Finite Group Databasebeta

← #72243458

Subgroups — #72243458

Order 4374.

6211 conjugacy classes of subgroups; only the 638 normal subgroups are shown.

normal subgroups only

This group has 6211 conjugacy classes of subgroups — too many to draw. Only its 638 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
1C3×H(3,𝔽3)×3−1+2C211yesmaximal, characteristic, complemented, Sylow
2C3×C3⋊(C3⋊AGL(1,ℤ/9))—34nomaximal
3C3×C3⋊(C3⋊#272)—31nomaximal
4#13091511—327nomaximal
5C3⋊#92302—38nomaximal
6C3⋊(C9⋊#272)C311yesmaximal, characteristic, complemented
7C33×3−1+2D314yescomplemented
8C33×H(3,𝔽3)D311yescharacteristic
9H(3,𝔽3)×3−1+2—227no—
10H(3,𝔽3)×3−1+2D3127yescomplemented
11C3×C9⋊H(3,𝔽3)—28no—
12C3×C9⋊H(3,𝔽3)—24no—
13C3×#6891D318yescomplemented
14C3×C9×H(3,𝔽3)—21no—
15C3×C9×H(3,𝔽3)C2×C311yescharacteristic, complemented
16C32×C3⋊(C3⋊D3)—94no—
17C3⋊(C3⋊#272)—31no—
18C3⋊(C3⋊AGL(1,ℤ/9))—312no—
19C3×C3⋊#272—927no—
20C3×C3⋊#272—99no—
21C3×C3⋊(C3⋊D9)—34no—
22C3×C3⋊AGL(1,ℤ/9)—98no—
23C3×C3⋊AGL(1,ℤ/9)—91no—
24C3×C3⋊AGL(1,ℤ/9)—936no—
25C9⋊#272—327no—
26C3⋊#4181—38no—
27#92302—972no—
28C35C3⋊D314yes—
29C32×3−1+2—28no—
30C32×3−1+2—24no—
31C32×3−1+2—236no—
32C32×3−1+2—312no—
33C32×3−1+2—636no—
34C32×3−1+2—612no—
35C32×3−1+2C3⋊D311yescharacteristic
36C32×3−1+2C3⋊D318yes—
37C32×3−1+2C3⋊D3136yescomplemented
38C32×H(3,𝔽3)—24no—
39C32×H(3,𝔽3)—29no—
40C32×H(3,𝔽3)—327no—
41C32×H(3,𝔽3)C3×D311yescharacteristic
42C32×H(3,𝔽3)C3⋊D319yes—
43C33×C9—24no—
44C33×C9C3×D314yescomplemented
45C9⋊H(3,𝔽3)—2108no—
46C9⋊H(3,𝔽3)—2216no—
47#6891—272no—
48#6891C3⋊D3172yescomplemented
49C9×H(3,𝔽3)—227no—
50C9×H(3,𝔽3)C3×D3127yescomplemented
51C3×C32⋊C9—28no—
52C3×C32⋊C9—224no—
53C3×C32⋊C9C3×D318yescomplemented
54C3×C3⋊(C3⋊D3)—912no—
55C3×C3⋊(C3⋊D3)—94no—
56C32×C3⋊D3—2736no—
57C32×C3⋊D3—2712no—
58C32×C3⋊D3—271no—
59C3⋊#272—927no—
60C3⋊#272—99no—
61C3⋊(C3⋊D9)—34no—
62C3⋊AGL(1,ℤ/9)—924no—
63C3⋊AGL(1,ℤ/9)—9108no—
64C3⋊AGL(1,ℤ/9)—93no—
65C3×#272—2781no—
66C3×C3⋊D9—98no—
67C3×C3⋊D9—91no—
68C3×C3⋊D9—936no—
69C3×AGL(1,ℤ/9)—2724no—
70C3×AGL(1,ℤ/9)—273no—
71#4181—972no—
72C34—24no—
73C34—212no—
74C34—312no—
75C34—336no—
76C34—612no—
77C34—1812no—
78C34C3×C3⋊D314yes—
79C34C3⋊(C3⋊D3)11yescharacteristic
80C34C3⋊(C3⋊D3)112yes—
81C3×3−1+2—23no—
82C3×3−1+2—29no—
83C3×3−1+2—23no—
84C3×3−1+2—224no—
85C3×3−1+2—272no—
86C3×3−1+2—236no—
87C3×3−1+2—324no—
88C3×3−1+2—3108no—
89C3×3−1+2—624no—
90C3×3−1+2—648no—
91C3×3−1+2—68no—
92C3×3−1+2—624no—
93C3×3−1+2—6108no—
94C3×3−1+2—6108no—
95C3×3−1+2—6216no—
96C3×3−1+2—624no—
97C3×3−1+2—64no—
98C3×3−1+2#27213yescomplemented
99C3×3−1+2C3⋊(C3⋊D3)13yes—
100C3×3−1+2C3⋊(C3⋊D3)124yes—
101C3×H(3,𝔽3)—236no—
102C3×H(3,𝔽3)—381no—
103C3×H(3,𝔽3)—636no—
104C3×H(3,𝔽3)—681no—
105C3×H(3,𝔽3)AGL(1,ℤ/9)127yescomplemented
106C3×H(3,𝔽3)C3×C3⋊D319yes—
107C32×C9—21no—
108C32×C9—28no—
109C32×C9—236no—
110C32×C9—34no—
111C32×C9—612no—
112C32×C9—64no—
113C32×C9—612no—
114C32×C9—624no—
115C32×C9C3×C3⋊D311yescharacteristic
116C32×C9C3×C3⋊D318yes—
117C32×C9C3×C3⋊D3136yescomplemented
118C32⋊C9—272no—
119C32⋊C9—2216no—
120C32⋊C9C3×C3⋊D3172yescomplemented
121C3⋊(C3⋊D3)—94no—
122C3×C3⋊D3—2736no—
123C3×C3⋊D3—27108no—
124C3×C3⋊D3—2736no—
125C3×C3⋊D3—273no—
126C3×C3⋊D3—2712no—
127C3×C3⋊D3—271no—
128C32×D3—8136no—
129C32×D3—813no—
130C32×D3—811no—
131#272—2781no—
132C3⋊D9—98no—
133C3⋊D9—936no—
134C3⋊D9—91no—
135AGL(1,ℤ/9)—2772no—
136AGL(1,ℤ/9)—279no—
137C3×D9—2724no—
138C3×D9—273no—
139C33—23no—
140C33—21no—
141C33—212no—
142C33—34no—
143C33—33no—
144C33—336no—
145C33—336no—
146C33—64no—
147C33—63no—
148C33—612no—
149C33—636no—
150C33—68no—
151C33—624no—
152C33—64no—
153C33—612no—
154C33—636no—
155C33—9108no—
156C33—1812no—
157C33—1824no—
158C33—18108no—
159C33C3⋊AGL(1,ℤ/9)136yescomplemented
160C33C3⋊#27213yes—
161C33C3×C3⋊(C3⋊D3)11yescharacteristic
162C33C3×C3⋊(C3⋊D3)112yes—
163C33C3⋊(C3⋊(C3⋊D3))11yescharacteristic
1643−1+2—218no—
1653−1+2—29no—
1663−1+2—29no—
1673−1+2—372no—
1683−1+2—6216no—
1693−1+2—672no—
1703−1+2—672no—
1713−1+2—636no—
1723−1+2C3⋊#27219yescomplemented
173H(3,𝔽3)—6108no—
174H(3,𝔽3)C3⋊AGL(1,ℤ/9)181yescomplemented
175C3×C9—26no—
176C3×C9—23no—
177C3×C9—21no—
178C3×C9—23no—
179C3×C9—23no—
180C3×C9—224no—
181C3×C9—38no—
182C3×C9—336no—
183C3×C9—636no—
184C3×C9—6108no—
185C3×C9—68no—
186C3×C9—616no—
187C3×C9—68no—
188C3×C9C3×#27211yescharacteristic, complemented
189C3×C9C3×C3⋊(C3⋊D3)13yes—
190C3×C9C3×C3⋊(C3⋊D3)124yes—
191C2×C32—2431no—
192C3⋊D3—2712no—
193C3⋊D3—2736no—
194C3⋊D3—271no—
195C3×D3—819no—
196C3×D3—81108no—
197C3×D3—813no—
198C3×D3—8136no—
199C3×D3—813no—
200C3×D3—811no—
201D9—2724no—
202D9—273no—
203C32—21no—
204C32—23no—
205C32—23no—
206C32—312no—
207C32—31no—
208C32—336no—
209C32—39no—
210C32—61no—
211C32—636no—
212C32—612no—
213C32—69no—
214C32—62no—
215C32—624no—
216C32—61no—
217C32—612no—
218C32—612no—
219C32—9108no—
220C32—1824no—
221C32—18108no—
222C32—1812no—
223C32C3×C3⋊#27211yescharacteristic
224C32C3⋊(C3⋊AGL(1,ℤ/9))13yes—
225C32C3⋊(C3⋊AGL(1,ℤ/9))136yescomplemented
226C32C3⋊(C3⋊#272)13yes—
227C32C3×C3⋊(C3⋊(C3⋊D3))11yescharacteristic
228C9—29no—
229C9—23no—
230C9—324no—
231C9—624no—
232C9C3×C3⋊#27213yescomplemented
233C2×C3—2433no—
234C2×C3—2431no—
235D3—8136no—
236D3—811no—
237D3—813no—
238C3—21no—
239C3—23no—
240C3—336no—
241C3—33no—
242C3—63no—
243C3—1812no—
244C3#1309151113yescomplemented
245C3C3×C3⋊(C3⋊#272)11yescharacteristic
246C3C3⋊(C3⋊(C3⋊AGL(1,ℤ/9)))11yescharacteristic
247C2—2431noSylow
subgroupG/H
center, hypercenterC3C3⋊(C3⋊(C3⋊AGL(1,ℤ/9)))
commutator, cosocle, hypocenterC3×C9×H(3,𝔽3)C2×C3
FrattiniC32C3×C3⋊(C3⋊(C3⋊D3))
FittingC3×H(3,𝔽3)×3−1+2C2
socleC33C3×C3⋊(C3⋊D3)
LCSG ▷ C3×C9×H(3,𝔽3)
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×H(3,𝔽3)×3−1+2C2semidirect
6. C3⋊(C9⋊#272)C3semidirect
7. C33×3−1+2D3semidirect
10. H(3,𝔽3)×3−1+2D3semidirect
13. C3×#6891D3semidirect
15. C3×C9×H(3,𝔽3)C2×C3semidirect
37. C32×3−1+2C3⋊D3semidirect
44. C33×C9C3×D3semidirect
48. #6891C3⋊D3semidirect
50. C9×H(3,𝔽3)C3×D3semidirect
53. C3×C32⋊C9C3×D3semidirect
98. C3×3−1+2#272semidirect
105. C3×H(3,𝔽3)AGL(1,ℤ/9)semidirect
117. C32×C9C3×C3⋊D3semidirect
120. C32⋊C9C3×C3⋊D3semidirect
159. C33C3⋊AGL(1,ℤ/9)semidirect
172. 3−1+2C3⋊#272semidirect
174. H(3,𝔽3)C3⋊AGL(1,ℤ/9)semidirect
188. C3×C9C3×#272semidirect
225. C32C3⋊(C3⋊AGL(1,ℤ/9))semidirect
232. C9C3×C3⋊#272semidirect
244. C3#13091511semidirect

Maximal subgroups

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

subgroupindex
C3×H(3,𝔽3)×3−1+22
C3×C3⋊(C3⋊AGL(1,ℤ/9))3
C3×C3⋊(C3⋊#272)3
#130915113
C3⋊#923023
C3⋊(C9⋊#272)3