Finite Group Databasebeta

← C2×C3×#626 #89009

Subgroups — C2×C3×#626 #89009

Order 432.

306 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 80 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×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×C32×D3C211yesmaximal, characteristic, complemented
3C2×C32×Dic3C211yesmaximal, characteristic, complemented
4C3×#626C214yesmaximal, direct product factor
5C2×C3×#67—31nomaximal
6C2×C3×D12—31nomaximal
7C2×#626C311yesmaximal, characteristic, direct product factor
8C22×C33C2211yescharacteristic
9C2×C3×C3⋊D3—22no—
10C2×C3×C3⋊D3C2212yescomplemented
11C2×C32×D3—22no—
12C2×C32×D3C2212yescomplemented
13C32×Dic3C2212yescomplemented
14C23×C32—31no—
15C22×C3⋊D3C2×C311yescharacteristic, complemented
16C22×C3×D3—21no—
17C22×C3×D3—31no—
18C22×C3×D3—31no—
19C22×C3×D3—61no—
20C22×C3×D3D311yescharacteristic, complemented
21C22×C3×D3C2×C311yescharacteristic, complemented
22C2×C32×C4—31no—
23C3×#67—34no—
24C2×C3×Dic3—21no—
25C2×C3×Dic3D311yescharacteristic, complemented
26C2×C3×Dic3C2×C311yescharacteristic, complemented
27C3×D12—34no—
28#626C2×C314yesdirect product factor
29C2×C33D412yescomplemented
30C2×C33C2311yescharacteristic
31C3×C3⋊D3—22no—
32C32×D3—22no—
33C2×C3×D4—91no—
34C2×#67—31no—
35C2×D12—31no—
36C22×C32—21no—
37C22×C32—21no—
38C22×C32—21no—
39C22×C32—32no—
40C22×C32—41no—
41C22×C32—62no—
42C22×C32C2×D311yescharacteristic
43C22×C32C2×D311yescharacteristic
44C22×C32C22×C311yescharacteristic
45C2×C3⋊D3—22no—
46C2×C3⋊D3C22×C312yescomplemented
47C2×C3×D3—24no—
48C2×C3×D3—22no—
49C2×C3×D3—22no—
50C2×C3×D3—22no—
51C2×C3×D3—32no—
52C2×C3×D3—32no—
53C2×C3×D3—62no—
54C2×C3×D3—64no—
55C2×C3×D3—62no—
56C2×C3×D3—62no—
57C2×C3×D3C2×D312yescomplemented
58C2×C3×D3C22×C312yescomplemented
59C32×C4—32no—
60C3×Dic3—22no—
61C3×Dic3C2×D312yescomplemented
62C3×Dic3C22×C312yescomplemented
63C33C2×D411yescharacteristic, complemented, Sylow
64C23×C3—31no—
65C23×C3—31no—
66C23×C3—61no—
67C23×C3—91no—
68C22×D3—31no—
69C22×D3—31no—
70C22×D3—61no—
71C22×D3C3×D311yescharacteristic, complemented
72C3×D4—94no—
73C2×C3×C4—31no—
74C2×C3×C4—31no—
75C2×C3×C4—61no—
76#67—34no—
77C2×Dic3C3×D311yescharacteristic, complemented
78D12—34no—
79C2×C32—22no—
80C2×C32—22no—
81C2×C32—22no—
82C2×C32—21no—
83C2×C32—21no—
84C2×C32—21no—
85C2×C32—42no—
86C2×C32—41no—
87C2×C32—62no—
88C2×C32D1212yescomplemented
89C2×C32#6712yescomplemented
90C2×C32C3×D412yescomplemented
91C2×C32C22×D311yescharacteristic
92C2×C32C22×D311yescharacteristic
93C2×C32C23×C311yescharacteristic
94C3⋊D3—22no—
95C3×D3—24no—
96C3×D3—22no—
97C3×D3—22no—
98C3×D3—64no—
99C3×D3—62no—
100C3×D3—62no—
101C2×D4—91noSylow
102C22×C3—21no—
103C22×C3—21no—
104C22×C3—21no—
105C22×C3—32no—
106C22×C3—32no—
107C22×C3—41no—
108C22×C3—62no—
109C22×C3—64no—
110C22×C3—62no—
111C22×C3—62no—
112C22×C3—92no—
113C22×C3—182no—
114C22×C3D3211yescharacteristic
115C22×C3C2×C3×D311yescharacteristic
116C22×C3C2×C3×D311yescharacteristic
117C2×D3—22no—
118C2×D3—32no—
119C2×D3—32no—
120C2×D3—62no—
121C2×D3—64no—
122C2×D3—62no—
123C2×D3—62no—
124C2×D3C2×C3×D312yescomplemented
125C3×C4—32no—
126C3×C4—32no—
127C3×C4—62no—
128Dic3C2×C3×D312yescomplemented
129C32—21no—
130C32—21no—
131C32—21no—
132C32—41no—
133C32C2×D1211yescharacteristic, complemented
134C32C2×#6711yescharacteristic, complemented
135C32C2×C3×D411yescharacteristic, complemented
136C23—31no—
137C23—91no—
138D4—94no—
139C2×C4—31no—
140C2×C3—22no—
141C2×C3—22no—
142C2×C3—22no—
143C2×C3—21no—
144C2×C3—21no—
145C2×C3—21no—
146C2×C3—42no—
147C2×C3—41no—
148C2×C3—64no—
149C2×C3—62no—
150C2×C3—62no—
151C2×C3—182no—
152C2×C3#62612yesdirect product factor
153C2×C3C3×D1212yescomplemented
154C2×C3C3×#6712yescomplemented
155C2×C3C2×D3211yescharacteristic
156C2×C3C22×C3×D311yescharacteristic
157C2×C3C22×C3×D311yescharacteristic
158D3—22no—
159D3—64no—
160D3—62no—
161D3—62no—
162C22—32no—
163C22—62no—
164C22—92no—
165C22—182no—
166C22C3×D3211yescharacteristic
167C4—32no—
168C3—21no—
169C3—21no—
170C3—21no—
171C3—41no—
172C3C2×#62611yescharacteristic, direct product factor
173C3C2×C3×D1211yescharacteristic, complemented
174C3C2×C3×#6711yescharacteristic, complemented
175C2—62no—
176C2—182no—
177C2C3×#62612yesdirect product factor
178C2C2×C3×D3211yescharacteristic
subgroupG/H
center, hypercenterC22×C3D32
commutator, cosocleC2×C32C23×C3
FrattiniC2C2×C3×D32
Fitting, socleC22×C33C22
hypocenterC32C2×C3×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×C3⋊D3C2semidirect
2. C22×C32×D3C2semidirect
3. C2×C32×Dic3C2semidirect
4. C3×#626C2direct (N × H)
7. C2×#626C3direct (N × H)
10. C2×C3×C3⋊D3C22semidirect
12. C2×C32×D3C22semidirect
13. C32×Dic3C22semidirect
15. C22×C3⋊D3C2×C3semidirect
20. C22×C3×D3D3semidirect
21. C22×C3×D3C2×C3semidirect
25. C2×C3×Dic3D3semidirect
26. C2×C3×Dic3C2×C3semidirect
28. #626C2×C3direct (N × H)
29. C2×C33D4semidirect
46. C2×C3⋊D3C22×C3semidirect
57. C2×C3×D3C2×D3semidirect
58. C2×C3×D3C22×C3semidirect
61. C3×Dic3C2×D3semidirect
62. C3×Dic3C22×C3semidirect
63. C33C2×D4semidirect
71. C22×D3C3×D3semidirect
77. C2×Dic3C3×D3semidirect
88. C2×C32D12semidirect
89. C2×C32#67semidirect
90. C2×C32C3×D4semidirect
124. C2×D3C2×C3×D3semidirect
128. Dic3C2×C3×D3semidirect
133. C32C2×D12semidirect
134. C32C2×#67semidirect
135. C32C2×C3×D4semidirect
152. C2×C3#626direct (N × H)
153. C2×C3C3×D12semidirect
154. C2×C3C3×#67semidirect
172. C3C2×#626direct (N × H)
173. C3C2×C3×D12semidirect
174. C3C2×C3×#67semidirect
177. C2C3×#626direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊D32
C22×C32×D32
C2×C32×Dic32
C3×#6262
C2×C3×#673
C2×C3×D123
C2×#6263