Finite Group Databasebeta

← C3×#13049364 #72210855

Subgroups — C3×#13049364 #72210855

Order 3888.

324 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 14 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×SL(2,ℤ/9)C211yesmaximal, characteristic, complemented
2C3×#88942—31nomaximal
3#13049364C311yesmaximal, characteristic, direct product factor
4C2×C3×#4183—41nomaximal
5C3×#6296D311yescharacteristic, complemented
6C3×#6295—31no—
7C32×#622—31no—
8SL(2,ℤ/9)—21no—
9SL(2,ℤ/9)C2×C311yescharacteristic, complemented
10C2×C3×AΣL(1,𝔽27)—41no—
11C3×#4183—42no—
12#88942—31no—
13C3×#3775—91no—
14C2×C32×C3⋊D3—61no—
15C32×C3⋊Dic3—31no—
16C2×C3×AGL(1,ℤ/9)—122no—
17C2×C3×(C3⋊D3)⋊C3—121no—
18C2×#4183—41no—
19C3×AΣL(1,𝔽27)—41noSylow
20#6296C3×D311yescharacteristic, complemented
21#6295—31no—
22C3×#627—91no—
23C3×#626—181no—
24C3×#625—91no—
25C3×#622—31no—
26C3×#622—31no—
27C3×#622—61no—
28C2×C34S411yescharacteristic
29C32×C3⋊D3—121no—
30C2×C3×3−1+2—42no—
31C2×C3×H(3,𝔽3)—41no—
32C3×AGL(1,ℤ/9)—124no—
33C3×(C3⋊D3)⋊C3—122no—
34C2×AΣL(1,𝔽27)—41no—
35C2×AΣL(1,𝔽27)—81no—
36C2×AΣL(1,𝔽27)—82no—
37C2×AΣL(1,𝔽27)—81no—
38#4183—42no—
39C3×GL(2,𝔽3)—271nomaximal
40#3775—91no—
41C3×#204—271no—
42C2×C3×C3⋊D3—61no—
43C2×C3×C3⋊D3—61no—
44C2×C3×C3⋊D3—121no—
45C2×C3×C3⋊D3—121no—
46C2×C32×D3—181no—
47C2×C32×D3—181no—
48C2×C32×D3—361no—
49C3×C3⋊Dic3—31no—
50C3×C3⋊Dic3—31no—
51C3×C3⋊Dic3—61no—
52C32×Dic3—181no—
53C32×Dic3—181no—
54C2×AGL(1,ℤ/9)—122no—
55C2×AGL(1,ℤ/9)—124no—
56C2×(C3⋊D3)⋊C3—121no—
57C2×(C3⋊D3)⋊C3—122no—
58C2×C3×D9—122no—
59C34GL(2,𝔽3)11yescharacteristic, complemented
60C3×3−1+2—42no—
61C3×H(3,𝔽3)—41no—
62AΣL(1,𝔽27)—41no—
63AΣL(1,𝔽27)—81no—
64AΣL(1,𝔽27)—82no—
65AΣL(1,𝔽27)—81no—
66C3×#67—541no—
67C3×D12—271no—
68C3×Dic6—271no—
69C3×SL(2,𝔽3)—271no—
70#627—91no—
71#626—181no—
72#625—91no—
73#622—31no—
74C32×C8—271no—
75C2×C33—31no—
76C2×C33—41no—
77C2×C33—61no—
78C2×C33—61no—
79C2×C33—81no—
80C2×C33—121no—
81C2×C33—121no—
82C2×C33C3×S411yescharacteristic
83C3×C3⋊D3—121no—
84C3×C3⋊D3—122no—
85C3×C3⋊D3—121no—
86C3×C3⋊D3—122no—
87C32×D3—361no—
88C32×D3—362no—
89C32×D3—361no—
90C2×3−1+2—42no—
91C2×3−1+2—82no—
92C2×3−1+2—124no—
93C2×H(3,𝔽3)—41no—
94C2×H(3,𝔽3)—81no—
95C2×H(3,𝔽3)—122no—
96C2×C3×C9—122no—
97AGL(1,ℤ/9)—124no—
98AGL(1,ℤ/9)—128no—
99(C3⋊D3)⋊C3—122no—
100(C3⋊D3)⋊C3—124no—
101C3×D9—124no—
102GL(2,𝔽3)—271no—
103C3×SD16—811no—
104#204—271no—
105C22×C32—541no—
106C2×C3⋊D3—61no—
107C2×C3⋊D3—121no—
108C2×C3×D3—181no—
109C2×C3×D3—181no—
110C2×C3×D3—181no—
111C2×C3×D3—181no—
112C2×C3×D3—361no—
113C2×C3×D3—361no—
114C2×C3×D3—361no—
115C2×C3×D3—361no—
116C2×C3×D3—362no—
117C2×C3×D3—1081no—
118C32×C4—271no—
119C3⋊Dic3—31no—
120C3×Dic3—181no—
121C3×Dic3—181no—
122C3×Dic3—181no—
123C3×Dic3—181no—
124C3×Dic3—361no—
125C3×Dic3—361no—
126C2×D9—122no—
127C33—31no—
128C33—41no—
129C33—61no—
130C33—61no—
131C33—81no—
132C33—121no—
133C33—121no—
134C33C3×GL(2,𝔽3)11yescharacteristic, complemented
1353−1+2—42no—
1363−1+2—82no—
1373−1+2—124no—
138H(3,𝔽3)—41no—
139H(3,𝔽3)—81no—
140H(3,𝔽3)—122no—
141C3×C9—122no—
142C3×Q8—271no—
143C3×D4—811no—
144#67—541no—
145D12—271no—
146Dic6—271no—
147SL(2,𝔽3)—271no—
148SL(2,𝔽3)—541no—
149C3×C8—271no—
150C3×C8—271no—
151C3×C8—541no—
152C2×C32—31no—
153C2×C32—31no—
154C2×C32—41no—
155C2×C32—41no—
156C2×C32—42no—
157C2×C32—61no—
158C2×C32—61no—
159C2×C32—121no—
160C2×C32—121no—
161C2×C32—121no—
162C2×C32—121no—
163C2×C32—121no—
164C2×C32—241no—
165C2×C32—241no—
166C2×C32—241no—
167C2×C32—361no—
168C2×C32—721no—
169C2×C32—1081no—
170C3⋊D3—121no—
171C3⋊D3—122no—
172C3×D3—361no—
173C3×D3—361no—
174C3×D3—362no—
175C3×D3—362no—
176C3×D3—364no—
177C3×D3—361no—
178C3×D3—361no—
179C3×D3—362no—
180C3×D3—362no—
181C3×D3—1082no—
182C2×C9—122no—
183C2×C9—242no—
184D9—124no—
185SD16—811noSylow
186C22×C3—541no—
187C22×C3—541no—
188C22×C3—1081no—
189C2×D3—181no—
190C2×D3—181no—
191C2×D3—361no—
192C2×D3—1081no—
193C3×C4—271no—
194C3×C4—271no—
195C3×C4—541no—
196Dic3—181no—
197Dic3—181no—
198C32—31no—
199C32—31no—
200C32—41no—
201C32—41no—
202C32—42no—
203C32—61no—
204C32—61no—
205C32—121no—
206C32—121no—
207C32—121no—
208C32—121no—
209C32—121no—
210C32—241no—
211C32—241no—
212C32—241no—
213C32—361no—
214C32—721no—
215C9—122no—
216C9—242no—
217Q8—271no—
218D4—811no—
219C8—271no—
220C2×C3—31no—
221C2×C3—41no—
222C2×C3—61no—
223C2×C3—61no—
224C2×C3—81no—
225C2×C3—121no—
226C2×C3—361no—
227C2×C3—721no—
228C2×C3—1081no—
229C2×C3—1082no—
230C2×C3—1081no—
231C2×C3#1062018411yescharacteristic
232D3—361no—
233D3—362no—
234D3—361no—
235D3—1082no—
236C22—541no—
237C4—271no—
238C3—31no—
239C3—41no—
240C3—61no—
241C3—61no—
242C3—81no—
243C3—121no—
244C3—361no—
245C3—721no—
246C3#1304936411yescharacteristic, direct product factor
247C2—1081no—
248C2C3×#1062018411yescharacteristic
subgroupG/H
center, hypercenterC2×C3#10620184
commutator, cosocle, hypocenterSL(2,ℤ/9)C2×C3
FrattiniC2C3×#10620184
Fitting, socleC2×C34S4
LCSG ▷ SL(2,ℤ/9)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C3×SL(2,ℤ/9)C2semidirect
3. #13049364C3direct (N × H)
5. C3×#6296D3semidirect
9. SL(2,ℤ/9)C2×C3semidirect
20. #6296C3×D3semidirect
59. C34GL(2,𝔽3)semidirect
134. C33C3×GL(2,𝔽3)semidirect
246. C3#13049364direct (N × H)

Maximal subgroups

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

subgroupindex
C3×SL(2,ℤ/9)2
C3×#889423
#130493643
C2×C3×#41834
C3×GL(2,𝔽3)27