Finite Group Databasebeta

← C22×C3×#3755 #13179561

Subgroups — C22×C3×#3755 #13179561

Order 1728.

2412 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C33×Q8C211yesmaximal, characteristic, complemented
2C22×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3C2×C3×#3755C2112yesmaximal, direct product factor
4C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C22×#3755C311yesmaximal, characteristic, direct product factor
6C22×C3×#215—34nomaximal
7C23×C3×C3⋊D3—21no—
8C2×C33×Q8—23no—
9C2×C33×Q8C2216yescomplemented
10C22×C33×C4—21no—
11C22×C33×C4C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)—23no—
13C2×C3×C4⋊(C3⋊D3)C2216yes—
14C3×#3755C22116yesdirect product factor
15C2×C3×C3⋊(C3⋊C8)C2216yescomplemented
16C22×C32×Q8—24no—
17C22×C32×Q8D314yescomplemented
18C22×C32×Q8C2×C311yescharacteristic, complemented
19C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
20C22×C3×D12—34no—
21C2×#3755C2×C3112yesdirect product factor
22C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
23C2×C3×#215—348no—
24C22×C3×C3⋊C8—34no—
25C23×C33D411yescharacteristic
26C22×C3×C3⋊D3—26no—
27C22×C3×C3⋊D3—44no—
28C33×Q8—26no—
29C33×Q8C2314yescomplemented
30C2×C33×C4—26no—
31C2×C33×C4D413yes—
32C2×C33×C4C2313yes—
33C3×C4⋊(C3⋊D3)—26no—
34C3×C4⋊(C3⋊D3)C2314yes—
35C3×C3⋊(C3⋊C8)C2314yescomplemented
36C22×C3×SD16—91no—
37C22×#215—34no—
38C23×C3⋊D3—21no—
39C23×C3×D3—64no—
40C2×C32×Q8—224no—
41C2×C32×Q8—224no—
42C2×C32×Q8—212no—
43C2×C32×Q8—23no—
44C2×C32×Q8C2×D3124yescomplemented
45C2×C32×Q8C22×C316yescomplemented
46C22×C32×C4—28no—
47C22×C32×C4—24no—
48C22×C32×C4—24no—
49C22×C32×C4—21no—
50C22×C32×C4C2×D314yes—
51C22×C32×C4C22×C311yescharacteristic
52C2×C4⋊(C3⋊D3)—23no—
53C2×C4⋊(C3⋊D3)C22×C316yes—
54C2×C3×D12—324no—
55C2×C3×D12—612no—
56#3755C22×C3116yesdirect product factor
57C2×C3⋊(C3⋊C8)C22×C316yescomplemented
58C3×#215—364no—
59C2×C3×C3⋊C8—324no—
60C22×C33SD1614yescomplemented
61C22×C33C2×D413yes—
62C2×C3×C3⋊D3—24no—
63C2×C3×C3⋊D3—412no—
64C33×C4—24no—
65C33×C4C2×D413yes—
66C33×C4C2411yescharacteristic
67C22×C3×Q8—24no—
68C22×C3×Q8C3×D314yescomplemented
69C22×C3×Q8C3⋊D311yescharacteristic, complemented
70C22×C3×D4—91no—
71C22×D12—34no—
72C2×C3×SD16—912no—
73C22×C3×C8—91no—
74C2×#215—348no—
75C22×C3⋊C8—34no—
76C23×C32—24no—
77C23×C32#6714yes—
78C23×C32C3×D411yescharacteristic
79C22×C3⋊D3—26no—
80C22×C3⋊D3—44no—
81C22×C3×D3—624no—
82C22×C3×D3—1216no—
83C32×Q8—248no—
84C32×Q8—224no—
85C32×Q8—26no—
86C32×Q8—216no—
87C32×Q8C22×D3116yescomplemented
88C32×Q8C23×C314yescomplemented
89C2×C32×C4—248no—
90C2×C32×C4—224no—
91C2×C32×C4—26no—
92C2×C32×C4—212no—
93C2×C32×C4—212no—
94C2×C32×C4#67112yes—
95C2×C32×C4C3×D413yes—
96C2×C32×C4C22×D3112yes—
97C2×C32×C4C23×C313yes—
98C4⋊(C3⋊D3)—26no—
99C4⋊(C3⋊D3)C23×C314yes—
100C3×D12—316no—
101C3×D12—624no—
102C3⋊(C3⋊C8)C23×C314yescomplemented
103C3×C3⋊C8—316no—
104C22×SD16—91noSylow
105C2×C33C2×SD1616yescomplemented
106C2×C33C22×D411yescharacteristic
107C3×C3⋊D3—44no—
108C24×C3—181no—
109C23×D3—64no—
110C2×C3×Q8—224no—
111C2×C3×Q8—212no—
112C2×C3×Q8—23no—
113C2×C3×Q8—224no—
114C2×C3×Q8C2×C3×D3124yescomplemented
115C2×C3×Q8C2×C3⋊D316yescomplemented
116C2×C3×D4—96no—
117C2×C3×D4—183no—
118C22×C3×C4—28no—
119C22×C3×C4—24no—
120C22×C3×C4—21no—
121C22×C3×C4—24no—
122C22×C3×C4C2×C3×D314yes—
123C22×C3×C4C2×C3⋊D311yescharacteristic
124C2×D12—324no—
125C2×D12—612no—
126C3×SD16—916no—
127C2×C3×C8—96no—
128#215—364no—
129C2×C3⋊C8—324no—
130C22×C32—216no—
131C22×C32—212no—
132C22×C32#215116yescomplemented
133C22×C32C3×SD1614yescomplemented
134C22×C32C2×#67112yes—
135C22×C32C2×C3×D413yes—
136C2×C3⋊D3—24no—
137C2×C3⋊D3—412no—
138C2×C3×D3—616no—
139C2×C3×D3—1248no—
140C32×C4—232no—
141C32×C4—216no—
142C32×C4—24no—
143C32×C4—24no—
144C32×C4—212no—
145C32×C4C2×#67112yes—
146C32×C4C2×C3×D413yes—
147C32×C4C23×D314yes—
148C32×C4C24×C311yescharacteristic
149C22×Q8C3×C3⋊D311yescharacteristic, complemented
150C22×D4—91no—
151C2×SD16—912no—
152C22×C8—91no—
153C33C22×SD1611yescharacteristic, complemented, Sylow
154C23×C3—24no—
155C23×C3—186no—
156C23×C3—364no—
157C23×C3C3×#6714yes—
158C23×C3#63811yescharacteristic
159C22×D3—624no—
160C22×D3—1216no—
161C3×Q8—248no—
162C3×Q8—224no—
163C3×Q8—26no—
164C3×Q8—216no—
165C3×Q8C22×C3×D3116yescomplemented
166C3×Q8C22×C3⋊D314yescomplemented
167C3×D4—94no—
168C3×D4—186no—
169C2×C3×C4—248no—
170C2×C3×C4—224no—
171C2×C3×C4—26no—
172C2×C3×C4—212no—
173C2×C3×C4—212no—
174C2×C3×C4C3×#67112yes—
175C2×C3×C4#63813yes—
176C2×C3×C4C22×C3×D3112yes—
177C2×C3×C4C22×C3⋊D313yes—
178D12—316no—
179D12—624no—
180C3×C8—94no—
181C3⋊C8—316no—
182C2×C32—224no—
183C2×C32—24no—
184C2×C32C2×#215124yescomplemented
185C2×C32C2×C3×SD1616yescomplemented
186C2×C32C22×#6714yes—
187C2×C32C22×C3×D411yescharacteristic
188C3⋊D3—44no—
189C3×D3—1216no—
190C24—181no—
191C2×Q8—23no—
192C2×Q8C2×C3×C3⋊D316yescomplemented
193C2×D4—96no—
194C2×D4—183no—
195C22×C4—21no—
196C22×C4C2×C3×C3⋊D311yescharacteristic
197SD16—916no—
198C2×C8—96no—
199C22×C3—216no—
200C22×C3—212no—
201C22×C3—184no—
202C22×C3—3612no—
203C22×C3C3×#215116yescomplemented
204C22×C3#375514yesdirect product factor
205C22×C3C2×C3×#67112yes—
206C22×C3C2×#63813yes—
207C2×D3—616no—
208C2×D3—1248no—
209C3×C4—232no—
210C3×C4—216no—
211C3×C4—24no—
212C3×C4—24no—
213C3×C4—212no—
214C3×C4C2×C3×#67112yes—
215C3×C4C2×#63813yes—
216C3×C4C23×C3×D314yes—
217C3×C4C23×C3⋊D311yescharacteristic
218C32—24no—
219C32C22×#21514yescomplemented
220C32C22×C3×SD1611yescharacteristic, complemented
221C23—186no—
222C23—364no—
223C23C3×#63811yescharacteristic
224Q8—26no—
225Q8C22×C3×C3⋊D314yescomplemented
226D4—94no—
227D4—186no—
228C2×C4—26no—
229C2×C4C3×#63813yes—
230C2×C4C22×C3×C3⋊D313yes—
231C8—94no—
232C2×C3—224no—
233C2×C3—24no—
234C2×C3—364no—
235C2×C3C2×C3×#215124yescomplemented
236C2×C3C2×#375516yesdirect product factor
237C2×C3C22×C3×#6714yes—
238C2×C3C22×#63811yescharacteristic
239D3—1216no—
240C22—184no—
241C22—3612no—
242C22C3×#375514yesdirect product factor
243C22C2×C3×#63813yes—
244C4—24no—
245C4C2×C3×#63813yes—
246C4C23×C3×C3⋊D311yescharacteristic
247C3—24no—
248C3C22×C3×#21514yescomplemented
249C3C22×#375511yescharacteristic, direct product factor
250C2—364no—
251C2C2×C3×#375516yesdirect product factor
252C2C22×C3×#63811yescharacteristic
subgroupG/H
centerC23×C3#638
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC22×C33×Q8C2
socleC23×C33D4
prime radicalC22×C3×C4⋊(C3⋊D3)C2
hypercenterC22×C3×Q8C3⋊D3
hypocenterC32C22×C3×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C33×Q8C2semidirect
3. C2×C3×#3755C2direct (N × H)
4. C22×C3×C3⋊(C3⋊C8)C2semidirect
5. C22×#3755C3direct (N × H)
9. C2×C33×Q8C22semidirect
14. C3×#3755C22direct (N × H)
15. C2×C3×C3⋊(C3⋊C8)C22semidirect
17. C22×C32×Q8D3semidirect
18. C22×C32×Q8C2×C3semidirect
21. C2×#3755C2×C3direct (N × H)
22. C22×C3⋊(C3⋊C8)C2×C3semidirect
29. C33×Q8C23semidirect
35. C3×C3⋊(C3⋊C8)C23semidirect
44. C2×C32×Q8C2×D3semidirect
45. C2×C32×Q8C22×C3semidirect
56. #3755C22×C3direct (N × H)
57. C2×C3⋊(C3⋊C8)C22×C3semidirect
60. C22×C33SD16semidirect
68. C22×C3×Q8C3×D3semidirect
69. C22×C3×Q8C3⋊D3semidirect
87. C32×Q8C22×D3semidirect
88. C32×Q8C23×C3semidirect
102. C3⋊(C3⋊C8)C23×C3semidirect
105. C2×C33C2×SD16semidirect
114. C2×C3×Q8C2×C3×D3semidirect
115. C2×C3×Q8C2×C3⋊D3semidirect
132. C22×C32#215semidirect
133. C22×C32C3×SD16semidirect
149. C22×Q8C3×C3⋊D3semidirect
153. C33C22×SD16semidirect
165. C3×Q8C22×C3×D3semidirect
166. C3×Q8C22×C3⋊D3semidirect
184. C2×C32C2×#215semidirect
185. C2×C32C2×C3×SD16semidirect
192. C2×Q8C2×C3×C3⋊D3semidirect
203. C22×C3C3×#215semidirect
204. C22×C3#3755direct (N × H)
219. C32C22×#215semidirect
220. C32C22×C3×SD16semidirect
225. Q8C22×C3×C3⋊D3semidirect
235. C2×C3C2×C3×#215semidirect
236. C2×C3C2×#3755direct (N × H)
242. C22C3×#3755direct (N × H)
248. C3C22×C3×#215semidirect
249. C3C22×#3755direct (N × H)
251. C2C2×C3×#3755direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C33×Q82
C22×C3×C4⋊(C3⋊D3)2
C2×C3×#37552
C22×C3×C3⋊(C3⋊C8)2
C22×#37553
C22×C3×#2153