Finite Group Databasebeta

← C2×C3⋊(((C22⋊C9)⋊C2)⋊C3) #13049283

Subgroups — C2×C3⋊(((C22⋊C9)⋊C2)⋊C3) #13049283

Order 1296.

424 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 54 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×C3×#1094C211yesmaximal, characteristic, complemented
2C3⋊(((C22⋊C9)⋊C2)⋊C3)C212yesmaximal, direct product factor
3C2×C3×#638—31nomaximal
4C2×C3⋊((C22⋊C9)⋊C2)C311yesmaximal, characteristic, complemented
5C2×((C22⋊C9)⋊C2)⋊C3—33nomaximal
6C2×C3⋊AGL(1,ℤ/9)—41nomaximal
7C3×#1094C2211yescharacteristic, complemented
8C23×C33D311yescharacteristic
9C22×C3×C3⋊D3—31no—
10C3×#638—32no—
11C3×#638—32no—
12C2×C3×C3⋊Dic3—31no—
13C2×#1094—23no—
14C2×#1094D313yescomplemented
15C2×C3×C22⋊C9—21no—
16C2×C3×C22⋊C9C2×C311yescharacteristic, complemented
17C3⋊((C22⋊C9)⋊C2)C2×C312yescomplemented
18((C22⋊C9)⋊C2)⋊C3—36no—
19C2×C3×3−1+2—41no—
20C3⋊AGL(1,ℤ/9)—42no—
21C2×#638—31no—
22C2×C3×#67—93no—
23C2×C3×#67—91no—
24C2×(C22⋊C9)⋊C2—33no—
25C22×C33—31no—
26C22×C33—31no—
27C22×C33C2×D311yescharacteristic
28C2×C3×C3⋊D3—32no—
29C2×C3×C3⋊D3—61no—
30C2×C3×C3⋊D3—61no—
31C3×C3⋊Dic3—32no—
32C2×C3⋊D9—41no—
33C2×AGL(1,ℤ/9)—123no—
34#1094—23no—
35#1094C2×D313yescomplemented
36C3×C22⋊C9—21no—
37C3×C22⋊C9C22×C311yescharacteristic, complemented
38C3×3−1+2—41noSylow
39C23×C32—21no—
40C23×C32—33no—
41C23×C32C3×D311yescharacteristic
42C23×C32C3⋊D311yescharacteristic
43C22×C3⋊D3—31no—
44C22×C3×D3—93no—
45C22×C3×D3—91no—
46#638—32no—
47#638—32no—
48C2×C3⋊Dic3—31no—
49C3×#67—92no—
50C3×#67—96no—
51C3×#67—96no—
52C3×#67—92no—
53C2×C3×Dic3—93no—
54C2×C3×Dic3—91no—
55C2×C22⋊C9—23no—
56C2×C22⋊C9C3×D313yescomplemented
57(C22⋊C9)⋊C2—36no—
58C2×C33—31no—
59C2×C33—31no—
60C2×C33S411yescharacteristic
61C3×C3⋊D3—62no—
62C2×3−1+2—43no—
63C2×3−1+2—83no—
64C2×C3×C9—41no—
65C2×C3×C9—81no—
66C3⋊D9—42no—
67AGL(1,ℤ/9)—126no—
68C2×C3×D4—271no—
69C2×#67—93no—
70C2×#67—91no—
71C22×C32—21no—
72C22×C32—31no—
73C22×C32—31no—
74C22×C32—31no—
75C22×C32—33no—
76C22×C32—31no—
77C22×C32—33no—
78C22×C32—33no—
79C22×C32—61no—
80C22×C32—63no—
81C22×C32—63no—
82C22×C32—61no—
83C22×C32C2×C3×D311yescharacteristic
84C22×C32C2×C3⋊D311yescharacteristic
85C2×C3⋊D3—32no—
86C2×C3⋊D3—61no—
87C2×C3⋊D3—61no—
88C2×C3×D3—96no—
89C2×C3×D3—92no—
90C2×C3×D3—183no—
91C2×C3×D3—181no—
92C2×C3×D3—183no—
93C2×C3×D3—181no—
94C3⋊Dic3—32no—
95C3×Dic3—96no—
96C3×Dic3—92no—
97C2×D9—123no—
98C22⋊C9—23no—
99C22⋊C9C2×C3×D313yescomplemented
100C33C2×S411yescharacteristic
1013−1+2—43no—
1023−1+2—83no—
103C3×C9—41no—
104C3×C9—81no—
105C23×C3—31no—
106C23×C3—61no—
107C23×C3—271no—
108C23×C3AGL(1,ℤ/9)13yescomplemented
109C23×C3C3×C3⋊D311yescharacteristic
110C22×D3—93no—
111C22×D3—91no—
112C3×D4—272no—
113C3×D4—272no—
114C2×C3×C4—271no—
115#67—96no—
116#67—92no—
117#67—96no—
118#67—92no—
119C2×Dic3—93no—
120C2×Dic3—91no—
121C2×C32—21no—
122C2×C32—31no—
123C2×C32—31no—
124C2×C32—33no—
125C2×C32—33no—
126C2×C32—31no—
127C2×C32—31no—
128C2×C32—33no—
129C2×C32—61no—
130C2×C32—61no—
131C2×C32—63no—
132C2×C32—63no—
133C2×C32C3×S411yescharacteristic
134C2×C32C3⋊S411yescharacteristic
135C3⋊D3—62no—
136C3×D3—186no—
137C3×D3—182no—
138C2×C9—43no—
139C2×C9—83no—
140D9—126no—
141C2×D4—271noSylow
142C22×C3—33no—
143C22×C3—31no—
144C22×C3—31no—
145C22×C3—33no—
146C22×C3—31no—
147C22×C3—31no—
148C22×C3—31no—
149C22×C3—61no—
150C22×C3—61no—
151C22×C3—63no—
152C22×C3—63no—
153C22×C3—61no—
154C22×C3—272no—
155C22×C3—541no—
156C22×C3—541no—
157C22×C3C2×AGL(1,ℤ/9)13yescomplemented
158C22×C3C2×C3×C3⋊D311yescharacteristic
159C2×D3—92no—
160C2×D3—96no—
161C2×D3—181no—
162C2×D3—183no—
163C2×D3—183no—
164C2×D3—181no—
165C3×C4—272no—
166Dic3—92no—
167Dic3—96no—
168C32—21no—
169C32—33no—
170C32C2×C3×S411yescharacteristic
171C32C2×C3⋊S411yescharacteristic
172C9—43no—
173C9—83no—
174C23—271no—
175C23C3⋊AGL(1,ℤ/9)11yescharacteristic, complemented
176D4—272no—
177D4—272no—
178C2×C4—271no—
179C2×C3—31no—
180C2×C3—31no—
181C2×C3—31no—
182C2×C3—31no—
183C2×C3—33no—
184C2×C3—33no—
185C2×C3—31no—
186C2×C3—61no—
187C2×C3—61no—
188C2×C3—63no—
189C2×C3—63no—
190C2×C3—61no—
191C2×C3—542no—
192C2×C3((C22⋊C9)⋊C2)⋊C313yescomplemented
193C2×C3C3×C3⋊S411yescharacteristic
194D3—182no—
195D3—186no—
196C22—31no—
197C22—31no—
198C22—272no—
199C22—541no—
200C22—541no—
201C22C2×C3⋊AGL(1,ℤ/9)11yescharacteristic, complemented
202C4—272no—
203C3—31no—
204C3—61no—
205C3C2×((C22⋊C9)⋊C2)⋊C313yescomplemented
206C3C2×C3×C3⋊S411yescharacteristic
207C2—31no—
208C2—31no—
209C2—542no—
210C2C3⋊(((C22⋊C9)⋊C2)⋊C3)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊(((C22⋊C9)⋊C2)⋊C3)
commutator, cosocle, hypocenterC3×C22⋊C9C22×C3
FrattiniC3C2×C3×C3⋊S4
FittingC23×C33D3
socleC23×C32C3×D3
LCSG ▷ C3×C22⋊C9
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1094C2semidirect
2. C3⋊(((C22⋊C9)⋊C2)⋊C3)C2direct (N × H)
4. C2×C3⋊((C22⋊C9)⋊C2)C3semidirect
7. C3×#1094C22semidirect
14. C2×#1094D3semidirect
16. C2×C3×C22⋊C9C2×C3semidirect
17. C3⋊((C22⋊C9)⋊C2)C2×C3semidirect
35. #1094C2×D3semidirect
37. C3×C22⋊C9C22×C3semidirect
56. C2×C22⋊C9C3×D3semidirect
99. C22⋊C9C2×C3×D3semidirect
108. C23×C3AGL(1,ℤ/9)semidirect
157. C22×C3C2×AGL(1,ℤ/9)semidirect
175. C23C3⋊AGL(1,ℤ/9)semidirect
192. C2×C3((C22⋊C9)⋊C2)⋊C3semidirect
201. C22C2×C3⋊AGL(1,ℤ/9)semidirect
205. C3C2×((C22⋊C9)⋊C2)⋊C3semidirect
210. C2C3⋊(((C22⋊C9)⋊C2)⋊C3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#10942
C3⋊(((C22⋊C9)⋊C2)⋊C3)2
C2×C3×#6383
C2×C3⋊((C22⋊C9)⋊C2)3
C2×((C22⋊C9)⋊C2)⋊C33
C2×C3⋊AGL(1,ℤ/9)4