Finite Group Databasebeta

← C2×#20260885 #27150917

Subgroups — C2×#20260885 #27150917

Order 12416.

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

normal subgroups only

This group has 996 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
1C23×D388C211yesmaximal, characteristic
2C22×#13095498C212yesmaximal, complemented
3#20260885C218yesmaximal, direct product factor
4C2×#15690730C212yesmaximal, complemented
5C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
6C2×C22⋊Q8×C97C211yesmaximal, characteristic, complemented
7C24×D97—21no—
8C22×D388—28no—
9C22×D388—21no—
10C22×D388—22no—
11C22×D388C2214yes—
12C22×D388C2212yes—
13C2×#13095498—28no—
14C2×#13095498C2218yescomplemented
15C22×#11715895C2212yescomplemented
16#15690730C2218yescomplemented
17C22⋊#11715895C2214yescomplemented
18C22×Q8×C97C2211yescharacteristic, complemented
19C23×C4×C97C2211yescharacteristic
20C22⋊Q8×C97—22no—
21C22⋊Q8×C97C2214yescomplemented
22C2×C4⋊C4×C97—21no—
23C2×C4⋊C4×C97C2211yescharacteristic, complemented
24C2×C22⋊C4×C97—21no—
25C23×D97—28no—
26C23×D97—24no—
27C23×D97—22no—
28C23×D97—44no—
29C23×D97—44no—
30C2×D388—28no—
31C2×D388—216no—
32C2×D388—24no—
33C2×D388—24no—
34C2×D388—22no—
35C2×D388—216no—
36C2×D388D418yes—
37C2×D388C2314yes—
38#13095498—216no—
39C2×#11715895—24no—
40C2×#11715895C2314yescomplemented
41C24×C97D411yescharacteristic
42C2×Q8×C97—21no—
43C2×Q8×C97—24no—
44C2×Q8×C97C2312yescomplemented
45C22×C4×C97—24no—
46C22×C4×C97—21no—
47C22×C4×C97—21no—
48C22×C4×C97D411yescharacteristic
49C22×C4×C97D412yes—
50C22×C4×C97C2312yes—
51C22×C4×C97C2311yescharacteristic
52C4⋊C4×C97—24no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97C2312yescomplemented
55C22⋊C4×C97—24no—
56C23×C97—22no—
57C23×C97—22no—
58C23×C97SD1614yescomplemented
59C23×C97C2×D412yes—
60C23×C97C2×D411yescharacteristic
61C22×D97—216no—
62C22×D97—28no—
63C22×D97—24no—
64C22×D97—48no—
65C22×D97—432no—
66C22×D97—44no—
67C22×D97—48no—
68C22×D97—44no—
69Q8×C97—24no—
70Q8×C97—42no—
71C2×C4×C97—22no—
72C2×C4×C97—24no—
73C2×C4×C97—24no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—42no—
77C2×C4×C97—42no—
78C2×C4×C97C2×D414yes—
79C2×C4×C97C2×D412yes—
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2411yescharacteristic
82D388—216no—
83D388—28no—
84D388—24no—
85D388C2×D418yes—
86#11715895—24no—
87C22×C97—22no—
88C22×C97—22no—
89C22×C97—28no—
90C22×C97#12012yes—
91C22×C97C2×SD1612yes—
92C22×C97C2×SD1614yescomplemented
93C22×C97AGL(1,ℤ/8)12yes—
94C22×C97C22×D411yescharacteristic
95C2×D97—28no—
96C2×D97—432no—
97C2×D97—44no—
98C2×D97—44no—
99C2×D97—416no—
100C4×C97—22no—
101C4×C97—42no—
102C4×C97—42no—
103C4×C97#12012yes—
104C4×C97C22×D412yes—
105C2×C97—24no—
106C2×C97#45014yescomplemented
107C2×C97C2×#12011yescharacteristic
108C2×C97C22×SD1611yescharacteristic
109C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
110D97—48no—
111C2×#450—971nomaximal, Sylow
112C97C2×#45011yescharacteristic, complemented, Sylow
113C23×D4—971no—
114C22×SD16—972no—
115C2×C22⋊Q8D9711yescharacteristic, complemented
116#450—978no—
117C2×#102—972no—
118C2×C22⋊C8—971no—
119C25—1941no—
120C22×Q8C2×D9711yescharacteristic, complemented
121C22×D4—972no—
122C22×D4—974no—
123C22×D4—1948no—
124C22×D4—1942no—
125C22×D4—1941no—
126C23×C4C2×D9711yescharacteristic
127C2×SD16—978no—
128C2×SD16—1948no—
129C22×C8—972no—
130C22⋊Q8—22no—
131C22⋊Q8C2×D9714yescomplemented
132C2×C4⋊C4—21no—
133C2×C4⋊C4C2×D9711yescharacteristic, complemented
134C2×C22⋊C4—21no—
135#102—978no—
136C22⋊C8—974no—
137C24—3884no—
138C24—3884no—
139C24—1948no—
140C24—1944no—
141C24—1942no—
142C24C22⋊D9711yescharacteristic
143C2×Q8—21no—
144C2×Q8—24no—
145C2×Q8C22×D9712yescomplemented
146C2×D4—978no—
147C2×D4—974no—
148C2×D4—19416no—
149C2×D4—1948no—
150C2×D4—19416no—
151C2×D4—1944no—
152C2×D4—1944no—
153C2×D4—1942no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4C22⋊D9712yes—
158C22×C4C22⋊D9711yescharacteristic
159C22×C4C22×D9712yes—
160C22×C4C22×D9711yescharacteristic
161SD16—19416no—
162C2×C8—974no—
163C2×C8—1944no—
164C4⋊C4—24no—
165C4⋊C4—21no—
166C4⋊C4C22×D9712yescomplemented
167C22⋊C4—24no—
168C23—22no—
169C23—22no—
170C23—38832no—
171C23—3888no—
172C23—3884no—
173C23—3888no—
174C23—3884no—
175C23—19416no—
176C23—1948no—
177C23—1944no—
178C23#1309549814yescomplemented
179C23C2×C22⋊D9712yes—
180C23C2×C22⋊D9711yescharacteristic
181Q8—24no—
182Q8—42no—
183D4—978no—
184D4—19416no—
185D4—1948no—
186D4—1944no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4D4×D9714yes—
195C2×C4C2×C22⋊D9712yes—
196C2×C4C2×C22⋊D9711yescharacteristic
197C2×C4C23×D9711yescharacteristic
198C8—1944no—
199C22—28no—
200C22—22no—
201C22—22no—
202C22—38832no—
203C22—38816no—
204C22—3884no—
205C22—3884no—
206C22—1948no—
207C22#1569077812yes—
208C22C2×#1309549812yes—
209C22C2×#1309549814yescomplemented
210C22#1569083512yes—
211C22C22×C22⋊D9711yescharacteristic
212C4—22no—
213C4—42no—
214C4—42no—
215C4#1569077812yes—
216C4C2×D4×D9712yes—
217C2—24no—
218C2—3888no—
219C2#2026088514yesdirect product factor
220C2C2×#1569077811yescharacteristic
221C2C22×#1309549811yescharacteristic
222C2C2×#1569083511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C22⋊Q8×C97C2
socleC23×C97C2×D4
prime radicalC23×D388C2
hypercenterC2×C22⋊Q8D97
hypocenterC97C2×#450
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×#13095498C2semidirect
3. #20260885C2direct (N × H)
4. C2×#15690730C2semidirect
5. C2×C22⋊#11715895C2semidirect
6. C2×C22⋊Q8×C97C2semidirect
14. C2×#13095498C22semidirect
15. C22×#11715895C22semidirect
16. #15690730C22semidirect
17. C22⋊#11715895C22semidirect
18. C22×Q8×C97C22semidirect
21. C22⋊Q8×C97C22semidirect
23. C2×C4⋊C4×C97C22semidirect
40. C2×#11715895C23semidirect
44. C2×Q8×C97C23semidirect
54. C4⋊C4×C97C23semidirect
58. C23×C97SD16semidirect
92. C22×C97C2×SD16semidirect
106. C2×C97#450semidirect
112. C97C2×#450semidirect
115. C2×C22⋊Q8D97semidirect
120. C22×Q8C2×D97semidirect
131. C22⋊Q8C2×D97semidirect
133. C2×C4⋊C4C2×D97semidirect
145. C2×Q8C22×D97semidirect
166. C4⋊C4C22×D97semidirect
178. C23#13095498semidirect
209. C22C2×#13095498semidirect
219. C2#20260885direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3882
C22×#130954982
#202608852
C2×#156907302
C2×C22⋊#117158952
C2×C22⋊Q8×C972
C2×#45097