Finite Group Databasebeta

← C2×C3×C3⋊((C22⋊C9)⋊C2) #13049282

Subgroups — C2×C3×C3⋊((C22⋊C9)⋊C2) #13049282

Order 1296.

448 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 70 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×C32×C22⋊C9C211yesmaximal, characteristic, complemented
2C3×C3⋊((C22⋊C9)⋊C2)C212yesmaximal, direct product factor
3C2×C3×#638—31nomaximal
4C2×C3⋊((C22⋊C9)⋊C2)C311yesmaximal, characteristic, direct product factor
5C2×C3×(C22⋊C9)⋊C2—33nomaximal
6C2×C3×C3⋊D9—41nomaximal
7C32×C22⋊C9C2211yescharacteristic, complemented
8C23×C33D311yescharacteristic
9C22×C3×C3⋊D3—31no—
10C3×#638—32no—
11C3×#638—32no—
12C2×C3×C3⋊Dic3—31no—
13C2×C3×C22⋊C9—21no—
14C2×C3×C22⋊C9—23no—
15C2×C3×C22⋊C9D313yescomplemented
16C2×C3×C22⋊C9C2×C311yescharacteristic, complemented
17C3⋊((C22⋊C9)⋊C2)C2×C312yesdirect product factor
18C3×(C22⋊C9)⋊C2—36no—
19C2×C32×C9—41no—
20C3×C3⋊D9—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×C3×D9—123no—
34C3×C22⋊C9—21no—
35C3×C22⋊C9—23no—
36C3×C22⋊C9C2×D313yescomplemented
37C3×C22⋊C9C22×C311yescharacteristic, complemented
38C32×C9—41noSylow
39C23×C32—23no—
40C23×C32—21no—
41C23×C32D913yescomplemented
42C23×C32C3×D311yescharacteristic
43C23×C32C3⋊D311yescharacteristic
44C22×C3⋊D3—31no—
45C22×C3×D3—93no—
46C22×C3×D3—91no—
47#638—32no—
48#638—32no—
49C2×C3⋊Dic3—31no—
50C3×#67—96no—
51C3×#67—92no—
52C3×#67—96no—
53C3×#67—92no—
54C2×C3×Dic3—93no—
55C2×C3×Dic3—91no—
56C2×C22⋊C9—23no—
57C2×C22⋊C9C3×D313yescomplemented
58(C22⋊C9)⋊C2—36no—
59C2×C33—31no—
60C2×C33—31no—
61C2×C33S411yescharacteristic
62C3×C3⋊D3—62no—
63C2×C3×C9—41no—
64C2×C3×C9—43no—
65C2×C3×C9—83no—
66C2×C3×C9—81no—
67C3⋊D9—42no—
68C3×D9—126no—
69C2×C3×D4—271no—
70C2×#67—93no—
71C2×#67—91no—
72C22×C32—23no—
73C22×C32—21no—
74C22×C32—31no—
75C22×C32—31no—
76C22×C32—33no—
77C22×C32—31no—
78C22×C32—33no—
79C22×C32—31no—
80C22×C32—63no—
81C22×C32—61no—
82C22×C32—63no—
83C22×C32—61no—
84C22×C32C2×D913yescomplemented
85C22×C32C2×C3×D311yescharacteristic
86C22×C32C2×C3⋊D311yescharacteristic
87C2×C3⋊D3—32no—
88C2×C3⋊D3—61no—
89C2×C3⋊D3—61no—
90C2×C3×D3—92no—
91C2×C3×D3—96no—
92C2×C3×D3—181no—
93C2×C3×D3—183no—
94C2×C3×D3—183no—
95C2×C3×D3—181no—
96C3⋊Dic3—32no—
97C3×Dic3—92no—
98C3×Dic3—96no—
99C2×D9—123no—
100C22⋊C9—23no—
101C22⋊C9C2×C3×D313yescomplemented
102C33C2×S411yescharacteristic
103C3×C9—41no—
104C3×C9—43no—
105C3×C9—83no—
106C3×C9—81no—
107C23×C3—23no—
108C23×C3—21no—
109C23×C3—271no—
110C23×C3C3×D913yescomplemented
111C23×C3C3⋊D911yescharacteristic, complemented
112C23×C3C3×C3⋊D311yescharacteristic
113C22×D3—93no—
114C22×D3—91no—
115C3×D4—272no—
116C3×D4—272no—
117C2×C3×C4—271no—
118#67—96no—
119#67—92no—
120#67—96no—
121#67—92no—
122C2×Dic3—93no—
123C2×Dic3—91no—
124C2×C32—23no—
125C2×C32—21no—
126C2×C32—31no—
127C2×C32—31no—
128C2×C32—31no—
129C2×C32—31no—
130C2×C32—33no—
131C2×C32—33no—
132C2×C32—61no—
133C2×C32—63no—
134C2×C32—61no—
135C2×C32—63no—
136C2×C32(C22⋊C9)⋊C213yescomplemented
137C2×C32C3×S411yescharacteristic
138C2×C32C3⋊S411yescharacteristic
139C3⋊D3—62no—
140C3×D3—182no—
141C3×D3—186no—
142C2×C9—43no—
143C2×C9—83no—
144D9—126no—
145C2×D4—271noSylow
146C22×C3—23no—
147C22×C3—21no—
148C22×C3—33no—
149C22×C3—31no—
150C22×C3—31no—
151C22×C3—33no—
152C22×C3—31no—
153C22×C3—31no—
154C22×C3—63no—
155C22×C3—61no—
156C22×C3—63no—
157C22×C3—61no—
158C22×C3—272no—
159C22×C3—541no—
160C22×C3—541no—
161C22×C3C2×C3×D913yescomplemented
162C22×C3C2×C3⋊D911yescharacteristic, complemented
163C22×C3C2×C3×C3⋊D311yescharacteristic
164C2×D3—92no—
165C2×D3—96no—
166C2×D3—181no—
167C2×D3—183no—
168C2×D3—183no—
169C2×D3—181no—
170C3×C4—272no—
171Dic3—92no—
172Dic3—96no—
173C32—21no—
174C32—23no—
175C32C2×(C22⋊C9)⋊C213yescomplemented
176C32C2×C3×S411yescharacteristic
177C32C2×C3⋊S411yescharacteristic
178C9—43no—
179C9—83no—
180C23—271no—
181C23C3×C3⋊D911yescharacteristic, complemented
182D4—272no—
183D4—272no—
184C2×C4—271no—
185C2×C3—23no—
186C2×C3—21no—
187C2×C3—31no—
188C2×C3—31no—
189C2×C3—33no—
190C2×C3—33no—
191C2×C3—31no—
192C2×C3—31no—
193C2×C3—61no—
194C2×C3—63no—
195C2×C3—61no—
196C2×C3—63no—
197C2×C3—542no—
198C2×C3C3×(C22⋊C9)⋊C213yescomplemented
199C2×C3C3⋊((C22⋊C9)⋊C2)11yescharacteristic, direct product factor
200C2×C3C3×C3⋊S411yescharacteristic
201D3—182no—
202D3—186no—
203C22—31no—
204C22—31no—
205C22—272no—
206C22—541no—
207C22—541no—
208C22C2×C3×C3⋊D911yescharacteristic, complemented
209C4—272no—
210C3—21no—
211C3—23no—
212C3C2×C3×(C22⋊C9)⋊C213yescomplemented
213C3C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic, direct product factor
214C3C2×C3×C3⋊S411yescharacteristic
215C2—31no—
216C2—31no—
217C2—542no—
218C2C3×C3⋊((C22⋊C9)⋊C2)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C3⋊((C22⋊C9)⋊C2)
commutator, cosocle, hypocenterC3×C22⋊C9C22×C3
FrattiniC3C2×C3×C3⋊S4
Fitting, socleC23×C33D3
LCSG ▷ C3×C22⋊C9
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C32×C22⋊C9C2semidirect
2. C3×C3⋊((C22⋊C9)⋊C2)C2direct (N × H)
4. C2×C3⋊((C22⋊C9)⋊C2)C3direct (N × H)
7. C32×C22⋊C9C22semidirect
15. C2×C3×C22⋊C9D3semidirect
16. C2×C3×C22⋊C9C2×C3semidirect
17. C3⋊((C22⋊C9)⋊C2)C2×C3direct (N × H)
36. C3×C22⋊C9C2×D3semidirect
37. C3×C22⋊C9C22×C3semidirect
41. C23×C32D9semidirect
57. C2×C22⋊C9C3×D3semidirect
84. C22×C32C2×D9semidirect
101. C22⋊C9C2×C3×D3semidirect
110. C23×C3C3×D9semidirect
111. C23×C3C3⋊D9semidirect
136. C2×C32(C22⋊C9)⋊C2semidirect
161. C22×C3C2×C3×D9semidirect
162. C22×C3C2×C3⋊D9semidirect
175. C32C2×(C22⋊C9)⋊C2semidirect
181. C23C3×C3⋊D9semidirect
198. C2×C3C3×(C22⋊C9)⋊C2semidirect
199. C2×C3C3⋊((C22⋊C9)⋊C2)direct (N × H)
208. C22C2×C3×C3⋊D9semidirect
212. C3C2×C3×(C22⋊C9)⋊C2semidirect
213. C3C2×C3⋊((C22⋊C9)⋊C2)direct (N × H)
218. C2C3×C3⋊((C22⋊C9)⋊C2)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊C92
C3×C3⋊((C22⋊C9)⋊C2)2
C2×C3×#6383
C2×C3⋊((C22⋊C9)⋊C2)3
C2×C3×(C22⋊C9)⋊C23
C2×C3×C3⋊D94