Finite Group Databasebeta

← #76870⋊C3 #11921174

Subgroups — #76870⋊C3 #11921174

Order 1152.

358 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 28 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
1D3×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#393⋊C3C211yesmaximal, characteristic, complemented
3C3⋊(#395⋊C3)C211yesmaximal, characteristic, complemented
4#76870C311yesmaximal, characteristic, complemented
5#2614⋊C3—31nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
7A4×#67—41nomaximal
8C22×D3×Q8C2×C311yescharacteristic, complemented
9C2×C22⋊SL(2,𝔽3)—31no—
10C2×#883—31no—
11C3×#393C2×C311yescharacteristic, complemented
12#5230C2×C311yescharacteristic, complemented
13#4947—31no—
14#4664—31no—
15#4656—31no—
16#395⋊C3—31no—
17#393⋊C3—21no—
18#393⋊C3D311yescharacteristic, complemented
19C22×C3×A4—41no—
20C2×D3×A4—41no—
21Dic3×A4—41no—
22D3×SL(2,𝔽3)—82no—
23#2614—31noSylow
24C22×C3×Q8C22×C311yescharacteristic
25C22×#67A411yescharacteristic
26C2×D3×Q8—62no—
27C2×D3×Q8—62no—
28C22×C4×D3—31no—
29C22×Dic6—31no—
30C22⋊SL(2,𝔽3)—21no—
31C22⋊SL(2,𝔽3)C2×D311yescharacteristic
32D4×A4—121no—
33C2×C3×C22⋊C4—31no—
34C2×#217—31no—
35C2×(C2×D3)⋊C4—31no—
36C2×C4⋊Dic3—31no—
37C2×Dic3⋊C4—31no—
38#883—62no—
39#883—62no—
40C3×#95—31no—
41C3⋊#95—32no—
42C3⋊#95—31no—
43C2×C3×A4—41no—
44C2×C3×A4—81no—
45D3×A4—81no—
46C3×#67—161no—
47C3×SL(2,𝔽3)—82no—
48C23×Q8—31no—
49C2×#123—91no—
50#395—31no—
51#395—91no—
52#393—91no—
53#393C3×D311yescharacteristic, complemented
54#381—91no—
55C24×C3C2×A411yescharacteristic
56C23×D3C2×A411yescharacteristic
57C22×A4—41no—
58C22×A4—81no—
59C22×A4—121no—
60C2×C3×Q8—62no—
61C22×C3×C4—31no—
62C22×C3×C4—31no—
63C2×#67—61no—
64C2×#67—61no—
65C22×Dic3—31no—
66C22×Dic3—31no—
67C22×Dic3C2×A411yescharacteristic
68D3×Q8—22no—
69D3×Q8—62no—
70D3×Q8—64no—
71D3×Q8—62no—
72D3×Q8—62no—
73C2×C4×D3—32no—
74C2×C4×D3—32no—
75C2×C4×D3—62no—
76C2×C4×D3—62no—
77C2×Dic6—64no—
78C2×Dic6—62no—
79C2×SL(2,𝔽3)—242no—
80C4×A4—121no—
81C3×C22⋊C4—62no—
82#217—62no—
83(C2×D3)⋊C4—61no—
84(C2×D3)⋊C4—61no—
85C4⋊Dic3—62no—
86Dic3⋊C4—62no—
87C22×C32—161no—
88C2×C3×D3—161no—
89C3×A4—41no—
90C3×Dic3—161no—
91C22×Q8—91no—
92C22×Q8—182no—
93C22×Q8—182no—
94C22×Q8C2×C3×D311yescharacteristic
95C22×D4—31no—
96C23×C4—91no—
97#123—182no—
98#123—182no—
99C2×C4⋊C4—91no—
100C2×C4⋊C4—91no—
101C2×C22⋊C4—31no—
102C2×C22⋊C4—91no—
103C2×C22⋊C4—91no—
104#95—31no—
105#95—91no—
106#95—92no—
107C23×C3—21no—
108C23×C3—61no—
109C23×C3—61no—
110C23×C3C22×A411yescharacteristic
111C22×D3—21no—
112C22×D3—31no—
113C22×D3—31no—
114C22×D3—61no—
115C2×A4—41no—
116C2×A4—81no—
117C2×A4—82no—
118C2×A4—81no—
119C2×A4—241no—
120C3×Q8—22no—
121C3×Q8—62no—
122C3×D4—481no—
123C2×C3×C4—32no—
124C2×C3×C4—62no—
125C2×C3×C4—61no—
126C2×C3×C4—62no—
127#67—41no—
128#67—121no—
129C2×Dic3—32no—
130C2×Dic3—61no—
131C2×Dic3—62no—
132C2×Dic3—61no—
133C2×Dic3—61no—
134C2×Dic3—61no—
135C4×D3—64no—
136C4×D3—62no—
137C4×D3—62no—
138Dic6—64no—
139Dic6—62no—
140Dic6—62no—
141SL(2,𝔽3)—82no—
142SL(2,𝔽3)—162no—
143C2×C32—161no—
144C2×C32—321no—
145C3×D3—321no—
146C24—31no—
147C24D3×A411yescharacteristic
148C2×Q8—62no—
149C2×Q8—62no—
150C2×Q8—182no—
151C2×Q8—184no—
152C2×Q8—182no—
153C2×Q8—182no—
154C2×Q8—184no—
155C2×Q8—182no—
156C2×D4—181no—
157C2×D4—181no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—92no—
162C22×C4—92no—
163C22×C4—91no—
164C22×C4—91no—
165C22×C4—182no—
166C22×C4—182no—
167C4⋊C4—182no—
168C4⋊C4—182no—
169C22⋊C4—62no—
170C22⋊C4—181no—
171C22⋊C4—181no—
172C22⋊C4—182no—
173C22×C3—31no—
174C22×C3—31no—
175C22×C3—41no—
176C22×C3—121no—
177C22×C3—121no—
178C22×C3—161no—
179C22×C3—321no—
180C22×C3—481no—
181C22×C32−1+4⋊C311yescharacteristic
182C2×D3—31no—
183C2×D3—61no—
184C2×D3—61no—
185C2×D3—61no—
186C2×D3—61no—
187C2×D3#120⋊C311yescharacteristic
188A4—41no—
189A4—81no—
190C3×C4—62no—
191C3×C4—121no—
192C3×C4—481no—
193Dic3—41no—
194Dic3—62no—
195Dic3—121no—
196C32—161noSylow
197C23—21no—
198C23—61no—
199C23—61no—
200C23—61no—
201C23—91no—
202C23—91no—
203C23—181no—
204C23C2×D3×A411yescharacteristic
205Q8—22no—
206Q8—62no—
207Q8—182no—
208Q8—182no—
209Q8—184no—
210D4—121no—
211D4—361no—
212C2×C4—32no—
213C2×C4—62no—
214C2×C4—61no—
215C2×C4—62no—
216C2×C4—92no—
217C2×C4—182no—
218C2×C4—182no—
219C2×C4—184no—
220C2×C4—181no—
221C2×C4—182no—
222C2×C4—181no—
223C2×C4—181no—
224C2×C4—181no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—81no—
228C2×C3—161no—
229C2×C3—321no—
230C2×C3—322no—
231C2×C3—321no—
232C2×C3—961no—
233C2×C3C2×#120⋊C311yescharacteristic
234D3—21no—
235D3—61no—
236C22—31no—
237C22—31no—
238C22—31no—
239C22—41no—
240C22—91no—
241C22—121no—
242C22—121no—
243C22—181no—
244C22—181no—
245C22—181no—
246C22—181no—
247C22#1007⋊C311yescharacteristic
248C4—62no—
249C4—121no—
250C4—121no—
251C4—182no—
252C4—361no—
253C3—161no—
254C3—321no—
255C3#2614⋊C311yescharacteristic, complemented
256C2—31no—
257C2—31no—
258C2—61no—
259C2—81no—
260C2—181no—
261C2D3×#120⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2D3×#120⋊C3
commutator, cosocle, hypocenterC22×C3×Q8C22×C3
FrattiniC23C2×D3×A4
FittingC3×#393C2×C3
socleC23×C3C22×A4
LCSG ▷ C22×C3×Q8
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. D3×C22⋊SL(2,𝔽3)C2semidirect
2. C3×#393⋊C3C2semidirect
3. C3⋊(#395⋊C3)C2semidirect
4. #76870C3semidirect
8. C22×D3×Q8C2×C3semidirect
11. C3×#393C2×C3semidirect
12. #5230C2×C3semidirect
18. #393⋊C3D3semidirect
53. #393C3×D3semidirect
255. C3#2614⋊C3semidirect

Maximal subgroups

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

subgroupindex
D3×C22⋊SL(2,𝔽3)2
C3×#393⋊C32
C3⋊(#395⋊C3)2
#768703
#2614⋊C33
A4×#674