Finite Group Databasebeta

← #77256⋊C3 #11921182

Subgroups — #77256⋊C3 #11921182

Order 1152.

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

normal subgroups only

This group has 354 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×#395⋊C3C211yesmaximal, characteristic, complemented
3C3⋊(#393⋊C3)C211yesmaximal, characteristic, complemented
4#77256C311yesmaximal, characteristic, complemented
5#2614⋊C3—31nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
7A4×D12—41nomaximal
8C22×D3×Q8C2×C311yescharacteristic, complemented
9C2×C22⋊SL(2,𝔽3)—31no—
10C2×#894—31no—
11C3×#395C2×C311yescharacteristic, complemented
12#5231C2×C311yescharacteristic, complemented
13#4979—31no—
14#4673—31no—
15#4664—31no—
16#395⋊C3—21no—
17#395⋊C3D311yescharacteristic, complemented
18#393⋊C3—31no—
19C2×D3×A4—41no—
20C2×D3×A4—41no—
21C3×C4×A4—41no—
22D3×SL(2,𝔽3)—82no—
23#2614—31noSylow
24C22×C3×Q8C22×C311yescharacteristic
25C2×D3×Q8—62no—
26C2×D3×Q8—62no—
27C22×D12A411yescharacteristic
28C22×C4×D3—31no—
29C22×Dic6—31no—
30C22⋊SL(2,𝔽3)—21no—
31C22⋊SL(2,𝔽3)C2×D311yescharacteristic
32D4×A4—121no—
33C2×C3×C4⋊C4—31no—
34C2×(C2×D3)⋊C4—31no—
35C2×(C2×D3)⋊C4—31no—
36C2×(C2×D3)⋊C4—31no—
37C2×Dic3⋊C4—31no—
38#894—62no—
39#894—62no—
40C3×#95—31no—
41C3⋊#95—32no—
42C3⋊#95—31no—
43C2×C3×A4—41no—
44D3×A4—81no—
45D3×A4—81no—
46C3×D12—161no—
47C3×SL(2,𝔽3)—82no—
48C23×Q8—31no—
49C2×#123—91no—
50#395—91no—
51#395C3×D311yescharacteristic, complemented
52#393—31no—
53#393—91no—
54#381—91no—
55C23×D3C2×A411yescharacteristic
56C23×D3C2×A411yescharacteristic
57C22×A4—121no—
58C22×A4—121no—
59C2×C3×Q8—62no—
60C22×C3×C4—31no—
61C22×C3×C4—31no—
62C22×C3×C4C2×A411yescharacteristic
63C22×Dic3—31no—
64C22×Dic3—31no—
65D3×Q8—22no—
66D3×Q8—62no—
67D3×Q8—64no—
68D3×Q8—62no—
69D3×Q8—62no—
70C2×D12—61no—
71C2×D12—61no—
72C2×C4×D3—32no—
73C2×C4×D3—32no—
74C2×C4×D3—62no—
75C2×C4×D3—62no—
76C2×Dic6—64no—
77C2×Dic6—62no—
78C2×SL(2,𝔽3)—242no—
79C4×A4—41no—
80C4×A4—81no—
81C3×C4⋊C4—62no—
82(C2×D3)⋊C4—62no—
83(C2×D3)⋊C4—62no—
84(C2×D3)⋊C4—61no—
85(C2×D3)⋊C4—61no—
86Dic3⋊C4—62no—
87C2×C3×D3—161no—
88C2×C3×D3—161no—
89C3×A4—41no—
90C32×C4—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—31no—
100C2×C4⋊C4—91no—
101C2×C22⋊C4—91no—
102C2×C22⋊C4—91no—
103C2×C22⋊C4—91no—
104#95—31no—
105#95—91no—
106#95—92no—
107C23×C3C22×A411yescharacteristic
108C22×D3—21no—
109C22×D3—21no—
110C22×D3—31no—
111C22×D3—31no—
112C22×D3—61no—
113C22×D3—61no—
114C22×D3—61no—
115C2×A4—41no—
116C2×A4—81no—
117C2×A4—241no—
118C2×A4—241no—
119C3×Q8—22no—
120C3×Q8—62no—
121C3×D4—481no—
122C2×C3×C4—32no—
123C2×C3×C4—61no—
124C2×C3×C4—62no—
125C2×C3×C4—61no—
126C2×C3×C4—61no—
127C2×C3×C4—61no—
128C2×Dic3—32no—
129C2×Dic3—62no—
130C2×Dic3—61no—
131C2×Dic3—62no—
132D12—41no—
133D12—121no—
134C4×D3—64no—
135C4×D3—62no—
136C4×D3—62no—
137Dic6—64no—
138Dic6—62no—
139Dic6—62no—
140SL(2,𝔽3)—82no—
141SL(2,𝔽3)—162no—
142C2×C32—161no—
143C3×D3—321no—
144C3×D3—321no—
145C24—31no—
146C24—31no—
147C2×Q8—62no—
148C2×Q8—62no—
149C2×Q8—182no—
150C2×Q8—184no—
151C2×Q8—182no—
152C2×Q8—182no—
153C2×Q8—184no—
154C2×Q8—182no—
155C2×D4—181no—
156C2×D4—181no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—92no—
160C22×C4—92no—
161C22×C4—91no—
162C22×C4—91no—
163C22×C4—182no—
164C22×C4—182no—
165C22×C4D3×A411yescharacteristic
166C4⋊C4—62no—
167C4⋊C4—182no—
168C22⋊C4—182no—
169C22⋊C4—182no—
170C22⋊C4—181no—
171C22⋊C4—181no—
172C22×C3—31no—
173C22×C3—31no—
174C22×C3—481no—
175C22×C3—481no—
176C22×C32−1+4⋊C311yescharacteristic
177C2×D3—31no—
178C2×D3—41no—
179C2×D3—61no—
180C2×D3—61no—
181C2×D3—61no—
182C2×D3—61no—
183C2×D3—121no—
184C2×D3—121no—
185C2×D3#120⋊C311yescharacteristic
186A4—41no—
187A4—81no—
188C3×C4—41no—
189C3×C4—62no—
190C3×C4—121no—
191C3×C4—161no—
192C3×C4—321no—
193Dic3—62no—
194Dic3—121no—
195C32—161noSylow
196C23—61no—
197C23—61no—
198C23—91no—
199C23—91no—
200C23—181no—
201C23—181no—
202C23—181no—
203C23C2×D3×A411yescharacteristic
204Q8—22no—
205Q8—62no—
206Q8—182no—
207Q8—182no—
208Q8—184no—
209D4—121no—
210D4—361no—
211C2×C4—32no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—92no—
218C2×C4—182no—
219C2×C4—182no—
220C2×C4—184no—
221C2×C4—182no—
222C2×C4—181no—
223C2×C4—182no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—161no—
227C2×C3—321no—
228C2×C3—961no—
229C2×C3—961no—
230C2×C3C2×#120⋊C311yescharacteristic
231D3—21no—
232D3—61no—
233D3—81no—
234C22—31no—
235C22—31no—
236C22—31no—
237C22—91no—
238C22—121no—
239C22—181no—
240C22—181no—
241C22—181no—
242C22—181no—
243C22—361no—
244C22—361no—
245C22#1010⋊C311yescharacteristic
246C4—41no—
247C4—62no—
248C4—121no—
249C4—182no—
250C4—361no—
251C3—161no—
252C3—321no—
253C3#2614⋊C311yescharacteristic, complemented
254C2—31no—
255C2—31no—
256C2—61no—
257C2—181no—
258C2—241no—
259C2D3×#120⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2D3×#120⋊C3
commutator, cosocle, hypocenterC22×C3×Q8C22×C3
FrattiniC23C2×D3×A4
FittingC3×#395C2×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×#395⋊C3C2semidirect
3. C3⋊(#393⋊C3)C2semidirect
4. #77256C3semidirect
8. C22×D3×Q8C2×C3semidirect
11. C3×#395C2×C3semidirect
12. #5231C2×C3semidirect
17. #395⋊C3D3semidirect
51. #395C3×D3semidirect
253. 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×#395⋊C32
C3⋊(#393⋊C3)2
#772563
#2614⋊C33
A4×D124