Finite Group Databasebeta

← C2×C3×#66331 #14517695

Subgroups — C2×C3×#66331 #14517695

Order 1944.

400 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 96 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×#731C211yesmaximal, characteristic
2C3×#66331C212yesmaximal, direct product factor
3C2×C3×3−1+2⋊C4—32nomaximal
4C2×C3×(C3⋊Dic3)⋊C3—31nomaximal
5C2×C32×Dic9—31nomaximal
6C2×#66331C311yesmaximal, characteristic, direct product factor
7C2×C3×#731C412yescomplemented
8C2×C3×#731C2211yescharacteristic
9C22×C3×3−1+2D312yes—
10C22×C3×H(3,𝔽3)D311yescharacteristic
11C22×C32×C9D311yescharacteristic
12C3×3−1+2⋊C4—34no—
13C3×(C3⋊Dic3)⋊C3—32no—
14C32×Dic9—32no—
15C22×#731—22no—
16C22×#731—21no—
17C22×#731—21no—
18C22×#731C2×C311yescharacteristic
19#66331C2×C312yesdirect product factor
20C3×#731C2×C411yescharacteristic, complemented, Sylow
21C2×C3×C3⋊Dic3—31no—
22C2×C32×Dic3—91no—
23C2×3−1+2⋊C4—34no—
24C2×3−1+2⋊C4—32no—
25C2×(C3⋊Dic3)⋊C3—32no—
26C2×(C3⋊Dic3)⋊C3—31no—
27C2×C3×Dic9—31no—
28C2×C3×Dic9—32no—
29C2×C3×Dic9—32no—
30C2×C3×Dic9—31no—
31C2×C3×3−1+2Dic314yescomplemented
32C2×C3×3−1+2C2×D312yes—
33C2×C3×H(3,𝔽3)Dic312yes—
34C2×C3×H(3,𝔽3)C2×D311yescharacteristic
35C2×C32×C9Dic312yescomplemented
36C2×C32×C9C2×D311yescharacteristic
37C2×#731—24no—
38C2×#731—22no—
39C2×#731—22no—
40C2×#731—21no—
41C2×#731—22no—
42C2×#731—21no—
43C2×#731C3×C412yescomplemented
44C2×#731C22×C311yescharacteristic
45C22×C33—31no—
46C22×C33C3⋊D311yescharacteristic
47C3×C3⋊Dic3—32no—
48C32×Dic3—92no—
49C22×3−1+2—22no—
50C22×3−1+2—34no—
51C22×3−1+2C3×D312yes—
52C22×H(3,𝔽3)—21no—
53C22×H(3,𝔽3)—32no—
54C22×H(3,𝔽3)C3×D311yescharacteristic
55C22×C3×C9—21no—
56C22×C3×C9—31no—
57C22×C3×C9—32no—
58C22×C3×C9—32no—
59C22×C3×C9C3×D311yescharacteristic
603−1+2⋊C4—38no—
613−1+2⋊C4—34no—
62(C3⋊Dic3)⋊C3—34no—
63(C3⋊Dic3)⋊C3—32no—
64C3×Dic9—32no—
65C3×Dic9—34no—
66C3×Dic9—34no—
67C3×Dic9—32no—
68C3×3−1+2C2×Dic312yescomplemented
69C3×H(3,𝔽3)C2×Dic311yescharacteristic
70C32×C9C2×Dic311yescharacteristic, complemented
71#731—22no—
72#731—21no—
73#731—21no—
74#731C2×C3×C411yescharacteristic, complemented
75C2×C32×C4—271no—
76C2×C3⋊Dic3—31no—
77C2×C3×Dic3—91no—
78C2×C3×Dic3—92no—
79C2×C3×Dic3—91no—
80C2×C3×Dic3—271no—
81C2×Dic9—31no—
82C2×Dic9—32no—
83C2×C33—32no—
84C2×C33—31no—
85C2×C33C3⋊Dic312yes—
86C2×C33C2×C3⋊D311yescharacteristic
87C2×3−1+2—24no—
88C2×3−1+2—22no—
89C2×3−1+2—38no—
90C2×3−1+2—34no—
91C2×3−1+2C3×Dic314yescomplemented
92C2×3−1+2C2×C3×D312yes—
93C2×H(3,𝔽3)—22no—
94C2×H(3,𝔽3)—21no—
95C2×H(3,𝔽3)—34no—
96C2×H(3,𝔽3)—32no—
97C2×H(3,𝔽3)C3×Dic312yes—
98C2×H(3,𝔽3)C2×C3×D311yescharacteristic
99C2×C3×C9—22no—
100C2×C3×C9—21no—
101C2×C3×C9—32no—
102C2×C3×C9—34no—
103C2×C3×C9—34no—
104C2×C3×C9—31no—
105C2×C3×C9—32no—
106C2×C3×C9—32no—
107C2×C3×C9C3×Dic312yescomplemented
108C2×C3×C9C2×C3×D311yescharacteristic
109C22×C32—31no—
110C22×C32—31no—
111C22×C32—61no—
112C22×C32—61no—
113C22×C32—91no—
114C22×C32—181no—
115C22×C32#27211yescharacteristic
116C22×C32#27212yes—
117C22×C32C3×C3⋊D311yescharacteristic
118C32×C4—272no—
119C3⋊Dic3—32no—
120C3×Dic3—92no—
121C3×Dic3—94no—
122C3×Dic3—92no—
123C3×Dic3—272no—
124C22×C9—31no—
125C22×C9—32no—
126C22×C9—61no—
127C22×C9—62no—
128Dic9—32no—
129Dic9—34no—
130C33—31no—
131C33C2×C3⋊Dic311yescharacteristic
1323−1+2—22no—
1333−1+2—34no—
1343−1+2C2×C3×Dic312yescomplemented
135H(3,𝔽3)—21no—
136H(3,𝔽3)—32no—
137H(3,𝔽3)C2×C3×Dic311yescharacteristic
138C3×C9—21no—
139C3×C9—31no—
140C3×C9—32no—
141C3×C9—32no—
142C3×C9C2×C3×Dic311yescharacteristic, complemented
143C2×C3×C4—272no—
144C2×C3×C4—271no—
145C2×C3×C4—271no—
146C2×Dic3—91no—
147C2×Dic3—271no—
148C2×C32—32no—
149C2×C32—31no—
150C2×C32—32no—
151C2×C32—31no—
152C2×C32—62no—
153C2×C32—61no—
154C2×C32—62no—
155C2×C32—61no—
156C2×C32—92no—
157C2×C32—91no—
158C2×C32—182no—
159C2×C32—181no—
160C2×C32#108412yes—
161C2×C32#108414yes—
162C2×C32C2×#27211yescharacteristic
163C2×C32C2×#27212yes—
164C2×C32C3×C3⋊Dic312yes—
165C2×C32C2×C3×C3⋊D311yescharacteristic
166C2×C9—32no—
167C2×C9—34no—
168C2×C9—31no—
169C2×C9—32no—
170C2×C9—62no—
171C2×C9—64no—
172C2×C9—61no—
173C2×C9—62no—
174C22×C3—21no—
175C22×C3—32no—
176C22×C3—31no—
177C22×C3—91no—
178C22×C3—181no—
179C22×C3#418411yescharacteristic
180C22×C3C3×#27211yescharacteristic
181C3×C4—274no—
182C3×C4—272no—
183C3×C4—272no—
184Dic3—92no—
185Dic3—272no—
186C32—31no—
187C32—31no—
188C32—61no—
189C32—61no—
190C32—91no—
191C32—181no—
192C32C2×#108411yescharacteristic
193C32C2×#108412yes—
194C32C2×C3×C3⋊Dic311yescharacteristic
195C9—31no—
196C9—32no—
197C9—61no—
198C9—62no—
199C2×C4—271noSylow
200C2×C3—22no—
201C2×C3—21no—
202C2×C3—34no—
203C2×C3—32no—
204C2×C3—32no—
205C2×C3—31no—
206C2×C3—92no—
207C2×C3—91no—
208C2×C3—182no—
209C2×C3—181no—
210C2×C3#6633112yesdirect product factor
211C2×C3C2×#418411yescharacteristic
212C2×C3C3×#108412yes—
213C2×C3C2×C3×#27211yescharacteristic
214C22C3×#418411yescharacteristic
215C4—272no—
216C3—21no—
217C3—32no—
218C3—31no—
219C3—91no—
220C3—181no—
221C3C2×#6633111yescharacteristic, direct product factor
222C3C2×C3×#108411yescharacteristic
223C2C3×#6633112yesdirect product factor
224C2C2×C3×#418411yescharacteristic
subgroupG/H
center, hypercenterC22×C3#4184
commutator, hypocenter#731C2×C3×C4
FrattiniC2×C32C2×C3×C3⋊D3
FittingC22×C3×#731C2
socleC22×C32#272
prime radicalC22×C3×H(3,𝔽3)D3
cosocleC2×#731C22×C3
LCSG ▷ #731
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×#66331C2direct (N × H)
6. C2×#66331C3direct (N × H)
7. C2×C3×#731C4semidirect
19. #66331C2×C3direct (N × H)
20. C3×#731C2×C4semidirect
31. C2×C3×3−1+2Dic3semidirect
35. C2×C32×C9Dic3semidirect
43. C2×#731C3×C4semidirect
68. C3×3−1+2C2×Dic3semidirect
70. C32×C9C2×Dic3semidirect
74. #731C2×C3×C4semidirect
91. C2×3−1+2C3×Dic3semidirect
107. C2×C3×C9C3×Dic3semidirect
134. 3−1+2C2×C3×Dic3semidirect
142. C3×C9C2×C3×Dic3semidirect
210. C2×C3#66331direct (N × H)
221. C3C2×#66331direct (N × H)
223. C2C3×#66331direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7312
C3×#663312
C2×C3×3−1+2⋊C43
C2×C3×(C3⋊Dic3)⋊C33
C2×C32×Dic93
C2×#663313