Finite Group Databasebeta

← C3×#13091511 #72243430

Subgroups — C3×#13091511 #72243430

Order 4374.

5476 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 5476 conjugacy classes of subgroups — too many to draw. Only its 266 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
2C32×C3⋊AGL(1,ℤ/9)—34nomaximal
3C32×C3⋊#272—31nomaximal
4#13091511C313yesmaximal, direct product factor
5C3×#92302—38nomaximal
6C3×C9⋊#272C311yesmaximal, characteristic, complemented
7C33×3−1+2D314yescomplemented
8C33×H(3,𝔽3)D311yescharacteristic
9H(3,𝔽3)×3−1+2—224no—
10H(3,𝔽3)×3−1+2—212no—
11H(3,𝔽3)×3−1+2—23no—
12H(3,𝔽3)×3−1+2C2×C313yescomplemented
13C3×C9⋊H(3,𝔽3)—28no—
14C3×C9⋊H(3,𝔽3)—24no—
15C3×#6891D318yescomplemented
16C3×C9×H(3,𝔽3)—21no—
17C3×C9×H(3,𝔽3)C2×C311yescharacteristic, complemented
18C33×C3⋊D3—94no—
19C3×C3⋊#272—31no—
20C3×C3⋊#272—33no—
21C3×C3⋊AGL(1,ℤ/9)—336no—
22C3×C3⋊AGL(1,ℤ/9)—312no—
23C32×#272—99no—
24C32×C3⋊D9—34no—
25C32×AGL(1,ℤ/9)—98no—
26C32×AGL(1,ℤ/9)—91no—
27C9⋊#272C3211yescharacteristic, complemented
28#92302—324no—
29C3×#4181—38no—
30C35C3⋊D314yes—
31C32×3−1+2—212no—
32C32×3−1+2—224no—
33C32×3−1+2—212no—
34C32×3−1+2—28no—
35C32×3−1+2—24no—
36C32×3−1+2—336no—
37C32×3−1+2—636no—
38C32×3−1+2C3×D3112yescomplemented
39C32×3−1+2C3⋊D311yescharacteristic
40C32×3−1+2C3⋊D318yes—
41C32×H(3,𝔽3)—24no—
42C32×H(3,𝔽3)—212no—
43C32×H(3,𝔽3)—39no—
44C32×H(3,𝔽3)—69no—
45C32×H(3,𝔽3)C3×D311yescharacteristic
46C32×H(3,𝔽3)C3×D313yes—
47C33×C9—24no—
48C33×C9C3×D314yescomplemented
49C9⋊H(3,𝔽3)—224no—
50C9⋊H(3,𝔽3)—248no—
51C9⋊H(3,𝔽3)—2144no—
52C9⋊H(3,𝔽3)—224no—
53C9⋊H(3,𝔽3)—248no—
54C9⋊H(3,𝔽3)—212no—
55C9⋊H(3,𝔽3)—224no—
56#6891—224no—
57#6891—272no—
58#6891C3×D3124yescomplemented
59C9×H(3,𝔽3)—224no—
60C9×H(3,𝔽3)—23no—
61C9×H(3,𝔽3)—21no—
62C9×H(3,𝔽3)—24no—
63C9×H(3,𝔽3)—28no—
64C9×H(3,𝔽3)C2×C3211yescharacteristic, complemented
65C3×C32⋊C9—28no—
66C3×C32⋊C9—224no—
67C3×C32⋊C9C3×D318yescomplemented
68C32×C3⋊D3—936no—
69C32×C3⋊D3—94no—
70C32×C3⋊D3—912no—
71C33×D3—2712no—
72C33×D3—271no—
73C3⋊#272—31no—
74C3⋊AGL(1,ℤ/9)—336no—
75C3×#272—927no—
76C3×#272—99no—
77C3×C3⋊D9—312no—
78C3×C3⋊D9—34no—
79C3×AGL(1,ℤ/9)—972no—
80C3×AGL(1,ℤ/9)—99no—
81C3×AGL(1,ℤ/9)—924no—
82C3×AGL(1,ℤ/9)—93no—
83C32×D9—98no—
84C32×D9—91no—
85#4181—38no—
86C34—24no—
87C34—212no—
88C34—312no—
89C34—336no—
90C34—612no—
91C34—1812no—
92C34C3×C3⋊D314yes—
93C34C3×C3⋊D3112yes—
94C34C3⋊(C3⋊D3)11yescharacteristic
95C3×3−1+2—29no—
96C3×3−1+2—23no—
97C3×3−1+2—248no—
98C3×3−1+2—224no—
99C3×3−1+2—224no—
100C3×3−1+2—224no—
101C3×3−1+2—212no—
102C3×3−1+2—372no—
103C3×3−1+2—336no—
104C3×3−1+2—672no—
105C3×3−1+2—624no—
106C3×3−1+2—636no—
107C3×3−1+2—6216no—
108C3×3−1+2—6108no—
109C3×3−1+2—672no—
110C3×3−1+2—636no—
111C3×3−1+2—612no—
112C3×3−1+2#27219yescomplemented
113C3×3−1+2C3×C3⋊D313yes—
114C3×3−1+2C3×C3⋊D3124yes—
115C3×H(3,𝔽3)—24no—
116C3×H(3,𝔽3)—29no—
117C3×H(3,𝔽3)—224no—
118C3×H(3,𝔽3)—212no—
119C3×H(3,𝔽3)—327no—
120C3×H(3,𝔽3)—6108no—
121C3×H(3,𝔽3)—624no—
122C3×H(3,𝔽3)—612no—
123C3×H(3,𝔽3)AGL(1,ℤ/9)19yescomplemented
124C3×H(3,𝔽3)C32×D311yescharacteristic
125C32×C9—21no—
126C32×C9—224no—
127C32×C9—212no—
128C32×C9—24no—
129C32×C9—28no—
130C32×C9—28no—
131C32×C9—24no—
132C32×C9—312no—
133C32×C9—612no—
134C32×C9—636no—
135C32×C9C32×D314yescomplemented
136C32×C9C3×C3⋊D311yescharacteristic
137C32×C9C3×C3⋊D318yes—
138C32⋊C9—248no—
139C32⋊C9—2144no—
140C32⋊C9—224no—
141C32⋊C9—28no—
142C32⋊C9—272no—
143C32⋊C9—224no—
144C32⋊C9C32×D318yescomplemented
145C2×C33—811no—
146C3×C3⋊D3—912no—
147C3×C3⋊D3—936no—
148C3×C3⋊D3—94no—
149C32×D3—2736no—
150C32×D3—27108no—
151C32×D3—2712no—
152C32×D3—279no—
153C32×D3—271no—
154C32×D3—273no—
155#272—99no—
156C3⋊D9—34no—
157AGL(1,ℤ/9)—972no—
158AGL(1,ℤ/9)—99no—
159C3×D9—924no—
160C3×D9—93no—
161C3×D9—98no—
162C3×D9—91no—
163C33—212no—
164C33—24no—
165C33—212no—
166C33—31no—
167C33—312no—
168C33—336no—
169C33—336no—
170C33—61no—
171C33—63no—
172C33—612no—
173C33—636no—
174C33—68no—
175C33—624no—
176C33—636no—
177C33—612no—
178C33—64no—
179C33—9108no—
180C33—1812no—
181C33—1824no—
182C33—18108no—
183C33C3⋊AGL(1,ℤ/9)112yescomplemented
184C33C3⋊#27219yes—
185C33C32×C3⋊D314yes—
186C33C3×C3⋊(C3⋊D3)11yescharacteristic
187C33C3×C3⋊(C3⋊D3)13yes—
1883−1+2—227no—
1893−1+2—29no—
1903−1+2—372no—
1913−1+2—672no—
1923−1+2—6144no—
1933−1+2—672no—
1943−1+2—624no—
1953−1+2—672no—
1963−1+2—612no—
1973−1+2C3×#27219yescomplemented
198H(3,𝔽3)—236no—
199H(3,𝔽3)—672no—
200H(3,𝔽3)—636no—
201H(3,𝔽3)C3×AGL(1,ℤ/9)19yescomplemented
202C3×C9—29no—
203C3×C9—23no—
204C3×C9—23no—
205C3×C9—21no—
206C3×C9—224no—
207C3×C9—28no—
208C3×C9—34no—
209C3×C9—324no—
210C3×C9—64no—
211C3×C9—624no—
212C3×C9—612no—
213C3×C9—636no—
214C3×C9—672no—
215C3×C9—68no—
216C3×C9—624no—
217C3×C9—64no—
218C3×C9C3×#27213yescomplemented
219C3×C9C32×C3⋊D311yescharacteristic
220C3×C9C32×C3⋊D318yes—
221C2×C32—813no—
222C2×C32—819no—
223C2×C32—811no—
224C3⋊D3—94no—
225C3×D3—2736no—
226C3×D3—27108no—
227C3×D3—2712no—
228C3×D3—273no—
229C3×D3—279no—
230C3×D3—271no—
231D9—98no—
232D9—91no—
233C32—21no—
234C32—23no—
235C32—212no—
236C32—34no—
237C32—33no—
238C32—39no—
239C32—336no—
240C32—69no—
241C32—63no—
242C32—64no—
243C32—636no—
244C32—68no—
245C32—624no—
246C32—64no—
247C32—612no—
248C32—612no—
249C32—9108no—
250C32—1824no—
251C32—18108no—
252C32—1812no—
253C32C3×C3⋊AGL(1,ℤ/9)112yescomplemented
254C32C3×C3⋊#27213yes—
255C32C3×C3⋊#27219yes—
256C32C3⋊(C3⋊AGL(1,ℤ/9))11yescharacteristic
257C32C32×C3⋊(C3⋊D3)11yescharacteristic
258C9—29no—
259C9—23no—
260C9—21no—
261C9—38no—
262C9—68no—
263C9—624no—
264C9C32×#27211yescharacteristic, complemented
265C2×C3—819no—
266C2×C3—811no—
267C2×C3—813no—
268D3—2712no—
269D3—271no—
270C3—21no—
271C3—23no—
272C3—312no—
273C3—39no—
274C3—612no—
275C3—1812no—
276C3#1309151113yesdirect product factor
277C3C32×C3⋊#27211yescharacteristic
278C3C3×C3⋊(C3⋊AGL(1,ℤ/9))11yescharacteristic
279C2—811noSylow
subgroupG/H
center, hypercenterC32C3⋊(C3⋊AGL(1,ℤ/9))
commutator, cosocle, hypocenterC9×H(3,𝔽3)C2×C32
FrattiniC32C32×C3⋊(C3⋊D3)
FittingC3×H(3,𝔽3)×3−1+2C2
socleC33C3×C3⋊(C3⋊D3)
LCSG ▷ C9×H(3,𝔽3)
UCSC1 ◁ C32

Complements

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

NH ≅ G/Ntype
1. C3×H(3,𝔽3)×3−1+2C2semidirect
4. #13091511C3direct (N × H)
6. C3×C9⋊#272C3semidirect
7. C33×3−1+2D3semidirect
12. H(3,𝔽3)×3−1+2C2×C3semidirect
15. C3×#6891D3semidirect
17. C3×C9×H(3,𝔽3)C2×C3semidirect
27. C9⋊#272C32semidirect
38. C32×3−1+2C3×D3semidirect
48. C33×C9C3×D3semidirect
58. #6891C3×D3semidirect
64. C9×H(3,𝔽3)C2×C32semidirect
67. C3×C32⋊C9C3×D3semidirect
112. C3×3−1+2#272semidirect
123. C3×H(3,𝔽3)AGL(1,ℤ/9)semidirect
135. C32×C9C32×D3semidirect
144. C32⋊C9C32×D3semidirect
183. C33C3⋊AGL(1,ℤ/9)semidirect
197. 3−1+2C3×#272semidirect
201. H(3,𝔽3)C3×AGL(1,ℤ/9)semidirect
218. C3×C9C3×#272semidirect
253. C32C3×C3⋊AGL(1,ℤ/9)semidirect
264. C9C32×#272semidirect
276. C3#13091511direct (N × H)

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
C32×C3⋊AGL(1,ℤ/9)3
C32×C3⋊#2723
#130915113
C3×#923023
C3×C9⋊#2723