Finite Group Databasebeta

← #492×D181 #37544125

Subgroups — #492×D181 #37544125

Order 23168.

656 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 165 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×SD16×D181C214yesmaximal, complemented
2C4⋊Q8×D181C211yesmaximal, characteristic, complemented
3#127×D181C211yesmaximal, characteristic
4#26041653C211yesmaximal, characteristic, complemented
5#26041652C211yesmaximal, characteristic, complemented
6#26041651C211yesmaximal, characteristic, complemented
7#26041650C214yesmaximal, complemented
8C4×C8×D181C211yesmaximal, characteristic, complemented
9#492×C181C211yesmaximal, characteristic, complemented
10C2×Q8×D181C2212yes—
11C2×D4×D181—22no—
12C2×D4×D181C2212yes—
13C2×#15668438C2214yescomplemented
14C2×#15668437C2214yescomplemented
15C2×#15668436C2214yescomplemented
16SD16×D181—216no—
17C2×C8×D181C2212yes—
18#19153816C2211yescharacteristic, complemented
19#19153815—22no—
20#19153814C2212yescomplemented
21#19153813C2211yescharacteristic
22#19153812—22no—
23#19153812C2212yes—
24C4⋊C4×D181—22no—
25C42×D181C2211yescharacteristic
26C4×#13089678C2211yescharacteristic, complemented
27C8×Dic181C2212yescomplemented
28C2×SD16×C181C2214yescomplemented
29C4⋊Q8×C181C2211yescharacteristic, complemented
30#127×C181C2211yescharacteristic
31C4×C8×C181C2211yescharacteristic, complemented
32C23×D181—22no—
33C2×Dic362—22no—
34C2×Dic362C2312yes—
35Q8×D181—28no—
36C2×D724—22no—
37C2×D724C2312yes—
38C2×C22⋊D181—22no—
39C2×C22⋊D181—24no—
40D4×D181—28no—
41D4×D181—48no—
42C2×C4×D181—22no—
43C2×C4×D181D412yes—
44C2×C4×D181C2311yescharacteristic
45#15668438—28no—
46#15668437—28no—
47#15668436—28no—
48C2×#13089678C2312yes—
49C8×D181D418yescomplemented
50C4⋊Dic181—22no—
51#15668428—24no—
52C4×Dic181D412yes—
53C4×Dic181C2311yescharacteristic
54C2×Q8×C181C2312yes—
55C2×D4×C181—22no—
56C2×D4×C181C2312yes—
57SD16×C181—28no—
58C2×C8×C181C2312yes—
59C4⋊C4×C181—22no—
60C42×C181C2311yescharacteristic
61C22×D181—22no—
62C22×D181—44no—
63C22×D181—48no—
64C22×D181C2×D411yescharacteristic
65Dic362—24no—
66Dic362—44no—
67D724—24no—
68D724—44no—
69C22⋊D181—44no—
70C22⋊D181—48no—
71C2×Dic181—22no—
72C2×Dic181C2×D411yescharacteristic
73C2×Dic181C2×D412yes—
74C4×D181—44no—
75C4×D181SD1618yescomplemented
76C4×D181C2×D414yes—
77#13089678C2×D414yescomplemented
78C23×C181—22no—
79Q8×C181—24no—
80D4×C181—24no—
81D4×C181—44no—
82C2×C4×C181—22no—
83C2×C4×C181C2×D412yes—
84C2×C4×C181C2411yescharacteristic
85C8×C181C2×D414yescomplemented
86C22×C181—42no—
87C22×C181—44no—
88C22×C181C22×D411yescharacteristic
89C2×D181—42no—
90C2×D181—44no—
91C2×D181—88no—
92C2×D181#12712yes—
93C2×D181C2×SD1614yes—
94C4×C181—42no—
95C4×C181C2×SD1614yescomplemented
96C4×C181C22×D412yes—
97Dic181—42no—
98Dic181#12712yes—
99Dic181C2×SD1614yescomplemented
100C2×C181—82no—
101C2×C181C2×#12711yescharacteristic
102C2×C181C22×SD1612yes—
103D181—82no—
104D181#49214yesdirect product factor
105C181C2×#49211yescharacteristic, complemented, Sylow
106C2×#492—1811nomaximal, Sylow
107C22×SD16—1814no—
108C2×C4⋊Q8—1811no—
109C2×#127—1811no—
110#492—1814no—
111#492—1811no—
112#492—1811no—
113#492—1811no—
114#492D18111yescharacteristic, direct product factor
115C2×C4×C8—1811no—
116C22×Q8—1812no—
117C22×D4—3622no—
118C22×D4—1812no—
119C2×SD16—36216no—
120C2×SD16—1814no—
121C2×SD16—1814no—
122C2×SD16—1814no—
123C2×SD16C2×D18114yescomplemented
124C22×C8—1812no—
125C4⋊Q8—3622no—
126C4⋊Q8—1812no—
127C4⋊Q8—1811no—
128C4⋊Q8C2×D18111yescharacteristic, complemented
129#127—3622no—
130#127—1812no—
131#127—1811no—
132#127C2×D18111yescharacteristic
133C2×C4⋊C4—3622no—
134C2×C42—1811no—
135C4×C8—1812no—
136C4×C8—1811no—
137C4×C8C2×D18111yescharacteristic, complemented
138C24—3622no—
139C2×Q8—3622no—
140C2×Q8—3628no—
141C2×Q8—1812no—
142C2×Q8C22×D18112yes—
143C2×D4—22no—
144C2×D4—3628no—
145C2×D4—3622no—
146C2×D4—3624no—
147C2×D4—3622no—
148C2×D4—1812no—
149C2×D4—7248no—
150C2×D4C22×D18112yes—
151C22×C4—3622no—
152C22×C4—1812no—
153C22×C4—1811no—
154SD16—28no—
155SD16—3628no—
156SD16—3628no—
157SD16—3628no—
158C2×C8—1812no—
159C2×C8—1818no—
160C2×C8C22×D18112yes—
161C4⋊C4—22no—
162C4⋊C4—3624no—
163C4⋊C4—3622no—
164C42—1812no—
165C42—1811no—
166C42C22×D18111yescharacteristic
167C23—22no—
168C23—3622no—
169C23—1811no—
170C23—7244no—
171C23—7248no—
172Q8—24no—
173Q8—3624no—
174Q8—7244no—
175D4—24no—
176D4—44no—
177D4—3624no—
178D4—7244no—
179D4—7248no—
180D4—7244no—
181C2×C4—22no—
182C2×C4—3622no—
183C2×C4—1812no—
184C2×C4—1811no—
185C2×C4—1818no—
186C2×C4—1814no—
187C2×C4—7244no—
188C2×C4D4×D18112yes—
189C2×C4C23×D18111yescharacteristic
190C8—1814no—
191C8D4×D18114yescomplemented
192C22—44no—
193C22—42no—
194C22—14488no—
195C22—1814no—
196C22—1812no—
197C22—7244no—
198C22—7242no—
199C22C2×D4×D18111yescharacteristic
200C4—42no—
201C4—1814no—
202C4—1812no—
203C4—7242no—
204C4SD16×D18114yescomplemented
205C4C2×D4×D18112yes—
206C2—82no—
207C2—14482no—
208C2—1814no—
209C2#127×D18111yescharacteristic
210C2C2×SD16×D18112yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#492×C181C2
socleC22×C181C22×D4
prime radical#127×D181C2
hypercenter#492D181
hypocenterC181C2×#492
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D181C2semidirect
2. C4⋊Q8×D181C2semidirect
4. #26041653C2semidirect
5. #26041652C2semidirect
6. #26041651C2semidirect
7. #26041650C2semidirect
8. C4×C8×D181C2semidirect
9. #492×C181C2semidirect
13. C2×#15668438C22semidirect
14. C2×#15668437C22semidirect
15. C2×#15668436C22semidirect
18. #19153816C22semidirect
20. #19153814C22semidirect
26. C4×#13089678C22semidirect
27. C8×Dic181C22semidirect
28. C2×SD16×C181C22semidirect
29. C4⋊Q8×C181C22semidirect
31. C4×C8×C181C22semidirect
49. C8×D181D4semidirect
75. C4×D181SD16semidirect
77. #13089678C2×D4semidirect
85. C8×C181C2×D4semidirect
95. C4×C181C2×SD16semidirect
99. Dic181C2×SD16semidirect
104. D181#492direct (N × H)
105. C181C2×#492semidirect
114. #492D181direct (N × H)
123. C2×SD16C2×D181semidirect
128. C4⋊Q8C2×D181semidirect
137. C4×C8C2×D181semidirect
191. C8D4×D181semidirect
204. C4SD16×D181semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1812
C4⋊Q8×D1812
#127×D1812
#260416532
#260416522
#260416512
#260416502
C4×C8×D1812
#492×C1812
C2×#492181