Finite Group Databasebeta

← C32×#5752 #13179129

Subgroups — C32×#5752 #13179129

Order 1728.

1856 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#1009C212yesmaximal, complemented
2C33×#131C211yesmaximal, characteristic, complemented
3C32×#908C216yesmaximal, complemented
4C32×#904C213yesmaximal, complemented
5C2×C32×D24C213yesmaximal, complemented
6C32×#576—31nomaximal
7C3×#5752C314yesmaximal, direct product factor
8C33×#41C2211yescharacteristic
9C32×#239—22no—
10C32×D3×D4—26no—
11C32×#235C2216yescomplemented
12C2×C32×D12C2216yescomplemented
13C33×M16C2213yescomplemented
14C2×C33×C8C2213yescomplemented
15C32×Dic12C2211yescharacteristic, complemented
16C32×D24C2216yescomplemented
17C32×D24C2213yescomplemented
18C32×#204C2216yescomplemented
19C32×H(5,𝔽2)—32no—
20C3×#1009C2×C318yescomplemented
21C32×AGL(1,ℤ/8)—36no—
22C32×#135—33no—
23C2×C32×D8—33no—
24C32×#131—24no—
25C32×#131D311yescharacteristic, complemented
26C32×#131C2×C314yescomplemented
27C3×#908C2×C3124yescomplemented
28C3×#904C2×C3112yescomplemented
29C2×C3×D24C2×C3112yescomplemented
30C22×C32×D3—26no—
31C33×Q8D411yescharacteristic
32C33×D4D413yes—
33C2×C33×C4C2313yes—
34C32×#67—26no—
35C32×D12—26no—
36C32×D12C2316yes—
37C32×C4×D3—26no—
38C32×Dic6C2312yes—
39C33×C8C2311yescharacteristic
40C33×C8C2313yes—
41C3×#576—34no—
42#5752C3211yescharacteristic, direct product factor
43C32×#41—24no—
44C32×#41—36no—
45C32×#41—62no—
46C32×#41C2×D311yescharacteristic
47C32×#41C22×C314yes—
48C2×C32×D4—36no—
49C2×C32×D4—66no—
50C3×#239—28no—
51C3×D3×D4—224no—
52C3×#235C22×C3124yescomplemented
53C2×C3×D12C22×C3124yescomplemented
54C32×Q16—31no—
55C32×SD16—36no—
56C32×D8—33no—
57C32×D8—36no—
58C32×M16—212no—
59C32×M16C2×D313yescomplemented
60C32×M16C22×C3112yescomplemented
61C2×C32×C8—212no—
62C2×C32×C8C2×D313yescomplemented
63C2×C32×C8C22×C3112yescomplemented
64C3×Dic12C22×C314yescomplemented
65C3×D24C22×C3112yescomplemented
66C3×D24C22×C3124yescomplemented
67C3×#204C22×C3124yescomplemented
68C22×C33C2×D413yes—
69C2×C32×D3—26no—
70C2×C32×D3—86no—
71C33×C4C2×D413yes—
72C33×C4C2411yescharacteristic
73C32×Dic3—22no—
74C3×H(5,𝔽2)—38no—
75#1009C2×C3212yescomplemented
76C3×AGL(1,ℤ/8)—324no—
77C3×#135—312no—
78C2×C3×D8—312no—
79C3×#131—24no—
80C3×#131C3×D314yescomplemented
81C3×#131C2×C3211yescharacteristic, complemented
82#908C2×C3216yescomplemented
83#904C2×C3213yescomplemented
84C2×D24C2×C3213yescomplemented
85C23×C32—66no—
86C22×C3×D3—224no—
87C32×Q8—24no—
88C32×Q8—32no—
89C32×Q8D1211yescharacteristic
90C32×Q8C3×D414yes—
91C32×D4—212no—
92C32×D4—36no—
93C32×D4—66no—
94C32×D4—66no—
95C32×D4D1213yes—
96C32×D4C3×D4112yes—
97C2×C32×C4—212no—
98C2×C32×C4—66no—
99C2×C32×C4C22×D313yes—
100C2×C32×C4C23×C3112yes—
101C3×#67—224no—
102C3×D12—224no—
103C3×D12C23×C3124yes—
104C3×C4×D3—224no—
105C3×Dic6C23×C318yes—
106C32×C8—24no—
107C32×C8—212no—
108C32×C8C22×D311yescharacteristic
109C32×C8C22×D313yes—
110C32×C8C23×C3112yes—
111C32×C8C23×C314yes—
112#576—31noSylow
113C2×C33—23no—
114C2×C33C22×D411yescharacteristic
115C32×D3—46no—
116C3×#41—24no—
117C3×#41—324no—
118C3×#41—68no—
119C3×#41C2×C3×D314yes—
120C3×#41C22×C3211yescharacteristic
121C2×C3×D4—324no—
122C2×C3×D4—624no—
123#239—22no—
124D3×D4—26no—
125#235C22×C3216yescomplemented
126C2×D12C22×C3216yescomplemented
127C3×Q16—34no—
128C3×SD16—324no—
129C3×D8—312no—
130C3×D8—324no—
131C3×M16—212no—
132C3×M16C2×C3×D3112yescomplemented
133C3×M16C22×C3213yescomplemented
134C2×C3×C8—212no—
135C2×C3×C8C2×C3×D3112yescomplemented
136C2×C3×C8C22×C3213yescomplemented
137Dic12C22×C3211yescharacteristic, complemented
138D24C22×C3213yescomplemented
139D24C22×C3216yescomplemented
140#204C22×C3216yescomplemented
141C22×C32—212no—
142C22×C32—66no—
143C22×C32—246no—
144C22×C32C2×D1213yes—
145C22×C32C2×C3×D4112yes—
146C2×C3×D3—224no—
147C2×C3×D3—824no—
148C32×C4—24no—
149C32×C4—212no—
150C32×C4—62no—
151C32×C4C2×D1213yes—
152C32×C4C2×C3×D4112yes—
153C32×C4C23×D311yescharacteristic
154C32×C4C24×C314yes—
155C3×Dic3—28no—
156H(5,𝔽2)—32no—
157AGL(1,ℤ/8)—36no—
158#135—33no—
159C2×D8—33no—
160#131C32×D311yescharacteristic, complemented
161C33#57611yescharacteristic, complemented, Sylow
162C23×C3—624no—
163C22×D3—26no—
164C3×Q8—24no—
165C3×Q8—38no—
166C3×Q8C3×D1214yes—
167C3×Q8C32×D411yescharacteristic
168C3×D4—212no—
169C3×D4—324no—
170C3×D4—624no—
171C3×D4—624no—
172C3×D4C3×D12112yes—
173C3×D4C32×D413yes—
174C2×C3×C4—212no—
175C2×C3×C4—624no—
176C2×C3×C4C22×C3×D3112yes—
177C2×C3×C4C23×C3213yes—
178#67—26no—
179D12—26no—
180D12C23×C3216yes—
181C4×D3—26no—
182Dic6C23×C3212yes—
183C3×C8—24no—
184C3×C8—212no—
185C3×C8C22×C3×D314yes—
186C3×C8C22×C3×D3112yes—
187C3×C8C23×C3213yes—
188C3×C8C23×C3211yescharacteristic
189C2×C32—23no—
190C2×C32—24no—
191C2×C32—212no—
192C2×C32—412no—
193C2×C32—126no—
194C2×C32C22×D1211yescharacteristic
195C2×C32C22×C3×D414yes—
196C3×D3—424no—
197#41—36no—
198#41—62no—
199#41C2×C32×D311yescharacteristic
200C2×D4—36no—
201C2×D4—66no—
202Q16—31no—
203SD16—36no—
204D8—33no—
205D8—36no—
206M16C2×C32×D313yescomplemented
207C2×C8C2×C32×D313yescomplemented
208C22×C3—212no—
209C22×C3—624no—
210C22×C3—2424no—
211C22×C3C2×C3×D12112yes—
212C22×C3C2×C32×D413yes—
213C2×D3—26no—
214C2×D3—86no—
215C3×C4—24no—
216C3×C4—212no—
217C3×C4—68no—
218C3×C4C2×C3×D12112yes—
219C3×C4C2×C32×D413yes—
220C3×C4C23×C3×D314yes—
221C3×C4C24×C3211yescharacteristic
222Dic3—22no—
223C32—24no—
224C32#575211yescharacteristic, direct product factor
225C32C3×#57614yescomplemented
226C23—66no—
227Q8—32no—
228Q8C32×D1211yescharacteristic
229D4—36no—
230D4—66no—
231D4—66no—
232D4C32×D1213yes—
233C2×C4—66no—
234C2×C4C22×C32×D313yes—
235C8C22×C32×D311yescharacteristic
236C8C22×C32×D313yes—
237C2×C3—212no—
238C2×C3—23no—
239C2×C3—24no—
240C2×C3—412no—
241C2×C3—1224no—
242C2×C3C22×C3×D1214yes—
243C2×C3C22×C32×D411yescharacteristic
244D3—46no—
245C22—66no—
246C22—246no—
247C22C2×C32×D1213yes—
248C4—62no—
249C4C2×C32×D1213yes—
250C4C23×C32×D311yescharacteristic
251C3—24no—
252C3C3×#575214yesdirect product factor
253C3C32×#57611yescharacteristic, complemented
254C2—23no—
255C2—126no—
256C2C22×C32×D1211yescharacteristic
subgroupG/H
centerC2×C32C22×D12
commutator, cosocleC3×C4C24×C32
FrattiniC4C23×C32×D3
FittingC33×#131C2
socleC2×C33C22×D4
hypercenterC32×#131D3
hypocenterC3C32×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C32×#41 ◁ C32×#131

Complements

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

NH ≅ G/Ntype
1. C32×#1009C2semidirect
2. C33×#131C2semidirect
3. C32×#908C2semidirect
4. C32×#904C2semidirect
5. C2×C32×D24C2semidirect
7. C3×#5752C3direct (N × H)
11. C32×#235C22semidirect
12. C2×C32×D12C22semidirect
13. C33×M16C22semidirect
14. C2×C33×C8C22semidirect
15. C32×Dic12C22semidirect
16. C32×D24C22semidirect
17. C32×D24C22semidirect
18. C32×#204C22semidirect
20. C3×#1009C2×C3semidirect
25. C32×#131D3semidirect
26. C32×#131C2×C3semidirect
27. C3×#908C2×C3semidirect
28. C3×#904C2×C3semidirect
29. C2×C3×D24C2×C3semidirect
42. #5752C32direct (N × H)
52. C3×#235C22×C3semidirect
53. C2×C3×D12C22×C3semidirect
59. C32×M16C2×D3semidirect
60. C32×M16C22×C3semidirect
62. C2×C32×C8C2×D3semidirect
63. C2×C32×C8C22×C3semidirect
64. C3×Dic12C22×C3semidirect
65. C3×D24C22×C3semidirect
66. C3×D24C22×C3semidirect
67. C3×#204C22×C3semidirect
75. #1009C2×C32semidirect
80. C3×#131C3×D3semidirect
81. C3×#131C2×C32semidirect
82. #908C2×C32semidirect
83. #904C2×C32semidirect
84. C2×D24C2×C32semidirect
125. #235C22×C32semidirect
126. C2×D12C22×C32semidirect
132. C3×M16C2×C3×D3semidirect
133. C3×M16C22×C32semidirect
135. C2×C3×C8C2×C3×D3semidirect
136. C2×C3×C8C22×C32semidirect
137. Dic12C22×C32semidirect
138. D24C22×C32semidirect
139. D24C22×C32semidirect
140. #204C22×C32semidirect
160. #131C32×D3semidirect
161. C33#576semidirect
206. M16C2×C32×D3semidirect
207. C2×C8C2×C32×D3semidirect
224. C32#5752direct (N × H)
225. C32C3×#576semidirect
252. C3C3×#5752direct (N × H)
253. C3C32×#576semidirect

Maximal subgroups

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

subgroupindex
C32×#10092
C33×#1312
C32×#9082
C32×#9042
C2×C32×D242
C32×#5763
C3×#57523