Finite Group Databasebeta

← #37550200

Subgroups — #37550200

Order 23168.

816 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 210 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×C22⋊#10624382C212yesmaximal
2#26042226C218yesmaximal, complemented
3#26042221C214yesmaximal, complemented
4C2×C22⋊C4×D181C211yesmaximal, characteristic
5C23×#10624382C2214yes—
6C2×C4⋊#10624382C2218yescomplemented
7C2×C22⋊#10624382—28no—
8C2×C22⋊#10624382C2214yes—
9C2×C22⋊#10624382C2214yes—
10(C2×#10624382)⋊C4C2218yescomplemented
11C24×D181C2211yescharacteristic
12C22×C4×D181C2212yes—
13C2×C22⋊Dic181C411yescharacteristic, complemented
14C2×#15668426C412yescomplemented
15C22⋊C4×D181C414yescomplemented
16C22⋊C4×D181C2214yes—
17C2×C22⋊C4×C181C411yescharacteristic, complemented
18C22×#10624382—28no—
19C22×#10624382—48no—
20C22×#10624382C2318yes—
21C4⋊#10624382—48no—
22C22⋊#10624382—28no—
23C22⋊#10624382—28no—
24C23×D181—44no—
25C23×D181C2×C412yes—
26C23×D181C2×C411yescharacteristic
27C23×D181D414yes—
28C23×D181Q814yes—
29C23×D181C2312yes—
30C23×D181C2311yescharacteristic
31C22×Dic181C2×C412yescomplemented
32C2×C4×D181—44no—
33C2×C4×D181C2×C414yescomplemented
34C2×C4×D181C2314yes—
35C22⋊Dic181C2×C414yescomplemented
36#15668426C2×C418yescomplemented
37C24×C181C2×C411yescharacteristic
38C22×C4×C181C2×C412yescomplemented
39C22⋊C4×C181C2×C414yescomplemented
40C2×#10624382—28no—
41C2×#10624382—216no—
42C2×#10624382—416no—
43C22×D181—24no—
44C22×D181—28no—
45C22×D181—48no—
46C22×D181—48no—
47C22×D181—44no—
48C22×D181—42no—
49C22×D181C4⋊C418yescomplemented
50C22×D181C4⋊C414yescomplemented
51C22×D181C22×C412yes—
52C22×D181C22×C411yescharacteristic
53C22×D181C2×D412yes—
54C22×D181C2×D414yes—
55C22×D181C2×Q814yes—
56C22×D181C2×Q812yes—
57C22×D181C2411yescharacteristic
58C2×Dic181—42no—
59C2×Dic181C22×C414yes—
60C4×D181—28no—
61C4×D181—44no—
62C23×C181—42no—
63C23×C181C4⋊C414yescomplemented
64C23×C181C22×C412yes—
65C23×C181C22×C411yescharacteristic
66C2×C4×C181—42no—
67C2×C4×C181C22×C414yes—
68C22×C181—22no—
69C22×C181—24no—
70C22×C181—44no—
71C22×C181C2×C4⋊C414yes—
72C22×C181C2×C4⋊C412yes—
73C22×C181C23×C411yescharacteristic
74C2×D181—22no—
75C2×D181—24no—
76C2×D181—24no—
77C2×D181—28no—
78C2×D181—24no—
79C2×D181—28no—
80C2×D181—48no—
81C2×D181—44no—
82C2×D181C2×C4⋊C414yes—
83C2×D181C2×C4⋊C412yes—
84C2×D181C22×D411yescharacteristic
85C2×D181C22×Q811yescharacteristic
86C2×D181H(5,𝔽2)14yes—
87#10624382—48no—
88C4×C181—24no—
89Dic181—24no—
90C2×C181—22no—
91C2×C181—24no—
92C2×C181C22×C4⋊C411yescharacteristic
93C2×C181#51812yes—
94D181—22no—
95D181—24no—
96D181#53412yes—
97D181#54312yes—
98C181#231511yescharacteristic, complemented, Sylow
99#2315—1811nomaximal, Sylow
100C22×C22⋊C4—1812no—
101C22×C22⋊C4—1811no—
102#385—1818no—
103#380—1814no—
104C25—1811no—
105C23×C4—1814no—
106C23×C4—1812no—
107C2×C4⋊C4—1818no—
108C2×C22⋊C4—3628no—
109C2×C22⋊C4—1814no—
110C2×C22⋊C4—1814no—
111C2×C22⋊C4—1811no—
112C2×C22⋊C4—1812no—
113C2×C22⋊C4—1814no—
114C2×C22⋊C4—1814no—
115C2×C22⋊C4#1062438211yescharacteristic, complemented
116#95—1818no—
117C24—1811no—
118C24—1812no—
119C24—1814no—
120C24—1814no—
121C24—1811no—
122C24—1812no—
123C24—7244no—
124C24C2×#1062438211yescharacteristic
125C22×C4—3628no—
126C22×C4—1818no—
127C22×C4—1814no—
128C22×C4—1812no—
129C22×C4—1814no—
130C22×C4—7244no—
131C22×C4—7248no—
132C22×C4C2×#1062438212yescomplemented
133C4⋊C4—7248no—
134C22⋊C4—3628no—
135C22⋊C4—3628no—
136C22⋊C4—1814no—
137C22⋊C4—1818no—
138C22⋊C4C2×#1062438214yescomplemented
139C23—42no—
140C23—3624no—
141C23—3628no—
142C23—1812no—
143C23—1814no—
144C23—1812no—
145C23—1814no—
146C23—1812no—
147C23—1811no—
148C23—1811no—
149C23—1818no—
150C23—1814no—
151C23—7242no—
152C23—7244no—
153C23—7248no—
154C23—7248no—
155C23C4⋊#1062438214yescomplemented
156C23C22×#1062438212yes—
157C23C22×#1062438211yescharacteristic
158C2×C4—42no—
159C2×C4—3628no—
160C2×C4—3628no—
161C2×C4—36216no—
162C2×C4—1814no—
163C2×C4—7244no—
164C2×C4—7242no—
165C2×C4—72416no—
166C2×C4C22×#1062438214yes—
167C22—22no—
168C22—24no—
169C22—44no—
170C22—3628no—
171C22—3624no—
172C22—3624no—
173C22—3628no—
174C22—3622no—
175C22—3624no—
176C22—1812no—
177C22—1814no—
178C22—1811no—
179C22—1811no—
180C22—1814no—
181C22—7244no—
182C22—7248no—
183C22C2×C4⋊#1062438212yes—
184C22C2×C4⋊#1062438214yes—
185C22C23×#1062438211yescharacteristic
186C4—24no—
187C4—3624no—
188C4—7248no—
189C2—22no—
190C2—24no—
191C2—3624no—
192C2—3622no—
193C2—1812no—
194C2—1812no—
195C2C22×C4⋊#1062438211yescharacteristic
196C2#2604234612yes—
subgroupG/H
center, FrattiniC22C23×#10624382
commutator, socleC22×C181C23×C4
FittingC2×C22⋊C4×C181C4
prime radicalC24×D181C22
cosocleC22×D181C24
hypercenterC2×C22⋊C4#10624382
hypocenterC181#2315
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #26042226C2semidirect
3. #26042221C2semidirect
6. C2×C4⋊#10624382C22semidirect
10. (C2×#10624382)⋊C4C22semidirect
13. C2×C22⋊Dic181C4semidirect
14. C2×#15668426C4semidirect
15. C22⋊C4×D181C4semidirect
17. C2×C22⋊C4×C181C4semidirect
31. C22×Dic181C2×C4semidirect
33. C2×C4×D181C2×C4semidirect
35. C22⋊Dic181C2×C4semidirect
36. #15668426C2×C4semidirect
38. C22×C4×C181C2×C4semidirect
39. C22⋊C4×C181C2×C4semidirect
49. C22×D181C4⋊C4semidirect
50. C22×D181C4⋊C4semidirect
63. C23×C181C4⋊C4semidirect
98. C181#2315semidirect
115. C2×C22⋊C4#10624382semidirect
132. C22×C4C2×#10624382semidirect
138. C22⋊C4C2×#10624382semidirect
155. C23C4⋊#10624382semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#106243822
#260422262
#260422212
C2×C22⋊C4×D1812
#2315181