Finite Group Databasebeta

← C22×C3×#3745 #13179491

Subgroups — C22×C3×#3745 #13179491

Order 1728.

2332 conjugacy classes of subgroups; only the 558 normal subgroups are shown.

normal subgroups only

This group has 2332 conjugacy classes of subgroups — too many to draw. Only its 558 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×C3×C4⋊(C3⋊D3)C212yesmaximal, complemented
2C22×C33×C8C211yesmaximal, characteristic, complemented
3C2×C3×#3745C2112yesmaximal, direct product factor
4C22×#3745C311yesmaximal, characteristic, direct product factor
5C22×C3×D24—34nomaximal
6C23×C3×C3⋊D3—22no—
7C22×C33×C4C2211yescharacteristic
8C2×C3×C4⋊(C3⋊D3)—26no—
9C2×C3×C4⋊(C3⋊D3)C22112yescomplemented
10C2×C33×C8C2216yescomplemented
11C3×#3745C22116yesdirect product factor
12C22×C4⋊(C3⋊D3)C2×C312yescomplemented
13C22×C3×D12—38no—
14C22×C32×C8—24no—
15C22×C32×C8D314yescomplemented
16C22×C32×C8C2×C311yescharacteristic, complemented
17C2×#3745C2×C3112yesdirect product factor
18C2×C3×D24—348no—
19C23×C33D411yescharacteristic
20C22×C3×C3⋊D3—212no—
21C22×C3×C3⋊D3—48no—
22C2×C33×C4D413yes—
23C2×C33×C4C2313yes—
24C3×C4⋊(C3⋊D3)—212no—
25C3×C4⋊(C3⋊D3)C2318yescomplemented
26C33×C8C2314yescomplemented
27C22×C3×D8—91no—
28C22×D24—34no—
29C23×C3⋊D3—22no—
30C23×C3×D3—68no—
31C22×C32×C4—24no—
32C22×C32×C4C2×D314yes—
33C22×C32×C4C22×C311yescharacteristic
34C2×C4⋊(C3⋊D3)—26no—
35C2×C4⋊(C3⋊D3)C22×C3112yescomplemented
36C2×C3×D12—348no—
37C2×C3×D12—624no—
38C2×C32×C8—224no—
39C2×C32×C8C2×D3124yescomplemented
40C2×C32×C8C22×C316yescomplemented
41#3745C22×C3116yesdirect product factor
42C3×D24—364no—
43C22×C33D814yescomplemented
44C22×C33C2×D413yes—
45C2×C3×C3⋊D3—28no—
46C2×C3×C3⋊D3—424no—
47C33×C4C2×D413yes—
48C33×C4C2411yescharacteristic
49C22×C3×D4—92no—
50C22×D12—38no—
51C2×C3×D8—912no—
52C22×C3×C8—24no—
53C22×C3×C8C3×D314yescomplemented
54C22×C3×C8C3⋊D311yescharacteristic, complemented
55C2×D24—348no—
56C23×C32—24no—
57C23×C32D1214yes—
58C23×C32C3×D411yescharacteristic
59C22×C3⋊D3—212no—
60C22×C3⋊D3—48no—
61C22×C3×D3—648no—
62C22×C3×D3—1232no—
63C2×C32×C4—212no—
64C2×C32×C4—212no—
65C2×C32×C4D12112yes—
66C2×C32×C4C3×D413yes—
67C2×C32×C4C22×D3112yes—
68C2×C32×C4C23×C313yes—
69C4⋊(C3⋊D3)—212no—
70C4⋊(C3⋊D3)C23×C318yescomplemented
71C3×D12—332no—
72C3×D12—648no—
73C32×C8—216no—
74C32×C8C22×D3116yescomplemented
75C32×C8C23×C314yescomplemented
76C22×D8—91noSylow
77C2×C33C2×D816yescomplemented
78C2×C33C22×D411yescharacteristic
79C3×C3⋊D3—48no—
80C24×C3—182no—
81C23×D3—68no—
82C2×C3×D4—912no—
83C2×C3×D4—186no—
84C22×C3×C4—24no—
85C22×C3×C4C2×C3×D314yes—
86C22×C3×C4C2×C3⋊D311yescharacteristic
87C2×D12—348no—
88C2×D12—624no—
89C3×D8—916no—
90C2×C3×C8—224no—
91C2×C3×C8C2×C3×D3124yescomplemented
92C2×C3×C8C2×C3⋊D316yescomplemented
93D24—364no—
94C22×C32—216no—
95C22×C32—212no—
96C22×C32D24116yescomplemented
97C22×C32C3×D814yescomplemented
98C22×C32C2×D12112yes—
99C22×C32C2×C3×D413yes—
100C2×C3⋊D3—28no—
101C2×C3⋊D3—424no—
102C2×C3×D3—632no—
103C2×C3×D3—1296no—
104C32×C4—24no—
105C32×C4—212no—
106C32×C4C2×D12112yes—
107C32×C4C2×C3×D413yes—
108C32×C4C23×D314yes—
109C32×C4C24×C311yescharacteristic
110C22×D4—92no—
111C2×D8—912no—
112C22×C8C3×C3⋊D311yescharacteristic, complemented
113C33C22×D811yescharacteristic, complemented, Sylow
114C23×C3—24no—
115C23×C3—1812no—
116C23×C3—368no—
117C23×C3C3×D1214yes—
118C23×C3C4⋊(C3⋊D3)11yescharacteristic
119C22×D3—648no—
120C22×D3—1232no—
121C3×D4—98no—
122C3×D4—1812no—
123C2×C3×C4—212no—
124C2×C3×C4—212no—
125C2×C3×C4C3×D12112yes—
126C2×C3×C4C4⋊(C3⋊D3)13yes—
127C2×C3×C4C22×C3×D3112yes—
128C2×C3×C4C22×C3⋊D313yes—
129D12—332no—
130D12—648no—
131C3×C8—216no—
132C3×C8C22×C3×D3116yescomplemented
133C3×C8C22×C3⋊D314yescomplemented
134C2×C32—224no—
135C2×C32—24no—
136C2×C32C2×D24124yescomplemented
137C2×C32C2×C3×D816yescomplemented
138C2×C32C22×D1214yes—
139C2×C32C22×C3×D411yescharacteristic
140C3⋊D3—48no—
141C3×D3—1232no—
142C24—182no—
143C2×D4—912no—
144C2×D4—186no—
145C22×C4C2×C3×C3⋊D311yescharacteristic
146D8—916no—
147C2×C8C2×C3×C3⋊D316yescomplemented
148C22×C3—216no—
149C22×C3—212no—
150C22×C3—188no—
151C22×C3—3624no—
152C22×C3C3×D24116yescomplemented
153C22×C3#374514yesdirect product factor
154C22×C3C2×C3×D12112yes—
155C22×C3C2×C4⋊(C3⋊D3)13yes—
156C2×D3—632no—
157C2×D3—1296no—
158C3×C4—24no—
159C3×C4—212no—
160C3×C4C2×C3×D12112yes—
161C3×C4C2×C4⋊(C3⋊D3)13yes—
162C3×C4C23×C3×D314yes—
163C3×C4C23×C3⋊D311yescharacteristic
164C32—24no—
165C32C22×D2414yescomplemented
166C32C22×C3×D811yescharacteristic, complemented
167C23—1812no—
168C23—368no—
169C23C3×C4⋊(C3⋊D3)11yescharacteristic
170D4—98no—
171D4—1812no—
172C2×C4C3×C4⋊(C3⋊D3)13yes—
173C2×C4C22×C3×C3⋊D313yes—
174C8C22×C3×C3⋊D314yescomplemented
175C2×C3—224no—
176C2×C3—24no—
177C2×C3—368no—
178C2×C3C2×C3×D24124yescomplemented
179C2×C3C2×#374516yesdirect product factor
180C2×C3C22×C3×D1214yes—
181C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
182D3—1232no—
183C22—188no—
184C22—3624no—
185C22C3×#374514yesdirect product factor
186C22C2×C3×C4⋊(C3⋊D3)13yes—
187C4C2×C3×C4⋊(C3⋊D3)13yes—
188C4C23×C3×C3⋊D311yescharacteristic
189C3—24no—
190C3C22×C3×D2414yescomplemented
191C3C22×#374511yescharacteristic, direct product factor
192C2—368no—
193C2C2×C3×#374516yesdirect product factor
194C2C22×C3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC23×C3C4⋊(C3⋊D3)
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC22×C33×C8C2
socleC23×C33D4
hypercenterC22×C3×C8C3⋊D3
hypocenterC32C22×C3×D8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×C8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊(C3⋊D3)C2semidirect
2. C22×C33×C8C2semidirect
3. C2×C3×#3745C2direct (N × H)
4. C22×#3745C3direct (N × H)
9. C2×C3×C4⋊(C3⋊D3)C22semidirect
10. C2×C33×C8C22semidirect
11. C3×#3745C22direct (N × H)
12. C22×C4⋊(C3⋊D3)C2×C3semidirect
15. C22×C32×C8D3semidirect
16. C22×C32×C8C2×C3semidirect
17. C2×#3745C2×C3direct (N × H)
25. C3×C4⋊(C3⋊D3)C23semidirect
26. C33×C8C23semidirect
35. C2×C4⋊(C3⋊D3)C22×C3semidirect
39. C2×C32×C8C2×D3semidirect
40. C2×C32×C8C22×C3semidirect
41. #3745C22×C3direct (N × H)
43. C22×C33D8semidirect
53. C22×C3×C8C3×D3semidirect
54. C22×C3×C8C3⋊D3semidirect
70. C4⋊(C3⋊D3)C23×C3semidirect
74. C32×C8C22×D3semidirect
75. C32×C8C23×C3semidirect
77. C2×C33C2×D8semidirect
91. C2×C3×C8C2×C3×D3semidirect
92. C2×C3×C8C2×C3⋊D3semidirect
96. C22×C32D24semidirect
97. C22×C32C3×D8semidirect
112. C22×C8C3×C3⋊D3semidirect
113. C33C22×D8semidirect
132. C3×C8C22×C3×D3semidirect
133. C3×C8C22×C3⋊D3semidirect
136. C2×C32C2×D24semidirect
137. C2×C32C2×C3×D8semidirect
147. C2×C8C2×C3×C3⋊D3semidirect
152. C22×C3C3×D24semidirect
153. C22×C3#3745direct (N × H)
165. C32C22×D24semidirect
166. C32C22×C3×D8semidirect
174. C8C22×C3×C3⋊D3semidirect
178. C2×C3C2×C3×D24semidirect
179. C2×C3C2×#3745direct (N × H)
185. C22C3×#3745direct (N × H)
190. C3C22×C3×D24semidirect
191. C3C22×#3745direct (N × H)
193. C2C2×C3×#3745direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C22×C33×C82
C2×C3×#37452
C22×#37453
C22×C3×D243