Finite Group Databasebeta

← #13048337

Subgroups — #13048337

Order 1296.

402 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#10619902C211yesmaximal, characteristic, complemented
2#10619890C211yesmaximal, characteristic, complemented
3#10619887C212yesmaximal, complemented
4#6289⋊C3C212yesmaximal, complemented
5C3×#6196C211yesmaximal, characteristic, complemented
6#88965—32nomaximal
7#88650—31nomaximal
8C2×C3⋊#272C2211yescharacteristic
9C2×C3⋊((C3⋊D3)⋊C3)C2212yes—
10C3×#1084C2211yescharacteristic
11Dic3×H(3,𝔽3)C2211yescharacteristic
12C3×(C3⋊Dic3)⋊C3C2212yes—
13#6295—32no—
14#6292—32no—
15#6292—32no—
16#6289D312yescomplemented
17C3×#627—32no—
18C3×#624—32no—
19C3×#624—32no—
20C3×#624—61no—
21C4×#272—31no—
22C4×(C3⋊D3)⋊C3—32no—
23#6200—32no—
24#6200—62no—
25#6198—31no—
26#6196D311yescharacteristic, complemented
27C2×C3×H(3,𝔽3)C2311yescharacteristic
28C3⋊#272—21no—
29C3⋊((C3⋊D3)⋊C3)—22no—
30#3799—91no—
31#3799—92no—
32#3798—92no—
33C2×C3⋊(C3⋊D3)C2×D312yes—
34C2×C3×C3⋊D3—32no—
35C2×C3×C3⋊D3—32no—
36C2×C3×C3⋊D3—61no—
37C3×C3⋊Dic3C2×D312yes—
38C32×Dic3—21no—
39C32×Dic3—32no—
40C32×Dic3—32no—
41C32×Dic3—61no—
42C32×Dic3C2×D312yes—
43C2×#272—31no—
44C2×#272—62no—
45C2×#272—121no—
46C2×(C3⋊D3)⋊C3—32no—
47C2×(C3⋊D3)⋊C3—62no—
48C4×H(3,𝔽3)—31no—
49#1084C2×D311yescharacteristic
50(C3⋊Dic3)⋊C3—22no—
51(C3⋊Dic3)⋊C3C2×D312yes—
52C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
53C4⋊(C3⋊D3)—91no—
54C4⋊(C3⋊D3)—92no—
55C4×C3⋊D3—32no—
56C3×C4×D3—92no—
57C3×C4×D3—92no—
58C3×C4×D3—92no—
59C3×C4×D3—91no—
60C3×C4×D3—181no—
61C3×Dic6—91no—
62C3×Dic6—92no—
63#627—32no—
64#626—92no—
65#626—92no—
66#626—92no—
67#626—92no—
68#626—92no—
69#626—182no—
70#626—182no—
71#625—92no—
72#624—32no—
73#624—32no—
74#624—61no—
75#624—92no—
76C2×C33—21no—
77C2×C33C22×D312yes—
78C3⋊(C3⋊D3)—22no—
79C3×C3⋊D3—62no—
80C3×C3⋊D3—62no—
81C3×C3⋊D3—62no—
82C2×H(3,𝔽3)—22no—
83C2×H(3,𝔽3)—41no—
84C2×H(3,𝔽3)C22×D311yescharacteristic
85#272—61no—
86#272—122no—
87#272—122no—
88(C3⋊D3)⋊C3—62no—
89(C3⋊D3)⋊C3—122no—
90#239—271no—
91#235—271no—
92#235—272no—
93C2×C3⋊D3—32no—
94C2×C3⋊D3—32no—
95C2×C3⋊D3—32no—
96C2×C3⋊D3—61no—
97C2×C3⋊D3—62no—
98C2×C3⋊D3—92no—
99C2×C3⋊D3—182no—
100C2×C3×D3—92no—
101C2×C3×D3—92no—
102C2×C3×D3—92no—
103C2×C3×D3—91no—
104C2×C3×D3—181no—
105C2×C3×D3—182no—
106C2×C3×D3—182no—
107C2×C3×D3—361no—
108C32×C4—32no—
109C32×C4—61no—
110C32×C4—92no—
111C32×C4—91no—
112C3⋊Dic3D3212yes—
113C3×Dic3—32no—
114C3×Dic3—32no—
115C3×Dic3—32no—
116C3×Dic3—32no—
117C3×Dic3—32no—
118C3×Dic3—62no—
119C3×Dic3—62no—
120C3×Dic3—61no—
121C3×Dic3—61no—
122C3×Dic3—61no—
123C3×Dic3—92no—
124C3×Dic3—121no—
125C3×Dic3D3211yescharacteristic
126C33—21no—
127C33#23512yescomplemented
128H(3,𝔽3)—22no—
129H(3,𝔽3)—41no—
130H(3,𝔽3)#23911yescharacteristic, complemented
131C3×Q8—271no—
132C2×C3×C4—271no—
133C2×C3×C4—272no—
134#67—272no—
135#67—272no—
136#67—272no—
137D12—271no—
138D12—271no—
139D12—272no—
140D12—272no—
141D12—541no—
142D12—542no—
143C4×D3—92no—
144C4×D3—92no—
145C4×D3—91no—
146C4×D3—92no—
147C4×D3—181no—
148C4×D3—272no—
149Dic6—91no—
150Dic6—92no—
151C2×C32—22no—
152C2×C32—21no—
153C2×C32—32no—
154C2×C32—41no—
155C2×C32—62no—
156C2×C32—61no—
157C2×C32—121no—
158C2×C32C2×D3211yescharacteristic
159C2×C32C2×D3212yes—
160C3⋊D3—62no—
161C3⋊D3—62no—
162C3⋊D3—62no—
163C3⋊D3—62no—
164C3⋊D3—122no—
165C3⋊D3—182no—
166C3⋊D3—362no—
167C3×D3—182no—
168C3×D3—182no—
169C3×D3—181no—
170C3×D3—182no—
171C3×D3—182no—
172C3×D3—362no—
173C3×D3—362no—
174C3×D3—362no—
175#41—811noSylow
176C22×C3—272no—
177C22×C3—271no—
178C2×D3—92no—
179C2×D3—91no—
180C2×D3—92no—
181C2×D3—92no—
182C2×D3—181no—
183C2×D3—182no—
184C2×D3—182no—
185C2×D3—272no—
186C2×D3—361no—
187C2×D3—542no—
188C3×C4—31no—
189C3×C4—92no—
190C3×C4—92no—
191C3×C4—91no—
192C3×C4—92no—
193C3×C4—91no—
194C3×C4—181no—
195C3×C4—181no—
196C3×C4—182no—
197Dic3—32no—
198Dic3—32no—
199Dic3—61no—
200Dic3—92no—
201Dic3((C3⋊D3)⋊C3)⋊C211yescharacteristic
202C32—21no—
203C32—22no—
204C32—32no—
205C32—41no—
206C32—62no—
207C32—61no—
208C32—121no—
209C32#379811yescharacteristic
210C32#379912yescomplemented
211Q8—271no—
212D4—811no—
213D4—812no—
214C2×C4—271no—
215C2×C4—272no—
216C2×C3—21no—
217C2×C3—32no—
218C2×C3—61no—
219C2×C3—62no—
220C2×C3—121no—
221C2×C3—542no—
222C2×C3—541no—
223C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
224C2×C3D3311yescharacteristic
225D3—182no—
226D3—182no—
227D3—181no—
228D3—182no—
229D3—182no—
230D3—362no—
231D3—362no—
232D3—362no—
233D3—542no—
234D3—1082no—
235C22—271no—
236C22—272no—
237C4—31no—
238C4—91no—
239C4—92no—
240C3—21no—
241C3—32no—
242C3—61no—
243C3—62no—
244C3—121no—
245C3#8865011yescharacteristic, complemented
246C3#8896511yescharacteristic
247C2—541no—
248C2—542no—
249C2D3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2D3×((C3⋊D3)⋊C3)⋊C2
commutator, Fitting, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3D33
socleC2×C32C2×D32
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #10619902C2semidirect
2. #10619890C2semidirect
3. #10619887C2semidirect
4. #6289⋊C3C2semidirect
5. C3×#6196C2semidirect
16. #6289D3semidirect
26. #6196D3semidirect
52. C3×H(3,𝔽3)#41semidirect
127. C33#235semidirect
130. H(3,𝔽3)#239semidirect
210. C32#3799semidirect
245. C3#88650semidirect

Maximal subgroups

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

subgroupindex
#106199022
#106198902
#106198872
#6289⋊C32
C3×#61962
#889653
#886503