Finite Group Databasebeta

← #10595386⋊C3 #13166136

Subgroups — #10595386⋊C3 #13166136

Order 1728.

467 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×C3⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C212yesmaximal, complemented
3#10595386C311yesmaximal, characteristic, complemented
4C4×C3⋊(#41⋊C3)—32nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C3⋊(C3⋊(#41⋊C3))C412yescomplemented
7C3×Dic32—41nomaximal
8C2×#3832C2×C311yescharacteristic
9C2×C3⋊(#41⋊C3)—32no—
10C2×C3⋊(#41⋊C3)—61no—
11C3×Q8×Dic3C2×C312yes—
12C3×C4×SL(2,𝔽3)—32no—
13#63894—32no—
14C4×#624—31no—
15#63849—31no—
16Dic3×SL(2,𝔽3)—22no—
17Dic3×SL(2,𝔽3)D312yescomplemented
18C2×C3×C3⋊Dic3—41no—
19C2×C32×Dic3—42no—
20C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
21C4×#239—32no—
22C4×#41⋊C3—91no—
23C2×C32×Q8C22×C311yescharacteristic
24#3832—32no—
25#3832C3×C412yescomplemented
26C2×C4⋊(C3⋊D3)—31no—
27C2×C4×C3⋊D3—31no—
28C2×C3×SL(2,𝔽3)—22no—
29C2×C3×SL(2,𝔽3)—21no—
30C2×C3×SL(2,𝔽3)—41no—
31C2×C3×SL(2,𝔽3)C2×D312yes—
32C2×#624—32no—
33C3⋊(#41⋊C3)—34no—
34C3⋊(#41⋊C3)—62no—
35C3×C4⋊Dic3—32no—
36C3×C4×Dic3—32no—
37C3×C4×Dic3—122no—
38#3722—32no—
39Dic32—31no—
40Dic32A411yescharacteristic
41C22×C33—41no—
42C3×C3⋊Dic3—42no—
43C32×Dic3—44no—
44C2×#239—32no—
45C2×#239—61no—
46C2×#41⋊C3—91no—
47C3×C4×Q8—32no—
48Q8×Dic3C3×D312yes—
49#893—92no—
50#892—92no—
51C4×D12—92no—
52C42×D3—92no—
53C4×SL(2,𝔽3)—32no—
54C4×SL(2,𝔽3)—62no—
55C22×C3⋊D3—31no—
56C32×Q8—31no—
57C32×Q8C2×C3×C411yescharacteristic
58C2×C32×C4—31no—
59C2×C32×C4—122no—
60C2×C3⋊Dic3C2×A411yescharacteristic
61C4⋊(C3⋊D3)—32no—
62C4⋊(C3⋊D3)—32no—
63C4×C3⋊D3—32no—
64C4×C3⋊D3—32no—
65C2×C3×Dic3—32no—
66C2×C3×Dic3—42no—
67C2×C3×Dic3—82no—
68C2×C3×Dic3—122no—
69C2×C3×Dic3—241no—
70C2×C3×Dic3C2×A412yes—
71C3×SL(2,𝔽3)—22no—
72C3×SL(2,𝔽3)—21no—
73C3×SL(2,𝔽3)—41no—
74C3×SL(2,𝔽3)C4×D312yescomplemented
75#624—64no—
76C4×#41—91noSylow
77C2×C33—41no—
78C2×C33—42no—
79C2×C3×Q8—21no—
80C2×C3×Q8C2×C3×D312yes—
81#239—34no—
82#239—62no—
83#239—94no—
84#239—182no—
85C2×D12—92no—
86C2×D12—181no—
87C2×C4×D3—92no—
88C2×C4×D3—92no—
89C2×C4×D3—92no—
90C2×C4×D3—181no—
91#41⋊C3—92no—
92C2×SL(2,𝔽3)—22no—
93C2×SL(2,𝔽3)—41no—
94C2×SL(2,𝔽3)D3211yescharacteristic
95C3×C4⋊C4—92no—
96C3×C42—92no—
97C3×C42—361no—
98(C2×D3)⋊C4—94no—
99C4⋊Dic3—32no—
100C4×Dic3—32no—
101C4×Dic3—32no—
102C4×Dic3—92no—
103C22×C32—42no—
104C22×C32—81no—
105C22×C32—82no—
106C22×C32—161no—
107C22×C32C22×A411yescharacteristic
108C2×C3⋊D3—32no—
109C2×C3⋊D3—62no—
110C32×C4—31no—
111C32×C4—31no—
112C32×C4—124no—
113C3⋊Dic3C4×A412yes—
114C3×Dic3—44no—
115C3×Dic3—62no—
116C3×Dic3—84no—
117C3×Dic3—124no—
118C3×Dic3—242no—
119C3×Dic3#41⋊C314yescomplemented
120C2×#41—91no—
121C4×Q8—32no—
122C4×D4—272no—
123#117—271no—
124C2×C42—271no—
125C33—41noSylow
126C22×D3—92no—
127C22×D3—181no—
128C3×Q8—21no—
129C3×Q8—32no—
130C3×Q8—61no—
131C3×Q8C3×C4×D312yes—
132C2×C3×C4—32no—
133C2×C3×C4—32no—
134C2×C3×C4—61no—
135C2×C3×C4—92no—
136C2×C3×C4—122no—
137C2×C3×C4—242no—
138C2×C3×C4—361no—
139C2×Dic3—32no—
140C2×Dic3—32no—
141C2×Dic3—61no—
142C2×Dic3D3×A412yes—
143D12—94no—
144D12—94no—
145D12—182no—
146D12—182no—
147C4×D3—94no—
148C4×D3—94no—
149C4×D3—182no—
150C4×D3—182no—
151C4×D3—184no—
152C4×D3—184no—
153SL(2,𝔽3)—22no—
154SL(2,𝔽3)—41no—
155SL(2,𝔽3)#62411yescharacteristic, complemented
156C2×C32—42no—
157C2×C32—42no—
158C2×C32—42no—
159C2×C32—81no—
160C2×C32—82no—
161C2×C32—82no—
162C2×C32—82no—
163C2×C32—82no—
164C2×C32—161no—
165C2×C32—162no—
166C2×C32C2×C4×A411yescharacteristic
167C2×C32C2×#41⋊C312yes—
168C3⋊D3—62no—
169#41—92no—
170#41—272no—
171C2×Q8C3×D3211yescharacteristic
172C2×D4—271no—
173C22×C4—271no—
174C22×C4—272no—
175C4⋊C4—92no—
176C22⋊C4—272no—
177C42—92no—
178C42—91no—
179C42—271no—
180C22×C3—21no—
181C22×C3—41no—
182C22×C3—82no—
183C22×C3—161no—
184C22×C3C2×D3×A412yes—
185C2×D3—94no—
186C2×D3—182no—
187C2×D3—182no—
188C2×D3—182no—
189C2×D3—362no—
190C3×C4—32no—
191C3×C4—32no—
192C3×C4—34no—
193C3×C4—61no—
194C3×C4—61no—
195C3×C4—124no—
196C3×C4—182no—
197C3×C4—244no—
198C3×C4—362no—
199Dic3—34no—
200Dic3—62no—
201Dic3—62no—
202Dic3C3⋊(#41⋊C3)14yescomplemented
203C32—42no—
204C32—81no—
205C32—82no—
206C32—161no—
207C32C4×#41⋊C311yescharacteristic, complemented
208C23—271no—
209Q8—31no—
210Q8C3×#62411yescharacteristic
211D4—272no—
212D4—272no—
213C2×C4—31no—
214C2×C4—32no—
215C2×C4—91no—
216C2×C4—92no—
217C2×C4—272no—
218C2×C4—272no—
219C2×C4—544no—
220C2×C3—22no—
221C2×C3—21no—
222C2×C3—42no—
223C2×C3—41no—
224C2×C3—82no—
225C2×C3—82no—
226C2×C3—82no—
227C2×C3—162no—
228C2×C3—161no—
229C2×C3C4×D3×A412yes—
230C2×C3C2×C3⋊(#41⋊C3)12yes—
231C2×C3D3×#41⋊C312yes—
232D3—184no—
233D3—362no—
234C22—272no—
235C22—542no—
236C22D32×A411yescharacteristic
237C4—34no—
238C4—31no—
239C4—31no—
240C4—92no—
241C4—182no—
242C3—21no—
243C3—41no—
244C3—82no—
245C3—161no—
246C3C4×C3⋊(#41⋊C3)12yescomplemented
247C2—542no—
248C2D3×C3⋊(#41⋊C3)12yes—
249C2A4×#62411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, hypocenterC32×Q8C2×C3×C4
Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
prime radicalC2×C3⋊(C3⋊(#41⋊C3))C2
LCSG ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3×Dic3×SL(2,𝔽3)C2semidirect
3. #10595386C3semidirect
6. C3⋊(C3⋊(#41⋊C3))C4semidirect
17. Dic3×SL(2,𝔽3)D3semidirect
20. C32×SL(2,𝔽3)C2×C4semidirect
25. #3832C3×C4semidirect
74. C3×SL(2,𝔽3)C4×D3semidirect
119. C3×Dic3#41⋊C3semidirect
155. SL(2,𝔽3)#624semidirect
202. Dic3C3⋊(#41⋊C3)semidirect
207. C32C4×#41⋊C3semidirect
246. C3C4×C3⋊(#41⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊(#41⋊C3))2
C3×Dic3×SL(2,𝔽3)2
#105953863
C4×C3⋊(#41⋊C3)3
C3×Dic324