Finite Group Databasebeta

← C2×#20260934 #27151167

Subgroups — C2×#20260934 #27151167

Order 12416.

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

normal subgroups only

This group has 584 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×#13095499C211yesmaximal, characteristic, complemented
3C2×#15690806C211yesmaximal, characteristic
4C2×#15690775C211yesmaximal, characteristic, complemented
5#20260934C218yesmaximal, direct product factor
6C2×#15690739C211yesmaximal, characteristic, complemented
7C2×#15690730C211yesmaximal, characteristic, complemented
8C2×#102×C97C211yesmaximal, characteristic, complemented
9C22×Dic194C2211yescharacteristic
10C22×D388C2211yescharacteristic
11C22×C22⋊D97—21no—
12C22×C22⋊D97—21no—
13C2×#13095500—24no—
14C2×#13095500C2214yescomplemented
15C2×#13095499—24no—
16C2×#13095499C2214yescomplemented
17C22×#11715895C2211yescharacteristic
18#15690806—22no—
19#15690806C2214yes—
20#15690775—22no—
21#15690775C2214yescomplemented
22C2×#13095490—21no—
23C2×C4×Dic97—21no—
24C2×C4×Dic97C2211yescharacteristic
25#15690739C2214yescomplemented
26#15690730C2214yescomplemented
27C22×D4×C97C2211yescharacteristic
28C22×C8×C97C2211yescharacteristic
29C2×C4⋊C4×C97C2211yescharacteristic
30#102×C97C2214yescomplemented
31C23×D97—21no—
32C2×Dic194—21no—
33C2×Dic194—22no—
34C2×Dic194D414yescomplemented
35C2×Dic194C2312yes—
36C2×D388—22no—
37C2×D388—22no—
38C2×D388—21no—
39C2×D388C2312yes—
40C2×C22⋊D97—24no—
41C2×C22⋊D97—24no—
42C2×C22⋊D97—44no—
43C2×C22⋊D97—44no—
44C22×Dic97—21no—
45C22×Dic97D411yescharacteristic
46C22×Dic97D411yescharacteristic
47#13095500—28no—
48#13095499—28no—
49C2×#11715895—22no—
50C2×#11715895C2312yes—
51#13095490—24no—
52C4×Dic97—24no—
53C4×Dic97D412yes—
54C4×Dic97C2312yes—
55C24×C97—21no—
56C2×D4×C97—22no—
57C2×D4×C97—21no—
58C2×D4×C97—22no—
59C2×D4×C97C2312yes—
60C22×C4×C97—21no—
61C22×C4×C97C2311yescharacteristic
62C2×C8×C97—22no—
63C2×C8×C97C2312yes—
64C4⋊C4×C97—21no—
65C4⋊C4×C97C2312yes—
66C23×C97—22no—
67C23×C97—42no—
68C23×C97—42no—
69C23×C97—42no—
70C23×C97C2×D411yescharacteristic
71C22×D97—22no—
72C22×D97—42no—
73C22×D97—42no—
74C22×D97—42no—
75D4×C97—22no—
76D4×C97—22no—
77D4×C97—42no—
78C2×C4×C97—22no—
79C2×C4×C97—42no—
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D412yes—
82C2×C4×C97#4112yes—
83C2×C4×C97C2411yescharacteristic
84C22⋊D97—48no—
85C22⋊D97—48no—
86C2×Dic97—24no—
87C2×Dic97—22no—
88C2×Dic97—22no—
89C2×Dic97SD1614yescomplemented
90C2×Dic97C2×D412yes—
91C2×Dic97C2×D412yes—
92D388—22no—
93D388—22no—
94D388—42no—
95Dic194—24no—
96Dic194—22no—
97Dic194C2×D414yescomplemented
98C8×C97—22no—
99#11715895—22no—
100C22×C97—42no—
101C22×C97—42no—
102C22×C97—42no—
103C22×C97—84no—
104C22×C97#12112yes—
105C22×C97C2×SD1612yes—
106C22×C97AGL(1,ℤ/8)12yes—
107C22×C97C22×D411yescharacteristic
108C2×D97—42no—
109C2×D97—42no—
110C2×D97—42no—
111C2×D97—84no—
112C4×C97—42no—
113C4×C97#12112yes—
114C4×C97C22×D411yescharacteristic
115C4×C97C2×#4111yescharacteristic
116Dic97—24no—
117Dic97—22no—
118Dic97C2×SD1614yescomplemented
119C2×C97—82no—
120C2×C97#46014yescomplemented
121C2×C97C2×#12111yescharacteristic
122C2×C97C22×SD1611yescharacteristic
123C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
124D97—82no—
125C2×#460—971nomaximal, Sylow
126C97C2×#46011yescharacteristic, complemented, Sylow
127C22×SD16—971no—
128C22×SD16—971no—
129C2×#127—971no—
130C2×C4×Q8—971no—
131#460—978no—
132C2×C4⋊C8—971no—
133C2×#102—971no—
134C2×#102D9711yescharacteristic, complemented
135C22×Q8—971no—
136C22×D4—971no—
137C22×D4—1941no—
138C22×D4—1941no—
139C22×D4C2×D9711yescharacteristic
140C2×SD16—974no—
141C2×SD16—974no—
142C2×SD16—1944no—
143C2×SD16—1944no—
144C22×C8—971no—
145C22×C8C2×D9711yescharacteristic
146#127—974no—
147#127—1942no—
148C4×Q8—974no—
149C4×Q8—1942no—
150C2×C4⋊C4—1941no—
151C2×C4⋊C4C2×D9711yescharacteristic
152C2×C42—971no—
153C2×C42—1941no—
154C4⋊C8—974no—
155#102—974no—
156#102C2×D9714yescomplemented
157C24—21no—
158C24—1941no—
159C2×Q8—974no—
160C2×Q8—972no—
161C2×Q8—1941no—
162C2×Q8—1942no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—972no—
167C2×D4—3884no—
168C2×D4—3884no—
169C2×D4—1944no—
170C2×D4—1944no—
171C2×D4—1941no—
172C2×D4—1942no—
173C2×D4—1942no—
174C2×D4C22×D9712yes—
175C22×C4—21no—
176C22×C4—971no—
177C22×C4—971no—
178C22×C4—1941no—
179C22×C4C22×D9711yescharacteristic
180SD16—1948no—
181SD16—1948no—
182C2×C8—22no—
183C2×C8—972no—
184C2×C8—1942no—
185C2×C8C22×D9712yes—
186C4⋊C4—21no—
187C4⋊C4—1944no—
188C4⋊C4C22×D9712yes—
189C42—972no—
190C42—972no—
191C42—1944no—
192C23—22no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—3882no—
197C23—3882no—
198C23—3882no—
199C23—1942no—
200C23D4×D9711yescharacteristic
201Q8—974no—
202Q8—1944no—
203Q8—1942no—
204D4—22no—
205D4—22no—
206D4—42no—
207D4—3888no—
208D4—3888no—
209D4—3882no—
210D4—1942no—
211D4—1942no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—974no—
215C2×C4—972no—
216C2×C4—972no—
217C2×C4—1944no—
218C2×C4—1942no—
219C2×C4—1942no—
220C2×C4D4×D9711yescharacteristic
221C2×C4D4×D9712yes—
222C2×C4#1309551112yes—
223C2×C4C23×D9711yescharacteristic
224C8—22no—
225C8—1942no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—84no—
230C22—7764no—
231C22—3882no—
232C22—3882no—
233C22—3882no—
234C22#1569078312yes—
235C22SD16×D9712yes—
236C22#1569083412yes—
237C22C2×D4×D9711yescharacteristic
238C4—42no—
239C4—974no—
240C4—1944no—
241C4—1942no—
242C4#1569078312yes—
243C4C2×D4×D9711yescharacteristic
244C4C2×#1309551111yescharacteristic
245C2—82no—
246C2—7762no—
247C2#2026093414yesdirect product factor
248C2C2×#1569078311yescharacteristic
249C2C2×SD16×D9711yescharacteristic
250C2C2×#1569083411yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#102×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690806C2
hypercenterC2×#102D97
hypocenterC97C2×#460
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#13095500C2semidirect
2. C22×#13095499C2semidirect
4. C2×#15690775C2semidirect
5. #20260934C2direct (N × H)
6. C2×#15690739C2semidirect
7. C2×#15690730C2semidirect
8. C2×#102×C97C2semidirect
14. C2×#13095500C22semidirect
16. C2×#13095499C22semidirect
21. #15690775C22semidirect
25. #15690739C22semidirect
26. #15690730C22semidirect
30. #102×C97C22semidirect
34. C2×Dic194D4semidirect
89. C2×Dic97SD16semidirect
97. Dic194C2×D4semidirect
118. Dic97C2×SD16semidirect
120. C2×C97#460semidirect
126. C97C2×#460semidirect
134. C2×#102D97semidirect
156. #102C2×D97semidirect
247. C2#20260934direct (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×#130954992
C2×#156908062
C2×#156907752
#202609342
C2×#156907392
C2×#156907302
C2×#102×C972
C2×#46097