Finite Group Databasebeta

← C2×C32×#625 #13049617

Subgroups — C2×C32×#625 #13049617

Order 1296.

1348 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 240 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×C33×D3C212yesmaximal, complemented
2C2×C32×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C32×#625C214yesmaximal, direct product factor
4C2×C32×#67—32nomaximal
5C2×C3×#625C314yesmaximal, direct product factor
6C22×C34C2211yescharacteristic
7C2×C33×D3—24no—
8C2×C33×D3C2214yescomplemented
9C32×C3⋊Dic3C2212yescomplemented
10C23×C33—32no—
11C22×C32×D3—28no—
12C22×C32×D3D312yescomplemented
13C22×C32×D3C2×C318yescomplemented
14C2×C3×C3⋊Dic3C2×C314yescomplemented
15C32×#67—38no—
16C2×C32×Dic3—32no—
17C2×C32×Dic3—61no—
18C3×#625C2×C3116yesdirect product factor
19C2×C34D412yescomplemented
20C2×C34C2311yescharacteristic
21C33×D3—24no—
22C2×C32×D4—91no—
23C2×C3×#67—38no—
24C2×#625C3211yescharacteristic, direct product factor
25C22×C33—21no—
26C22×C33—28no—
27C22×C33—34no—
28C22×C33—44no—
29C22×C33—64no—
30C22×C33C2×D312yes—
31C22×C33C22×C314yes—
32C2×C32×D3—232no—
33C2×C32×D3—216no—
34C2×C32×D3—24no—
35C2×C32×D3—216no—
36C2×C32×D3C2×D314yescomplemented
37C2×C32×D3C22×C3116yescomplemented
38C3×C3⋊Dic3C22×C318yescomplemented
39C32×Dic3—34no—
40C32×Dic3—62no—
41C34C2×D411yescharacteristic, complemented, Sylow
42C23×C32—32no—
43C23×C32—38no—
44C23×C32—68no—
45C22×C3×D3—28no—
46C22×C3×D3C3×D318yescomplemented
47C22×C3×D3C2×C3212yescomplemented
48C32×D4—94no—
49C2×C32×C4—91no—
50C2×C3⋊Dic3C2×C3211yescharacteristic, complemented
51C3×#67—332no—
52C2×C3×Dic3—38no—
53C2×C3×Dic3—64no—
54#625C2×C3214yesdirect product factor
55C2×C33—22no—
56C2×C33—216no—
57C2×C33—21no—
58C2×C33—28no—
59C2×C33—48no—
60C2×C33—44no—
61C2×C33—64no—
62C2×C33#6714yescomplemented
63C2×C33C3×D418yescomplemented
64C2×C33C22×D312yes—
65C2×C33C23×C314yes—
66C32×D3—232no—
67C32×D3—24no—
68C32×D3—216no—
69C2×C3×D4—94no—
70C2×#67—32no—
71C22×C32—28no—
72C22×C32—24no—
73C22×C32—28no—
74C22×C32—34no—
75C22×C32—316no—
76C22×C32—44no—
77C22×C32—412no—
78C22×C32—44no—
79C22×C32—616no—
80C22×C32—632no—
81C22×C32—64no—
82C22×C32—616no—
83C22×C32D3211yescharacteristic
84C22×C32C2×C3×D318yes—
85C22×C32C22×C3211yescharacteristic
86C2×C3×D3—232no—
87C2×C3×D3—216no—
88C2×C3×D3—24no—
89C2×C3×D3—216no—
90C2×C3×D3C2×C3×D3116yescomplemented
91C2×C3×D3C22×C3214yescomplemented
92C32×C4—92no—
93C3⋊Dic3C22×C3212yescomplemented
94C3×Dic3—316no—
95C3×Dic3—68no—
96C33—21no—
97C33—28no—
98C33—44no—
99C33C2×#6712yescomplemented
100C33C2×C3×D414yescomplemented
101C23×C3—38no—
102C23×C3—32no—
103C23×C3—68no—
104C22×D3C32×D312yescomplemented
105C3×D4—916no—
106C2×C3×C4—94no—
107#67—38no—
108C2×Dic3—32no—
109C2×Dic3—61no—
110C2×C32—216no—
111C2×C32—28no—
112C2×C32—28no—
113C2×C32—216no—
114C2×C32—24no—
115C2×C32—28no—
116C2×C32—48no—
117C2×C32—44no—
118C2×C32—412no—
119C2×C32—424no—
120C2×C32—48no—
121C2×C32—44no—
122C2×C32—632no—
123C2×C32—64no—
124C2×C32—616no—
125C2×C32#62512yesdirect product factor
126C2×C32C3×#67116yescomplemented
127C2×C32C32×D412yescomplemented
128C2×C32C2×D3211yescharacteristic
129C2×C32C22×C3×D318yes—
130C2×C32C23×C3211yescharacteristic
131C3×D3—232no—
132C3×D3—216no—
133C3×D3—24no—
134C2×D4—91noSylow
135C22×C3—21no—
136C22×C3—28no—
137C22×C3—316no—
138C22×C3—34no—
139C22×C3—44no—
140C22×C3—616no—
141C22×C3—632no—
142C22×C3—616no—
143C22×C3—64no—
144C22×C3C3×D3214yes—
145C22×C3C2×C32×D312yes—
146C2×D3—24no—
147C2×D3C2×C32×D314yescomplemented
148C3×C4—98no—
149Dic3—34no—
150Dic3—62no—
151C32—28no—
152C32—24no—
153C32—28no—
154C32—44no—
155C32—412no—
156C32—44no—
157C32C2×#62511yescharacteristic, direct product factor
158C32C2×C3×#6718yescomplemented
159C32C2×C32×D411yescharacteristic, complemented
160C23—32no—
161D4—94no—
162C2×C4—91no—
163C2×C3—21no—
164C2×C3—28no—
165C2×C3—216no—
166C2×C3—22no—
167C2×C3—44no—
168C2×C3—48no—
169C2×C3—632no—
170C2×C3—616no—
171C2×C3—64no—
172C2×C3C3×#62518yesdirect product factor
173C2×C3C32×#6714yescomplemented
174C2×C3C2×C3×D3214yes—
175C2×C3C22×C32×D312yes—
176D3—24no—
177C22—34no—
178C22—64no—
179C22C32×D3211yescharacteristic
180C4—92no—
181C3—28no—
182C3—21no—
183C3—44no—
184C3C2×C3×#62514yesdirect product factor
185C3C2×C32×#6712yescomplemented
186C2—64no—
187C2C32×#62512yesdirect product factor
188C2C2×C32×D3211yescharacteristic
subgroupG/H
center, hypercenterC22×C32D32
commutator, cosocleC2×C32C23×C32
FrattiniC2C2×C32×D32
Fitting, socleC22×C34C22
hypocenterC32C2×C32×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
1. C22×C33×D3C2semidirect
2. C2×C32×C3⋊Dic3C2semidirect
3. C32×#625C2direct (N × H)
5. C2×C3×#625C3direct (N × H)
8. C2×C33×D3C22semidirect
9. C32×C3⋊Dic3C22semidirect
12. C22×C32×D3D3semidirect
13. C22×C32×D3C2×C3semidirect
14. C2×C3×C3⋊Dic3C2×C3semidirect
18. C3×#625C2×C3direct (N × H)
19. C2×C34D4semidirect
24. C2×#625C32direct (N × H)
36. C2×C32×D3C2×D3semidirect
37. C2×C32×D3C22×C3semidirect
38. C3×C3⋊Dic3C22×C3semidirect
41. C34C2×D4semidirect
46. C22×C3×D3C3×D3semidirect
47. C22×C3×D3C2×C32semidirect
50. C2×C3⋊Dic3C2×C32semidirect
54. #625C2×C32direct (N × H)
62. C2×C33#67semidirect
63. C2×C33C3×D4semidirect
90. C2×C3×D3C2×C3×D3semidirect
91. C2×C3×D3C22×C32semidirect
93. C3⋊Dic3C22×C32semidirect
99. C33C2×#67semidirect
100. C33C2×C3×D4semidirect
104. C22×D3C32×D3semidirect
125. C2×C32#625direct (N × H)
126. C2×C32C3×#67semidirect
127. C2×C32C32×D4semidirect
147. C2×D3C2×C32×D3semidirect
157. C32C2×#625direct (N × H)
158. C32C2×C3×#67semidirect
159. C32C2×C32×D4semidirect
172. C2×C3C3×#625direct (N × H)
173. C2×C3C32×#67semidirect
184. C3C2×C3×#625direct (N × H)
185. C3C2×C32×#67semidirect
187. C2C32×#625direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C33×D32
C2×C32×C3⋊Dic32
C32×#6252
C2×C32×#673
C2×C3×#6253