Finite Group Databasebeta

← C2×#476×C181 #37531014

Subgroups — C2×#476×C181 #37531014

Order 23168.

480 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 232 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×C4⋊Q8×C181C211yesmaximal, characteristic, complemented
2C2×C4×D4×C181C211yesmaximal, characteristic
3#476×C181C218yesmaximal, direct product factor
4C2×#106×C181C212yesmaximal, complemented
5C2×C4⋊C8×C181C211yesmaximal, characteristic, complemented
6C2×#102×C181C212yesmaximal
7C22×Q8×C181—21no—
8C22×D4×C181C2211yescharacteristic
9C23×C4×C181—21no—
10C22×C8×C181C2212yes—
11C4⋊Q8×C181—22no—
12C4⋊Q8×C181C2214yescomplemented
13C4×D4×C181—22no—
14C4×D4×C181C2214yes—
15C2×C4⋊C4×C181—21no—
16C2×C4⋊C4×C181C2211yescharacteristic
17C2×C4⋊C4×C181C2212yes—
18C2×C22⋊C4×C181—21no—
19C2×C42×C181C2211yescharacteristic
20#106×C181C2218yescomplemented
21C4⋊C8×C181C2214yescomplemented
22#102×C181C2218yes—
23C24×C181—21no—
24C2×Q8×C181—24no—
25C2×Q8×C181—44no—
26C2×D4×C181—21no—
27C2×D4×C181—22no—
28C2×D4×C181Q814yescomplemented
29C2×D4×C181C2312yes—
30C22×C4×C181—21no—
31C22×C4×C181—22no—
32C22×C4×C181—24no—
33C22×C4×C181—44no—
34C22×C4×C181D411yescharacteristic
35C22×C4×C181D411yescharacteristic
36C22×C4×C181C2311yescharacteristic
37C2×C8×C181—24no—
38C2×C8×C181C2314yes—
39C4⋊C4×C181—24no—
40C4⋊C4×C181—21no—
41C4⋊C4×C181—22no—
42C4⋊C4×C181C2314yes—
43C4⋊C4×C181C2312yes—
44C22⋊C4×C181—24no—
45C42×C181D412yes—
46C42×C181C2312yes—
47C23×C181—24no—
48C23×C181—22no—
49C23×C181—42no—
50C23×C181—42no—
51C23×C181C2×D411yescharacteristic
52Q8×C181—48no—
53D4×C181—24no—
54D4×C181—22no—
55D4×C181C2×Q814yescomplemented
56C2×C4×C181—22no—
57C2×C4×C181—22no—
58C2×C4×C181—24no—
59C2×C4×C181—42no—
60C2×C4×C181—44no—
61C2×C4×C181—48no—
62C2×C4×C181SD1614yes—
63C2×C4×C181C2×D412yes—
64C2×C4×C181C2×D411yescharacteristic
65C2×C4×C181C2×D412yes—
66C2×C4×C181C2×Q812yes—
67C2×C4×C181#4112yes—
68C2×C4×C181C2411yescharacteristic
69C8×C181—24no—
70C22×C181—24no—
71C22×C181—42no—
72C22×C181—42no—
73C22×C181—48no—
74C22×C181C22⋊Q812yes—
75C22×C181C2×SD1612yes—
76C22×C181AGL(1,ℤ/8)12yes—
77C22×C181C22×D411yescharacteristic
78C4×C181—22no—
79C4×C181—42no—
80C4×C181—44no—
81C4×C181C22⋊Q812yes—
82C4×C181C2×SD1614yes—
83C4×C181C22×Q811yescharacteristic
84C4×C181C2×#4111yescharacteristic
85C2×C181—44no—
86C2×C181#47614yesdirect product factor
87C2×C181C2×C22⋊Q811yescharacteristic
88C2×C181C22×SD1611yescharacteristic
89C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
90C181C2×#47611yescharacteristic, direct product factor, Sylow
91C2×#476C18111yesmaximal, characteristic, direct product factor, Sylow
92C2×C4⋊Q8C2×C18111yescharacteristic, complemented
93C2×C4×D4C2×C18111yescharacteristic
94#476C2×C18118yesdirect product factor
95C2×#106C2×C18112yescomplemented
96C2×C4⋊C8C2×C18111yescharacteristic, complemented
97C2×#102C2×C18112yes—
98C22×Q8—21no—
99C22×D4C22×C18111yescharacteristic
100C23×C4—21no—
101C22×C8C22×C18112yes—
102C4⋊Q8—22no—
103C4⋊Q8C22×C18114yescomplemented
104C4×D4—22no—
105C4×D4C22×C18114yes—
106C2×C4⋊C4—21no—
107C2×C4⋊C4C22×C18111yescharacteristic
108C2×C4⋊C4C22×C18112yes—
109C2×C22⋊C4—21no—
110C2×C42C22×C18111yescharacteristic
111#106C22×C18118yescomplemented
112C4⋊C8C22×C18114yescomplemented
113#102C22×C18118yes—
114C24—21no—
115C2×Q8—24no—
116C2×Q8—44no—
117C2×D4—21no—
118C2×D4—22no—
119C2×D4Q8×C18114yescomplemented
120C2×D4C23×C18112yes—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—44no—
125C22×C4D4×C18111yescharacteristic
126C22×C4D4×C18111yescharacteristic
127C22×C4C23×C18111yescharacteristic
128C2×C8—24no—
129C2×C8C23×C18114yes—
130C4⋊C4—24no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4C23×C18114yes—
134C4⋊C4C23×C18112yes—
135C22⋊C4—24no—
136C42D4×C18112yes—
137C42C23×C18112yes—
138C23—24no—
139C23—22no—
140C23—42no—
141C23—42no—
142C23C2×D4×C18111yescharacteristic
143Q8—48no—
144D4—24no—
145D4—22no—
146D4C2×Q8×C18114yescomplemented
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—42no—
151C2×C4—44no—
152C2×C4—48no—
153C2×C4SD16×C18114yes—
154C2×C4C2×D4×C18112yes—
155C2×C4C2×D4×C18111yescharacteristic
156C2×C4C2×D4×C18112yes—
157C2×C4C2×Q8×C18112yes—
158C2×C4#41×C18112yes—
159C2×C4C24×C18111yescharacteristic
160C8—24no—
161C22—24no—
162C22—48no—
163C22—42no—
164C22—42no—
165C22C22⋊Q8×C18112yes—
166C22C2×SD16×C18112yes—
167C22AGL(1,ℤ/8)×C18112yes—
168C22C22×D4×C18111yescharacteristic
169C4—22no—
170C4—42no—
171C4—44no—
172C4C22⋊Q8×C18112yes—
173C4C2×SD16×C18114yes—
174C4C22×Q8×C18111yescharacteristic
175C4C2×#41×C18111yescharacteristic
176C2—44no—
177C2#476×C18114yesdirect product factor
178C2C2×C22⋊Q8×C18111yescharacteristic
179C2C22×SD16×C18111yescharacteristic
180C2C2×AGL(1,ℤ/8)×C18111yescharacteristic
subgroupG/H
center, socleC23×C181C2×D4
commutator, Frattini, cosocleC2×C4C24×C181
prime radicalC22×D4×C181C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C181 ◁ C2×C42×C181 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×C181C2semidirect
3. #476×C181C2direct (N × H)
4. C2×#106×C181C2semidirect
5. C2×C4⋊C8×C181C2semidirect
12. C4⋊Q8×C181C22semidirect
20. #106×C181C22semidirect
21. C4⋊C8×C181C22semidirect
28. C2×D4×C181Q8semidirect
55. D4×C181C2×Q8semidirect
86. C2×C181#476direct (N × H)
90. C181C2×#476direct (N × H)
91. C2×#476C181direct (N × H)
92. C2×C4⋊Q8C2×C181semidirect
94. #476C2×C181direct (N × H)
95. C2×#106C2×C181semidirect
96. C2×C4⋊C8C2×C181semidirect
103. C4⋊Q8C22×C181semidirect
111. #106C22×C181semidirect
112. C4⋊C8C22×C181semidirect
119. C2×D4Q8×C181semidirect
146. D4C2×Q8×C181semidirect
177. C2#476×C181direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×C1812
C2×C4×D4×C1812
#476×C1812
C2×#106×C1812
C2×C4⋊C8×C1812
C2×#102×C1812
C2×#476181