Finite Group Databasebeta

← #37543518

Subgroups — #37543518

Order 23168.

440 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 153 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
1#26041836C211yesmaximal, characteristic, complemented
2#26041579C214yesmaximal, complemented
3#26041569C212yesmaximal, complemented
4#26041559C212yesmaximal, complemented
5#26041319C212yesmaximal, complemented
6#26041303C211yesmaximal, characteristic, complemented
7#26041284C212yesmaximal
8C2×C4×D4×C181C211yesmaximal, characteristic
9C2×#15668433C2212yes—
10#19153816C2211yescharacteristic, complemented
11#19153816C2211yescharacteristic, complemented
12#19153808C2212yescomplemented
13#19153795—21no—
14#19153795—21no—
15C4×Dic362—21no—
16C4×Dic362—21no—
17#19153775C2212yes—
18#19153749C2214yescomplemented
19#19153747C2214yescomplemented
20C4⋊#13089678C2212yescomplemented
21C4⋊#13089678C2212yescomplemented
22#19153737C2214yes—
23#19153737C2214yes—
24C22×D4×C181C2211yescharacteristic
25C23×C4×C181—21no—
26C4×D4×C181—21no—
27C4×D4×C181—21no—
28C4×D4×C181C2212yes—
29C4×D4×C181C2212yes—
30C2×C4⋊C4×C181C2211yescharacteristic
31C2×C22⋊C4×C181—21no—
32C2×C42×C181C2211yescharacteristic
33C2×Dic362—21no—
34C2×Dic362—21no—
35#15668433—24no—
36C2×#13089678C2314yes—
37C4⋊Dic181—21no—
38C4⋊Dic181—21no—
39C4⋊Dic181C2312yes—
40C4⋊Dic181C2312yes—
41#15668428—22no—
42#15668428—22no—
43C22⋊Dic181—22no—
44C4×Dic181—22no—
45C24×C181—21no—
46C2×D4×C181—21no—
47C2×D4×C181—22no—
48C2×D4×C181Q814yescomplemented
49C2×D4×C181C2312yes—
50C22×C4×C181—21no—
51C22×C4×C181—22no—
52C22×C4×C181—22no—
53C22×C4×C181—44no—
54C22×C4×C181D411yescharacteristic
55C22×C4×C181D411yescharacteristic
56C22×C4×C181C2311yescharacteristic
57C4⋊C4×C181—21no—
58C4⋊C4×C181C2312yes—
59C22⋊C4×C181—22no—
60C22⋊C4×C181—22no—
61C42×C181D411yescharacteristic
62C42×C181D411yescharacteristic
63C42×C181C2311yescharacteristic
64C42×C181C2311yescharacteristic
65Dic362—42no—
66Dic362—42no—
67C2×Dic181—22no—
68C2×Dic181—22no—
69#13089678—24no—
70C23×C181—24no—
71C23×C181—22no—
72C23×C181—42no—
73C23×C181—42no—
74C23×C181C2×D411yescharacteristic
75D4×C181—24no—
76D4×C181—22no—
77D4×C181C2×Q814yes—
78C2×C4×C181—22no—
79C2×C4×C181—22no—
80C2×C4×C181—41no—
81C2×C4×C181—44no—
82C2×C4×C181—42no—
83C2×C4×C181—44no—
84C2×C4×C181C2×D411yescharacteristic
85C2×C4×C181C2×D411yescharacteristic
86C2×C4×C181C2×D411yescharacteristic
87C2×C4×C181C2×D411yescharacteristic
88C2×C4×C181C2×D411yescharacteristic
89C2×C4×C181C2×Q812yes—
90C2×C4×C181#4112yes—
91C2×C4×C181C2411yescharacteristic
92C22×C181—21no—
93C22×C181—21no—
94C22×C181—24no—
95C22×C181—48no—
96C22×C181—42no—
97C22×C181—42no—
98C22×C181C22⋊Q812yes—
99C22×C181C22×D411yescharacteristic
100C4×C181—21no—
101C4×C181—21no—
102C4×C181—21no—
103C4×C181—42no—
104C4×C181C22⋊Q812yes—
105C4×C181AGL(1,ℤ/8)12yes—
106C4×C181C22×Q811yescharacteristic
107C4×C181C2×#4111yescharacteristic
108Dic181—42no—
109Dic181—42no—
110C2×C181—22no—
111C2×C181—44no—
112C2×C181C2×C22⋊Q811yescharacteristic
113C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
114C2×C181#57511yescharacteristic
115C181#307711yescharacteristic, complemented, Sylow
116#3077—1811nomaximal, Sylow
117#533—1811no—
118C2×C4×D4D18111yescharacteristic
119#478—1814no—
120#476—1812no—
121#474—1812no—
122#428—1812no—
123#423—1811no—
124#418—1812no—
125C22×D4C2×D18111yescharacteristic
126C23×C4—21no—
127C2×M16—1812no—
128C4⋊Q8—1811no—
129C4⋊Q8—1811no—
130#125—1812no—
131C22⋊Q8—3621no—
132C22⋊Q8—3621no—
133C4×Q8—3621no—
134C4×Q8—3621no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4C2×D18112yes—
138C4×D4C2×D18112yes—
139#117—1812no—
140C2×C4⋊C4C2×D18111yescharacteristic
141C2×C22⋊C4—21no—
142C2×C42C2×D18111yescharacteristic
143#107—1814no—
144#106—1814no—
145C4⋊C8—1812no—
146C4⋊C8—1812no—
147#102—1814no—
148#102—1814no—
149C24—21no—
150C2×Q8—3621no—
151C2×Q8—3621no—
152C2×D4—21no—
153C2×D4—22no—
154C2×D4Dic36214yescomplemented
155C2×D4C22×D18112yes—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4C22⋊D18111yescharacteristic
161C22×C4C22⋊D18111yescharacteristic
162C22×C4C22×D18111yescharacteristic
163M16—3624no—
164C2×C8—1814no—
165C4⋊C4—21no—
166C4⋊C4—3622no—
167C4⋊C4—3622no—
168C4⋊C4—3621no—
169C4⋊C4—3621no—
170C4⋊C4—1812no—
171C4⋊C4—1812no—
172C4⋊C4C22×D18112yes—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—3622no—
176C42—3622no—
177C42C22⋊D18111yescharacteristic
178C42C22⋊D18111yescharacteristic
179C42C22×D18111yescharacteristic
180C42C22×D18111yescharacteristic
181C23—24no—
182C23—22no—
183C23—42no—
184C23—42no—
185C23C2×C22⋊D18111yescharacteristic
186Q8—7242no—
187Q8—7242no—
188D4—24no—
189D4—22no—
190D4C2×Dic36214yes—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—3622no—
198C2×C4—3622no—
199C2×C4C2×C22⋊D18111yescharacteristic
200C2×C4C2×C22⋊D18111yescharacteristic
201C2×C4C2×C22⋊D18111yescharacteristic
202C2×C4C2×C22⋊D18111yescharacteristic
203C2×C4C2×C22⋊D18111yescharacteristic
204C2×C4C2×Dic36212yes—
205C2×C4#1566844912yes—
206C2×C4C23×D18111yescharacteristic
207C8—3624no—
208C22—21no—
209C22—21no—
210C22—24no—
211C22—48no—
212C22—42no—
213C22—42no—
214C22#1915379512yes—
215C22C22×C22⋊D18111yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—7242no—
221C4—7242no—
222C4#1915379512yes—
223C4#1915384212yes—
224C4C22×Dic36211yescharacteristic
225C4C2×#1566844911yescharacteristic
226C2—22no—
227C2—44no—
228C2C2×#1915379511yescharacteristic
229C2C2×#1915384211yescharacteristic
230C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×C4×D4×C181C2
socleC22×C181C22×D4
prime radicalC22×D4×C181C22
hypercenterC2×C4×D4D181
hypocenterC181#3077
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #26041836C2semidirect
2. #26041579C2semidirect
3. #26041569C2semidirect
4. #26041559C2semidirect
5. #26041319C2semidirect
6. #26041303C2semidirect
10. #19153816C22semidirect
11. #19153816C22semidirect
12. #19153808C22semidirect
18. #19153749C22semidirect
19. #19153747C22semidirect
20. C4⋊#13089678C22semidirect
21. C4⋊#13089678C22semidirect
48. C2×D4×C181Q8semidirect
115. C181#3077semidirect
154. C2×D4Dic362semidirect

Maximal subgroups

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

subgroupindex
#260418362
#260415792
#260415692
#260415592
#260413192
#260413032
#260412842
C2×C4×D4×C1812
#3077181