Finite Group Databasebeta

← #5711⋊C9 #13147225

Subgroups — #5711⋊C9 #13147225

Order 1728.

258 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 39 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×#126⋊C9C211yesmaximal, characteristic, complemented
2C3⋊(#126⋊C9)C211yesmaximal, characteristic, complemented
3C3⋊(#127⋊C9)C211yesmaximal, characteristic, complemented
4C3×#5711C311yesmaximal, characteristic
5#560⋊C9—31nomaximal
6C2×D3×C22⋊C9—41nomaximal
7C3×C42⋊C9C2211yescharacteristic, complemented
8C32×#126C2×C311yescharacteristic
9C3×#895—31no—
10C3×#894—31no—
11C3×#884—31no—
12C3×#882—31no—
13C3×#876C2×C311yescharacteristic
14C3×#874C2×C311yescharacteristic
15#127⋊C9—31no—
16#126⋊C9—21no—
17#126⋊C9—31no—
18#126⋊C9D311yescharacteristic, complemented
19C2×C3×C22⋊C9—41no—
20D3×C22⋊C9—41no—
21D3×C22⋊C9—41no—
22C3×#560—31no—
23#5711C911yescharacteristic, complemented
24C23×C3×D3A411yescharacteristic
25C2×C3×#67—31no—
26C2×C3×D12—31no—
27C2×C3×D12—31no—
28C2×C3×D12—31no—
29C2×C3×C4×D3—31no—
30C22×C22⋊C9—121no—
31C32×C4⋊C4—31no—
32C32×C22⋊C4—31no—
33C32×C42C22×C311yescharacteristic
34C3×(C2×D3)⋊C4—31no—
35C3×(C2×D3)⋊C4—31no—
36C3×(C2×D3)⋊C4—31no—
37C3×Dic3⋊C4—31no—
38C42⋊C9—21no—
39C42⋊C9C2×D311yescharacteristic, complemented
40C2×D3×C9—161no—
41C3×C22⋊C9—41no—
42C3×#127—31no—
43C3×#126—21no—
44C3×#126—31no—
45C3×#126C2×C911yescharacteristic, complemented
46C3×#126C3×D311yescharacteristic
47C3×#123—91no—
48C3×#121—91no—
49C3×#121—91no—
50C3×#120—91no—
51#895—31no—
52#894—31no—
53#884—31no—
54#882—31no—
55#876C2×C911yescharacteristic, complemented
56#874C2×C911yescharacteristic, complemented
57C23×C32C2×A411yescharacteristic
58C22×C3×D3—31no—
59C22×C3×D3—121no—
60C22×C3×D3C2×A411yescharacteristic
61C22×C3×D3C2×A411yescharacteristic
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C3×#67—121no—
65C2×C3×Dic3—31no—
66C3×D12—121no—
67C3×D12—121no—
68C3×D12—121no—
69C3×C4×D3—121no—
70C2×C22⋊C9—41no—
71C2×C22⋊C9—81no—
72C2×C22⋊C9—121no—
73C2×C22⋊C9—121no—
74#560—31noSylow
75C2×C3×C9—161no—
76D3×C9—161no—
77D3×C9—161no—
78C24×C3—31no—
79C23×D3C22⋊C911yescharacteristic
80C2×C3×D4—91no—
81C2×C3×D4—91no—
82C2×C3×D4—91no—
83C2×C3×D4—91no—
84C22×C3×C4—91no—
85C2×#67—31no—
86C2×D12—31no—
87C2×D12—31no—
88C2×D12—31no—
89C2×C4×D3—31no—
90C3×C4⋊C4—31no—
91C3×C4⋊C4—31no—
92C3×C4⋊C4—61no—
93C3×C4⋊C4—91no—
94C3×C22⋊C4—31no—
95C3×C22⋊C4—31no—
96C3×C22⋊C4—61no—
97C3×C22⋊C4—91no—
98C3×C22⋊C4—91no—
99C3×C22⋊C4—91no—
100C3×C42—21no—
101C3×C42C22×C911yescharacteristic, complemented
102C3×C42C2×C3×D311yescharacteristic
103(C2×D3)⋊C4—31no—
104(C2×D3)⋊C4—31no—
105(C2×D3)⋊C4—31no—
106Dic3⋊C4—31no—
107C22×C32—61no—
108C22×C32C22×A411yescharacteristic
109C2×C3×D3—41no—
110C2×C3×D3—61no—
111C2×C3×D3—61no—
112C2×C3×D3—61no—
113C2×C3×D3—61no—
114C2×C3×D3—61no—
115C2×C3×D3—121no—
116C32×C4—61no—
117C32×C4—61no—
118C3×Dic3—61no—
119C22×C9—481no—
120C22⋊C9—41no—
121C22⋊C9—81no—
122#127—31no—
123#126—31no—
124#126D3×C911yescharacteristic, complemented
125#123—91no—
126#121—91no—
127#121—91no—
128#120—91no—
129C3×C9—161noSylow
130C23×C3—21no—
131C23×C3—31no—
132C23×C3—31no—
133C23×C3—91no—
134C23×C3—361no—
135C23×C3C2×C22⋊C911yescharacteristic
136C23×C3D3×A411yescharacteristic
137C22×D3—31no—
138C22×D3—121no—
139C22×D3C2×C22⋊C911yescharacteristic
140C22×D3C2×C22⋊C911yescharacteristic
141C3×D4—361no—
142C3×D4—361no—
143C3×D4—361no—
144C3×D4—361no—
145C2×C3×C4—31no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—61no—
150C2×C3×C4—61no—
151C2×C3×C4—91no—
152C2×C3×C4—361no—
153#67—121no—
154C2×Dic3—31no—
155D12—121no—
156D12—121no—
157D12—121no—
158C4×D3—121no—
159C2×C32—31no—
160C2×C32—41no—
161C3×D3—41no—
162C3×D3—41no—
163C3×D3—121no—
164C2×C9—161no—
165C2×C9—321no—
166C2×C9—481no—
167C2×C9—481no—
168C24—31no—
169C2×D4—91no—
170C2×D4—91no—
171C2×D4—91no—
172C2×D4—91no—
173C22×C4—91no—
174C4⋊C4—31no—
175C4⋊C4—91no—
176C22⋊C4—31no—
177C22⋊C4—91no—
178C22⋊C4—91no—
179C22⋊C4—91no—
180C42C2×D3×C911yescharacteristic, complemented
181C22×C3—21no—
182C22×C3—61no—
183C22×C3—61no—
184C22×C3—121no—
185C22×C3—121no—
186C22×C3—181no—
187C22×C3—181no—
188C22×C3—181no—
189C22×C3—181no—
190C22×C3—181no—
191C22×C3—361no—
192C22×C3C22×C22⋊C911yescharacteristic
193C22×C3C2×D3×A411yescharacteristic
194C2×D3—41no—
195C2×D3—61no—
196C2×D3—61no—
197C2×D3—61no—
198C2×D3—61no—
199C2×D3—61no—
200C2×D3—121no—
201C3×C4—61no—
202C3×C4—61no—
203C3×C4—61no—
204C3×C4—61no—
205C3×C4—121no—
206C3×C4—121no—
207C3×C4—181no—
208Dic3—61no—
209C32#560⋊C311yescharacteristic
210C9—161no—
211C9—321no—
212C23—31no—
213C23—31no—
214C23—91no—
215C23—361no—
216C23D3×C22⋊C911yescharacteristic
217D4—361no—
218D4—361no—
219D4—361no—
220D4—361no—
221C2×C4—31no—
222C2×C4—31no—
223C2×C4—91no—
224C2×C4—361no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—41no—
228C2×C3—41no—
229C2×C3—61no—
230C2×C3—81no—
231C2×C3—121no—
232C2×C3—121no—
233C2×C3—361no—
234D3—41no—
235D3—41no—
236D3—121no—
237C22—61no—
238C22—121no—
239C22—181no—
240C22—181no—
241C22—181no—
242C22—181no—
243C22—181no—
244C22—361no—
245C22C2×D3×C22⋊C911yescharacteristic
246C4—61no—
247C4—61no—
248C4—181no—
249C3—21no—
250C3#560⋊C911yescharacteristic, complemented
251C3(#126⋊C3)⋊D311yescharacteristic
252C2—31no—
253C2—41no—
254C2—121no—
255C2—121no—
256C2—361no—
subgroupG/H
center, hypercenterC3(#126⋊C3)⋊D3
commutator, hypocenterC3×C42C22×C9
FrattiniC22×C3C2×D3×A4
FittingC32×#126C2×C3
socleC22×C32C22×A4
prime radicalC3×#5711C3
cosocleC32×C42C22×C3
LCSG ▷ C3×C42
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×#126⋊C9C2semidirect
2. C3⋊(#126⋊C9)C2semidirect
3. C3⋊(#127⋊C9)C2semidirect
7. C3×C42⋊C9C22semidirect
18. #126⋊C9D3semidirect
23. #5711C9semidirect
39. C42⋊C9C2×D3semidirect
45. C3×#126C2×C9semidirect
55. #876C2×C9semidirect
56. #874C2×C9semidirect
101. C3×C42C22×C9semidirect
124. #126D3×C9semidirect
180. C42C2×D3×C9semidirect
250. C3#560⋊C9semidirect

Maximal subgroups

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

subgroupindex
C3×#126⋊C92
C3⋊(#126⋊C9)2
C3⋊(#127⋊C9)2
C3×#57113
#560⋊C93
C2×D3×C22⋊C94