊D9711yescharacteristic
215C2×C4C23×D9711yescharacteristic
216C8—1944no—
217C22—22no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—82no—
222C22—7768no—
223C22—3884no—
224C22—1944no—
225C22#1569077811yescharacteristic
226C22#1569077811yescharacteristic
227C22C22×C22⋊D9711yescharacteristic
228C4—42no—
229C4—42no—
230C4—1944no—
231C4#1569077812yes—
232C4C2×D4×D9712yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—3884no—
237C2#2026091212yesdirect product factor
238C2C2×#1569077811yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#124×C97C2
socleC22×C97#120
prime radicalC2×#15690855C2
hypercenterC2×#124D97
hypocenterC97C2×#455
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#15690841C2semidirect
3. #20260912C2direct (N × H)
4. C2×#15690738C2semidirect
5. C2×#15690727C2semidirect
6. C2×#124×C97C2semidirect
15. #15690841C22semidirect
18. C2×#13095495C22semidirect
19. #15690738C22semidirect
20. #15690727C22semidirect
21. C22×Q8×C97C22semidirect
24. #124×C97C22semidirect
26. C2×C42×C97C22semidirect
47. #13095495C23semidirect
53. C2×Q8×C97C23semidirect
63. C42×C97C23semidirect
116. C2×C97#455semidirect
120. C97C2×#455semidirect
123. C2×#124D97semidirect
132. C22×Q8C2×D97semidirect
142. #124C2×D97semidirect
144. C2×C42C2×D97semidirect
159. C2×Q8C22×D97semidirect
183. C42C22×D97semidirect
237. C2#20260912direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908552
C2×#156908412
#202609122
C2×#156907382
C2×#156907272
C2×#124×C972
C2×#45597