Finite Group Databasebeta

← #13091513

Subgroups — #13091513

Order 1458.

497 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 61 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
1H(3,𝔽3)×3−1+2C211yesmaximal, characteristic, complemented, Sylow
2C3⋊(C3⋊AGL(1,ℤ/9))C311yesmaximal, characteristic, complemented
3C3×C3⋊AGL(1,ℤ/9)—31nomaximal
4C3×C3⋊((C3⋊D3)⋊C3)—31nomaximal
5#92303C312yesmaximal, complemented
6#92297C311yesmaximal, characteristic, complemented
7#4169⋊C3—32nomaximal
8C32×3−1+2—21no—
9C32×3−1+2D311yescharacteristic, complemented
10C32×3−1+2C2×C311yescharacteristic, complemented
11C32×H(3,𝔽3)D311yescharacteristic
12C9⋊H(3,𝔽3)—22no—
13C9⋊H(3,𝔽3)—22no—
14C9⋊H(3,𝔽3)—21no—
15C9⋊H(3,𝔽3)—24no—
16C9⋊H(3,𝔽3)—22no—
17C9⋊H(3,𝔽3)C2×C312yescomplemented
18#6891—21no—
19#6891—22no—
20#6891D312yescomplemented
21C9×H(3,𝔽3)—21no—
22C9×H(3,𝔽3)C2×C311yescharacteristic, complemented
23C3×C3⋊(C3⋊D3)—31no—
24C32×C3⋊D3—91no—
25C3⋊(C3⋊D9)C3211yescharacteristic, complemented
26C3⋊AGL(1,ℤ/9)—31no—
27C3⋊AGL(1,ℤ/9)—32no—
28C3⋊AGL(1,ℤ/9)—32no—
29C3⋊AGL(1,ℤ/9)—93no—
30C3⋊((C3⋊D3)⋊C3)—31no—
31C3⋊((C3⋊D3)⋊C3)—32no—
32C3×C3⋊D9—31no—
33C3×AGL(1,ℤ/9)—93no—
34C3×(C3⋊D3)⋊C3—93no—
35#4169—32no—
36#4169—34no—
37C34—21no—
38C34C3×D311yescharacteristic
39C34C3⋊D311yescharacteristic
40C3×3−1+2—24no—
41C3×3−1+2—22no—
42C3×3−1+2—21no—
43C3×3−1+2—22no—
44C3×3−1+2—22no—
45C3×3−1+2—22no—
46C3×3−1+2—21no—
47C3×3−1+2—33no—
48C3×3−1+2—33no—
49C3×3−1+2—63no—
50C3×3−1+2—63no—
51C3×3−1+2—66no—
52C3×3−1+2—63no—
53C3×3−1+2C3×D311yescharacteristic
54C3×3−1+2C3×D312yes—
55C3×3−1+2C3×D312yescomplemented
56C3×H(3,𝔽3)—22no—
57C3×H(3,𝔽3)—21no—
58C3×H(3,𝔽3)—33no—
59C3×H(3,𝔽3)—63no—
60C3×H(3,𝔽3)C3×D311yescharacteristic
61C3×H(3,𝔽3)C3×D312yes—
62C32×C9—22no—
63C32×C9—21no—
64C32×C9—21no—
65C32×C9—21no—
66C32×C9C3×D311yescharacteristic, complemented
67C32×C9C2×C3211yescharacteristic, complemented
68C32⋊C9—24no—
69C32⋊C9—24no—
70C32⋊C9—24no—
71C32⋊C9—24no—
72C32⋊C9—22no—
73C32⋊C9—22no—
74C32⋊C9—21no—
75C32⋊C9—22no—
76C32⋊C9—24no—
77C32⋊C9—22no—
78C32⋊C9—24no—
79C32⋊C9C3×D312yescomplemented
80C32⋊C9C3×D314yescomplemented
81C3⋊(C3⋊D3)—31no—
82C3×C3⋊D3—91no—
83C3×C3⋊D3—92no—
84C3×C3⋊D3—91no—
85C3×C3⋊D3—93no—
86C3×C3⋊D3—271no—
87C3×C3⋊D3—272no—
88C32×D3—271no—
89C32×D3—271no—
90C32×D3—272no—
91C3⋊D9—31no—
92C3⋊D9—32no—
93C3⋊D9—93no—
94AGL(1,ℤ/9)—96no—
95AGL(1,ℤ/9)—93no—
96AGL(1,ℤ/9)—272no—
97(C3⋊D3)⋊C3—96no—
98(C3⋊D3)⋊C3—93no—
99C3×D9—93no—
100C33—21no—
101C33—21no—
102C33—22no—
103C33—31no—
104C33—31no—
105C33—31no—
106C33—33no—
107C33—61no—
108C33—61no—
109C33—63no—
110C33—61no—
111C33—62no—
112C33—61no—
113C33—92no—
114C33—92no—
115C33—182no—
116C33C32×D311yescharacteristic
117C33C3×C3⋊D311yescharacteristic
118C33C3×C3⋊D311yescharacteristic
119C33C3×C3⋊D312yes—
1203−1+2—23no—
1213−1+2—32no—
1223−1+2—36no—
1233−1+2—66no—
1243−1+2—66no—
1253−1+2—612no—
1263−1+2—62no—
1273−1+2—62no—
1283−1+2—64no—
1293−1+2—61no—
1303−1+2—62no—
1313−1+2—63no—
1323−1+2—66no—
1333−1+2(C3⋊D3)⋊C313yescomplemented
134H(3,𝔽3)—23no—
135H(3,𝔽3)—36no—
136H(3,𝔽3)—66no—
137H(3,𝔽3)—62no—
138H(3,𝔽3)—61no—
139H(3,𝔽3)AGL(1,ℤ/9)13yescomplemented
140C3×C9—21no—
141C3×C9—22no—
142C3×C9—22no—
143C3×C9—21no—
144C3×C9—24no—
145C3×C9—22no—
146C3×C9—33no—
147C3×C9—33no—
148C3×C9—63no—
149C3×C9—63no—
150C3×C9—66no—
151C3×C9—63no—
152C3×C9C32×D312yes—
153C3×C9C32×D311yescharacteristic
154C2×C32—811no—
155C3⋊D3—91no—
156C3⋊D3—93no—
157C3⋊D3—271no—
158C3⋊D3—272no—
159C3×D3—271no—
160C3×D3—272no—
161C3×D3—272no—
162C3×D3—272no—
163C3×D3—274no—
164C3×D3—271no—
165C3×D3—272no—
166C3×D3—271no—
167C3×D3—271no—
168C3×D3—813no—
169D9—93no—
170D9—272no—
171C32—21no—
172C32—21no—
173C32—23no—
174C32—31no—
175C32—32no—
176C32—31no—
177C32—31no—
178C32—32no—
179C32—32no—
180C32—32no—
181C32—32no—
182C32—34no—
183C32—61no—
184C32—62no—
185C32—61no—
186C32—62no—
187C32—62no—
188C32—61no—
189C32—62no—
190C32—64no—
191C32—91no—
192C32—93no—
193C32—181no—
194C32—181no—
195C32—183no—
196C32—183no—
197C32—182no—
198C32—186no—
199C32C3×AGL(1,ℤ/9)13yescomplemented
200C32C3⋊((C3⋊D3)⋊C3)11yescharacteristic
201C32C3⋊AGL(1,ℤ/9)11yescharacteristic, complemented
202C32C32×C3⋊D311yescharacteristic
203C9—23no—
204C9—32no—
205C9—61no—
206C9—62no—
207C9—62no—
208C9—64no—
209C9—62no—
210C9C3×(C3⋊D3)⋊C313yescomplemented
211C2×C3—811no—
212C2×C3—811no—
213C2×C3—812no—
214D3—271no—
215D3—271no—
216D3—272no—
217D3—813no—
218C3—33no—
219C3—31no—
220C3—31no—
221C3—61no—
222C3—61no—
223C3—63no—
224C3—92no—
225C3—181no—
226C3—182no—
227C3C3×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
228C3C3×C3⋊AGL(1,ℤ/9)11yescharacteristic
229C3#4207⋊C312yes—
230C2—811noSylow
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC32×C9C2×C32
Frattini, socleC32C32×C3⋊D3
FittingH(3,𝔽3)×3−1+2C2
LCSG ▷ C32×C9
UCSC1

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×3−1+2C2semidirect
2. C3⋊(C3⋊AGL(1,ℤ/9))C3semidirect
5. #92303C3semidirect
6. #92297C3semidirect
9. C32×3−1+2D3semidirect
10. C32×3−1+2C2×C3semidirect
17. C9⋊H(3,𝔽3)C2×C3semidirect
20. #6891D3semidirect
22. C9×H(3,𝔽3)C2×C3semidirect
25. C3⋊(C3⋊D9)C32semidirect
55. C3×3−1+2C3×D3semidirect
66. C32×C9C3×D3semidirect
67. C32×C9C2×C32semidirect
79. C32⋊C9C3×D3semidirect
80. C32⋊C9C3×D3semidirect
133. 3−1+2(C3⋊D3)⋊C3semidirect
139. H(3,𝔽3)AGL(1,ℤ/9)semidirect
199. C32C3×AGL(1,ℤ/9)semidirect
201. C32C3⋊AGL(1,ℤ/9)semidirect
210. C9C3×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×3−1+22
C3⋊(C3⋊AGL(1,ℤ/9))3
C3×C3⋊AGL(1,ℤ/9)3
C3×C3⋊((C3⋊D3)⋊C3)3
#923033
#922973
#4169⋊C33