Finite Group Databasebeta

← D12×Dic9 #11721022

Subgroups — D12×Dic9 #11721022

Order 864.

308 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×C9×D12C211yesmaximal, characteristic
2C2×D3×Dic9C212yesmaximal, complemented
3C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
4C3×C4×Dic9C211yesmaximal, characteristic, complemented
5(C2×D3)⋊Dic9C212yesmaximal, complemented
6Dic3×D12—31nomaximal
7D4×Dic9—31nomaximal
8C22×D3×C9C2212yes—
9C2×C3×C4×C9C2211yescharacteristic
10C2×C3⋊Dic9C2212yes—
11C2×C3×Dic9C2211yescharacteristic
12C2×C3×Dic9C2211yescharacteristic
13C9×D12C414yescomplemented
14D3×Dic9—24no—
15C2×C3×D12D311yescharacteristic
16C2×D3×Dic3—32no—
17C4⋊(C3⋊Dic3)—31no—
18C3×C4×Dic3—31no—
19(C2×D3)⋊Dic3—32no—
20C2×D4×C9—31no—
21C22×Dic9—32no—
22#3676—32no—
23C4⋊Dic9—31no—
24C4⋊Dic9—61no—
25C4×Dic9D311yescharacteristic, complemented
26C22×C3×C9C2311yescharacteristic
27C2×D3×C9—22no—
28C2×D3×C9—22no—
29C2×D3×C9C2×C414yescomplemented
30C3×C4×C9C2×C412yescomplemented
31C3⋊Dic9—22no—
32C3×Dic9—21no—
33C3×Dic9D412yescomplemented
34D4×Dic3—91no—
35C4×D12—91no—
36C22×C3×D3C2×D312yes—
37C2×C32×C4C2×D311yescharacteristic
38C2×C3⋊Dic3—32no—
39C2×C3×Dic3—31no—
40C2×C3×Dic3—31no—
41C3×D12Dic314yes—
42D3×Dic3—64no—
43C23×C9—32no—
44D4×C9—34no—
45C2×C4×C9—21no—
46C2×C4×C9C2×D311yescharacteristic
47C2×Dic9—32no—
48C2×Dic9—62no—
49C2×Dic9—64no—
50C2×Dic9C2×D311yescharacteristic
51C2×Dic9C2×D311yescharacteristic
52C2×C3×C9C22×C411yescharacteristic
53C2×C3×C9C2×D411yescharacteristic
54C2×C3×C9#4111yescharacteristic
55D3×C9—24no—
56C2×C3×D4—31no—
57C22×Dic3—92no—
58C2×D12D911yescharacteristic
59C2×C4×D3—92no—
60C3×C42—91no—
61#217—92no—
62(C2×D3)⋊C4—92no—
63C4⋊Dic3—91no—
64C4⋊Dic3—91no—
65C4⋊Dic3—181no—
66C4×Dic3—31no—
67C22×C32C22×D311yescharacteristic
68C2×C3×D3—22no—
69C2×C3×D3—22no—
70C2×C3×D3C2×Dic314yes—
71C32×C4C2×Dic312yes—
72C3⋊Dic3—62no—
73C3×Dic3—32no—
74C3×Dic3—61no—
75C22×C9—21no—
76C22×C9—34no—
77C22×C9—62no—
78C22×C9—62no—
79C22×C9C22×D311yescharacteristic
80C4×C9—22no—
81C4×C9C4×D312yescomplemented
82Dic9—21no—
83Dic9—62no—
84Dic9—122no—
85Dic9D1212yesdirect product factor
86C4×D4—271noSylow
87C3×C9C4×D411yescharacteristic, complemented, Sylow
88C23×C3—32no—
89C22×D3C2×D912yes—
90C3×D4—34no—
91C2×C3×C4—21no—
92C2×C3×C4—91no—
93C2×C3×C4—91no—
94C2×C3×C4C2×D911yescharacteristic
95C2×C3×C4D3211yescharacteristic
96C2×Dic3—31no—
97C2×Dic3—31no—
98C2×Dic3—92no—
99C2×Dic3—92no—
100C2×Dic3—182no—
101C2×Dic3—184no—
102D12Dic914yesdirect product factor
103C4×D3—184no—
104C2×C32D3×D411yescharacteristic
105C2×C32#23711yescharacteristic
106C2×C32C22×Dic311yescharacteristic
107C3×D3—24no—
108C2×C9—21no—
109C2×C9—21no—
110C2×C9—21no—
111C2×C9—64no—
112C2×C9C2×C4×D311yescharacteristic
113C2×C9C2×D1211yescharacteristic
114C2×C9#23511yescharacteristic
115C2×D4—31no—
116C22×C4—272no—
117C4⋊C4—271no—
118C22⋊C4—272no—
119C42—91no—
120C22×C3—21no—
121C22×C3—34no—
122C22×C3—62no—
123C22×C3—62no—
124C22×C3C22×D911yescharacteristic
125C22×C3C2×D3211yescharacteristic
126C2×D3—22no—
127C2×D3—22no—
128C2×D3C2×Dic914yescomplemented
129C3×C4—22no—
130C3×C4—92no—
131C3×C4—181no—
132C3×C4C2×Dic912yescomplemented
133C3×C4D3×Dic312yes—
134Dic3—32no—
135Dic3—61no—
136Dic3—182no—
137Dic3—182no—
138Dic3—362no—
139C32D4×Dic311yescharacteristic
140C9—21no—
141C9C4×D1211yescharacteristic, complemented
142C23—32no—
143D4—34no—
144C2×C4—91no—
145C2×C4—91no—
146C2×C4—272no—
147C2×C4—544no—
148C2×C4D3×D911yescharacteristic
149C2×C3—21no—
150C2×C3—21no—
151C2×C3—21no—
152C2×C3—64no—
153C2×C3D4×D911yescharacteristic
154C2×C3#369911yescharacteristic
155C2×C3C22×Dic911yescharacteristic
156C2×C3#379511yescharacteristic
157C2×C3D3×D1211yescharacteristic
158C2×C3C2×D3×Dic311yescharacteristic
159D3—24no—
160C22—34no—
161C22—62no—
162C22—62no—
163C22C2×D3×D911yescharacteristic
164C4—92no—
165C4—181no—
166C4—542no—
167C4D3×Dic912yescomplemented
168C3—21no—
169C3D4×Dic911yescharacteristic, complemented
170C3Dic3×D1211yescharacteristic
171C2—64no—
172C2#8863811yescharacteristic
173C2D9×D1211yescharacteristic
174C2C2×D3×Dic911yescharacteristic
subgroupG/H
centerC22C2×D3×D9
commutatorC2×C3×C9C22×C4
FrattiniC22×C3C2×D32
FittingC2×C3×C4×C9C22
socleC22×C32C22×D3
prime radicalC2×C3×D12D3
cosocleC22×C3×C9C23
hypercenterC2×C4D3×D9
hypocenterC3×C9C4×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×D3×Dic9C2semidirect
3. C4⋊(C3⋊Dic9)C2semidirect
4. C3×C4×Dic9C2semidirect
5. (C2×D3)⋊Dic9C2semidirect
13. C9×D12C4semidirect
25. C4×Dic9D3semidirect
29. C2×D3×C9C2×C4semidirect
30. C3×C4×C9C2×C4semidirect
33. C3×Dic9D4semidirect
81. C4×C9C4×D3semidirect
85. Dic9D12direct (N × H)
87. C3×C9C4×D4semidirect
102. D12Dic9direct (N × H)
128. C2×D3C2×Dic9semidirect
132. C3×C4C2×Dic9semidirect
141. C9C4×D12semidirect
167. C4D3×Dic9semidirect
169. C3D4×Dic9semidirect

Maximal subgroups

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

subgroupindex
C2×C9×D122
C2×D3×Dic92
C4⋊(C3⋊Dic9)2
C3×C4×Dic92
(C2×D3)⋊Dic92
Dic3×D123
D4×Dic93