Finite Group Databasebeta

← C22×C3×#3832 #13181738

Subgroups — C22×C3×#3832 #13181738

Order 1728.

5964 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C33×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#3832C2124yesmaximal, direct product factor
3C22×C3×C4⋊(C3⋊D3)C213yesmaximal, complemented
4C22×C3×C4×C3⋊D3C213yesmaximal, complemented
5C22×#3832C311yesmaximal, characteristic, direct product factor
6C22×C3×#239—34nomaximal
7C23×C3×C3⋊D3C2213yes—
8C2×C33×Q8C22112yescomplemented
9C22×C33×C4C2213yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C3×#3832C22164yesdirect product factor
12C2×C3×C4⋊(C3⋊D3)C22136yescomplemented
13C2×C3×C4×C3⋊D3C22136yescomplemented
14C22×C32×Q8—24no—
15C22×C32×Q8D314yescomplemented
16C22×C32×Q8C2×C311yescharacteristic, complemented
17C2×#3832C2×C3124yesdirect product factor
18C22×C4⋊(C3⋊D3)C2×C313yescomplemented
19C22×C4×C3⋊D3C2×C313yescomplemented
20C2×C3×#239—396no—
21C22×C3×D12—312no—
22C22×C3×C4×D3—312no—
23C23×C33C2311yescharacteristic
24C22×C3×C3⋊D3—212no—
25C22×C3×C3⋊D3C23118yes—
26C33×Q8C23116yescomplemented
27C2×C33×C4C23118yes—
28C2×C3×C3⋊Dic3C2316yes—
29C3×C4⋊(C3⋊D3)C23148yescomplemented
30C3×C4×C3⋊D3C23148yescomplemented
31C22×C3×#41—91no—
32C22×#239—34no—
33C23×C3⋊D3C22×C313yes—
34C23×C3×D3—312no—
35C2×C32×Q8—248no—
36C2×C32×Q8C2×D3148yescomplemented
37C2×C32×Q8C22×C3112yescomplemented
38C22×C32×C4—212no—
39C22×C32×C4C2×D3112yes—
40C22×C32×C4C22×C313yes—
41C22×C3⋊Dic3C22×C311yescharacteristic
42#3832C22×C3164yesdirect product factor
43C2×C4⋊(C3⋊D3)C22×C3136yescomplemented
44C2×C4×C3⋊D3C22×C3136yescomplemented
45C22×C3×Dic3—34no—
46C3×#239—3256no—
47C2×C3×D12—3144no—
48C2×C3×C4×D3—3144no—
49C22×C33#4114yescomplemented
50C22×C33C2413yes—
51C2×C3×C3⋊D3—236no—
52C2×C3×C3⋊D3C24112yes—
53C33×C4C24112yes—
54C3×C3⋊Dic3C2414yes—
55C2×C3×#41—924no—
56C22×C3×Q8—24no—
57C22×C3×Q8C3×D314yescomplemented
58C22×C3×Q8C3⋊D311yescharacteristic, complemented
59C22×C3×D4—93no—
60C23×C3×C4—93no—
61C2×#239—396no—
62C22×D12—312no—
63C22×C4×D3—312no—
64C23×C32—24no—
65C23×C32C22×D314yes—
66C23×C32C23×C311yescharacteristic
67C22×C3⋊D3—212no—
68C22×C3⋊D3C23×C3118yes—
69C22×C3×D3—372no—
70C22×C3×D3—648no—
71C32×Q8—264no—
72C32×Q8C22×D3164yescomplemented
73C32×Q8C23×C3116yescomplemented
74C2×C32×C4—272no—
75C2×C32×C4C22×D3172yes—
76C2×C32×C4C23×C3118yes—
77C2×C3⋊Dic3C23×C316yes—
78C4⋊(C3⋊D3)C23×C3148yescomplemented
79C4×C3⋊D3C23×C3148yescomplemented
80C2×C3×Dic3—324no—
81C3×D12—3192no—
82C3×C4×D3—3192no—
83C22×#41—91noSylow
84C2×C33C2×#4116yescomplemented
85C2×C33C2511yescharacteristic
86C3×C3⋊D3—212no—
87C24×C3—93no—
88C23×D3—312no—
89C3×#41—964no—
90C2×C3×Q8—248no—
91C2×C3×Q8C2×C3×D3148yescomplemented
92C2×C3×Q8C2×C3⋊D3112yescomplemented
93C2×C3×D4—936no—
94C22×C3×C4—212no—
95C22×C3×C4—91no—
96C22×C3×C4—936no—
97C22×C3×C4C2×C3×D3112yes—
98C22×C3×C4C2×C3⋊D313yes—
99C22×Dic3—34no—
100#239—3256no—
101C2×D12—3144no—
102C2×C4×D3—3144no—
103C22×C32—212no—
104C22×C32—216no—
105C22×C32#239116yescomplemented
106C22×C32C3×#4114yescomplemented
107C22×C32C23×D3112yes—
108C22×C32C24×C313yes—
109C2×C3⋊D3—236no—
110C2×C3⋊D3C24×C3112yes—
111C2×C3×D3—348no—
112C2×C3×D3—6144no—
113C32×C4—248no—
114C32×C4C23×D3148yes—
115C32×C4C24×C3112yes—
116C3⋊Dic3C24×C314yes—
117C3×Dic3—316no—
118C2×#41—924no—
119C22×Q8C3×C3⋊D311yescharacteristic, complemented
120C22×D4—93no—
121C23×C4—93no—
122C33C22×#4111yescharacteristic, complemented, Sylow
123C23×C3—24no—
124C23×C3—918no—
125C23×C3—1812no—
126C23×C3C22×C3×D314yes—
127C23×C3C22×C3⋊D311yescharacteristic
128C22×D3—372no—
129C22×D3—648no—
130C3×Q8—264no—
131C3×Q8C22×C3×D3164yescomplemented
132C3×Q8C22×C3⋊D3116yescomplemented
133C3×D4—948no—
134C2×C3×C4—272no—
135C2×C3×C4—96no—
136C2×C3×C4—948no—
137C2×C3×C4C22×C3×D3172yes—
138C2×C3×C4C22×C3⋊D3118yes—
139C2×Dic3—324no—
140D12—3192no—
141C4×D3—3192no—
142C2×C32—224no—
143C2×C32—24no—
144C2×C32C2×#239124yescomplemented
145C2×C32C2×C3×#4116yescomplemented
146C2×C32C24×D314yes—
147C2×C32C25×C311yescharacteristic
148C3⋊D3—212no—
149C3×D3—648no—
150C24—93no—
151#41—964no—
152C2×Q8C2×C3×C3⋊D3112yescomplemented
153C2×D4—936no—
154C22×C4—91no—
155C22×C4—936no—
156C22×C4C2×C3×C3⋊D313yes—
157C22×C3—212no—
158C22×C3—216no—
159C22×C3—912no—
160C22×C3—1836no—
161C22×C3C3×#239116yescomplemented
162C22×C3#383214yesdirect product factor
163C22×C3C23×C3×D3112yes—
164C22×C3C23×C3⋊D313yes—
165C2×D3—348no—
166C2×D3—6144no—
167C3×C4—248no—
168C3×C4—94no—
169C3×C4C23×C3×D3148yes—
170C3×C4C23×C3⋊D3112yes—
171Dic3—316no—
172C32—24no—
173C32C22×#23914yescomplemented
174C32C22×C3×#4111yescharacteristic, complemented
175C23—918no—
176C23—1812no—
177C23C22×C3×C3⋊D311yescharacteristic
178Q8C22×C3×C3⋊D3116yescomplemented
179D4—948no—
180C2×C4—96no—
181C2×C4—948no—
182C2×C4C22×C3×C3⋊D3118yes—
183C2×C3—224no—
184C2×C3—24no—
185C2×C3—1812no—
186C2×C3C2×C3×#239124yescomplemented
187C2×C3C2×#383216yesdirect product factor
188C2×C3C24×C3×D314yes—
189C2×C3C24×C3⋊D311yescharacteristic
190D3—648no—
191C22—912no—
192C22—1836no—
193C22C3×#383214yesdirect product factor
194C22C23×C3×C3⋊D313yes—
195C4—94no—
196C4C23×C3×C3⋊D3112yes—
197C3—24no—
198C3C22×C3×#23914yescomplemented
199C3C22×#383211yescharacteristic, direct product factor
200C2—1812no—
201C2C2×C3×#383216yesdirect product factor
202C2C24×C3×C3⋊D311yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutator, cosocleC2×C32C25×C3
FrattiniC2C24×C3×C3⋊D3
FittingC22×C33×Q8C2
socleC23×C33C23
hypercenterC22×C3×Q8C3⋊D3
hypocenterC32C22×C3×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C33×Q8C2semidirect
2. C2×C3×#3832C2direct (N × H)
3. C22×C3×C4⋊(C3⋊D3)C2semidirect
4. C22×C3×C4×C3⋊D3C2semidirect
5. C22×#3832C3direct (N × H)
8. C2×C33×Q8C22semidirect
11. C3×#3832C22direct (N × H)
12. C2×C3×C4⋊(C3⋊D3)C22semidirect
13. C2×C3×C4×C3⋊D3C22semidirect
15. C22×C32×Q8D3semidirect
16. C22×C32×Q8C2×C3semidirect
17. C2×#3832C2×C3direct (N × H)
18. C22×C4⋊(C3⋊D3)C2×C3semidirect
19. C22×C4×C3⋊D3C2×C3semidirect
26. C33×Q8C23semidirect
29. C3×C4⋊(C3⋊D3)C23semidirect
30. C3×C4×C3⋊D3C23semidirect
36. C2×C32×Q8C2×D3semidirect
37. C2×C32×Q8C22×C3semidirect
42. #3832C22×C3direct (N × H)
43. C2×C4⋊(C3⋊D3)C22×C3semidirect
44. C2×C4×C3⋊D3C22×C3semidirect
49. C22×C33#41semidirect
57. C22×C3×Q8C3×D3semidirect
58. C22×C3×Q8C3⋊D3semidirect
72. C32×Q8C22×D3semidirect
73. C32×Q8C23×C3semidirect
78. C4⋊(C3⋊D3)C23×C3semidirect
79. C4×C3⋊D3C23×C3semidirect
84. C2×C33C2×#41semidirect
91. C2×C3×Q8C2×C3×D3semidirect
92. C2×C3×Q8C2×C3⋊D3semidirect
105. C22×C32#239semidirect
106. C22×C32C3×#41semidirect
119. C22×Q8C3×C3⋊D3semidirect
122. C33C22×#41semidirect
131. C3×Q8C22×C3×D3semidirect
132. C3×Q8C22×C3⋊D3semidirect
144. C2×C32C2×#239semidirect
145. C2×C32C2×C3×#41semidirect
152. C2×Q8C2×C3×C3⋊D3semidirect
161. C22×C3C3×#239semidirect
162. C22×C3#3832direct (N × H)
173. C32C22×#239semidirect
174. C32C22×C3×#41semidirect
178. Q8C22×C3×C3⋊D3semidirect
186. C2×C3C2×C3×#239semidirect
187. C2×C3C2×#3832direct (N × H)
193. C22C3×#3832direct (N × H)
198. C3C22×C3×#239semidirect
199. C3C22×#3832direct (N × H)
201. C2C2×C3×#3832direct (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
C2×C3×#38322
C22×C3×C4⋊(C3⋊D3)2
C22×C3×C4×C3⋊D32
C22×#38323
C22×C3×#2393