Finite Group Databasebeta

← C2×#10620047 #13049278

Subgroups — C2×#10620047 #13049278

Order 1296.

285 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 42 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×#66437C211yesmaximal, characteristic, complemented
2#10620047C212yesmaximal, direct product factor
3C2×C3⋊((C22⋊C9)⋊C2)C311yesmaximal, characteristic, complemented
4C2×C3×(C22⋊C9)⋊C2—31nomaximal
5C2×((C22⋊C9)⋊C2)⋊C3—32nomaximal
6C2×#638⋊C3—31nomaximal
7C2×#4207—41nomaximal
8#66437C2211yescharacteristic, complemented
9C23×H(3,𝔽3)D311yescharacteristic
10C22×(C3⋊D3)⋊C3—31no—
11C2×#1094—21no—
12C2×#1094—22no—
13C2×#1094D312yescomplemented
14C2×C3×C22⋊C9—21no—
15C2×C3×C22⋊C9D311yescharacteristic, complemented
16C2×C3×C22⋊C9C2×C311yescharacteristic, complemented
17C3⋊((C22⋊C9)⋊C2)C2×C312yescomplemented
18C3×(C22⋊C9)⋊C2—32no—
19((C22⋊C9)⋊C2)⋊C3—34no—
20#638⋊C3—32no—
21#638⋊C3—32no—
22C2×(C3⋊Dic3)⋊C3—31no—
23C2×#737—41no—
24#4207—42no—
25C2×#638—31no—
26C2×C3×#67—91no—
27C2×(C22⋊C9)⋊C2—32no—
28C2×(C22⋊C9)⋊C2—31no—
29C22×H(3,𝔽3)—31no—
30C22×H(3,𝔽3)—31no—
31C22×H(3,𝔽3)C2×D311yescharacteristic
32C2×C3⋊D9—41no—
33C2×AGL(1,ℤ/9)—122no—
34C2×(C3⋊D3)⋊C3—32no—
35C2×(C3⋊D3)⋊C3—61no—
36C2×(C3⋊D3)⋊C3—61no—
37C2×C3×D9—121no—
38#1094—21no—
39#1094—22no—
40#1094C2×D312yescomplemented
41C3×C22⋊C9—21no—
42C3×C22⋊C9C2×D311yescharacteristic, complemented
43C3×C22⋊C9C22×C311yescharacteristic, complemented
44(C3⋊Dic3)⋊C3—32no—
45#737—41noSylow
46C23×C32—21no—
47C23×C32C3×D311yescharacteristic
48C23×C32C3⋊D311yescharacteristic
49C22×C3⋊D3—31no—
50C22×C3×D3—91no—
51#638—32no—
52#638—32no—
53C2×C3⋊Dic3—31no—
54C3×#67—92no—
55C3×#67—92no—
56C2×C3×Dic3—91no—
57C2×C22⋊C9—22no—
58C2×C22⋊C9—21no—
59C2×C22⋊C9C3×D311yescharacteristic
60C2×C22⋊C9C3×D312yes—
61(C22⋊C9)⋊C2—34no—
62(C22⋊C9)⋊C2—32no—
63C2×3−1+2—42no—
64C2×3−1+2—82no—
65C2×3−1+2—81no—
66C2×H(3,𝔽3)—31no—
67C2×H(3,𝔽3)—31no—
68C2×H(3,𝔽3)S411yescharacteristic
69C2×C3×C9—41no—
70C2×C3×C9—41no—
71C2×C3×C9—81no—
72C3⋊D9—42no—
73AGL(1,ℤ/9)—124no—
74(C3⋊D3)⋊C3—62no—
75C3×D9—122no—
76C2×C3×D4—271no—
77C2×#67—91no—
78C2×#67—271no—
79C22×C32—21no—
80C22×C32—31no—
81C22×C32—31no—
82C22×C32—31no—
83C22×C32—31no—
84C22×C32—61no—
85C22×C32—61no—
86C22×C32C2×C3×D311yescharacteristic
87C22×C32C2×C3⋊D311yescharacteristic
88C2×C3⋊D3—32no—
89C2×C3⋊D3—61no—
90C2×C3⋊D3—61no—
91C2×C3×D3—92no—
92C2×C3×D3—181no—
93C2×C3×D3—181no—
94C3⋊Dic3—32no—
95C3×Dic3—92no—
96C2×D9—121no—
97C2×D9—122no—
98C22⋊C9—22no—
99C22⋊C9—21no—
100C22⋊C9C2×C3×D311yescharacteristic
101C22⋊C9C2×C3×D312yes—
1023−1+2—42no—
1033−1+2—82no—
1043−1+2—81no—
105H(3,𝔽3)C2×S411yescharacteristic
106C3×C9—41no—
107C3×C9—41no—
108C3×C9—81no—
109C23×C3—31no—
110C23×C3—31no—
111C23×C3—61no—
112C23×C3—271no—
113C23×C3C3×C3⋊D311yescharacteristic
114C22×D3—91no—
115C22×D3—271no—
116C3×D4—272no—
117C3×D4—272no—
118C2×C3×C4—271no—
119#67—92no—
120#67—92no—
121#67—272no—
122#67—272no—
123C2×Dic3—91no—
124C2×Dic3—271no—
125C2×C32—21no—
126C2×C32—31no—
127C2×C32—31no—
128C2×C32—31no—
129C2×C32—31no—
130C2×C32—61no—
131C2×C32—61no—
132C2×C32C3×S411yescharacteristic
133C2×C32C3⋊S411yescharacteristic
134C3⋊D3—62no—
135C3×D3—182no—
136C2×C9—41no—
137C2×C9—42no—
138C2×C9—81no—
139C2×C9—82no—
140D9—122no—
141D9—124no—
142C2×D4—271noSylow
143C22×C3—31no—
144C22×C3—31no—
145C22×C3—31no—
146C22×C3—31no—
147C22×C3—61no—
148C22×C3—91no—
149C22×C3—91no—
150C22×C3—91no—
151C22×C3—91no—
152C22×C3—181no—
153C22×C3—181no—
154C22×C3—272no—
155C22×C3—541no—
156C22×C3—541no—
157C22×C3C2×C3×C3⋊D311yescharacteristic
158C2×D3—92no—
159C2×D3—181no—
160C2×D3—181no—
161C2×D3—272no—
162C2×D3—541no—
163C2×D3—541no—
164C3×C4—272no—
165Dic3—92no—
166Dic3—272no—
167C32—21no—
168C32C2×C3×S411yescharacteristic
169C32C2×C3⋊S411yescharacteristic
170C9—41no—
171C9—42no—
172C9—81no—
173C9—82no—
174C23—271no—
175C23#420711yescharacteristic, complemented
176D4—272no—
177D4—272no—
178C2×C4—271no—
179C2×C3—31no—
180C2×C3—31no—
181C2×C3—31no—
182C2×C3—31no—
183C2×C3—61no—
184C2×C3—91no—
185C2×C3—91no—
186C2×C3—91no—
187C2×C3—91no—
188C2×C3—181no—
189C2×C3—181no—
190C2×C3—542no—
191C2×C3C3×C3⋊S411yescharacteristic
192D3—182no—
193D3—542no—
194C22—31no—
195C22—31no—
196C22—272no—
197C22—541no—
198C22—541no—
199C22C2×#420711yescharacteristic, complemented
200C4—272no—
201C3—31no—
202C3—31no—
203C3—61no—
204C3C2×C3×C3⋊S411yescharacteristic
205C2—31no—
206C2—31no—
207C2—542no—
208C2#1062004711yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#10620047
commutator, cosocle, hypocenterC3×C22⋊C9C22×C3
FrattiniC3C2×C3×C3⋊S4
FittingC23×H(3,𝔽3)D3
socleC23×C3C3×C3⋊D3
LCSG ▷ C3×C22⋊C9
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#66437C2semidirect
2. #10620047C2direct (N × H)
3. C2×C3⋊((C22⋊C9)⋊C2)C3semidirect
8. #66437C22semidirect
13. C2×#1094D3semidirect
15. C2×C3×C22⋊C9D3semidirect
16. C2×C3×C22⋊C9C2×C3semidirect
17. C3⋊((C22⋊C9)⋊C2)C2×C3semidirect
40. #1094C2×D3semidirect
42. C3×C22⋊C9C2×D3semidirect
43. C3×C22⋊C9C22×C3semidirect
175. C23#4207semidirect
199. C22C2×#4207semidirect
208. C2#10620047direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#664372
#106200472
C2×C3⋊((C22⋊C9)⋊C2)3
C2×C3×(C22⋊C9)⋊C23
C2×((C22⋊C9)⋊C2)⋊C33
C2×#638⋊C33
C2×#42074