Finite Group Databasebeta

← #84069⋊C3 #11921366

Subgroups — #84069⋊C3 #11921366

Order 1152.

364 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 29 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×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3⋊(#393⋊C3)C211yesmaximal, characteristic, complemented
3C3⋊(#395⋊C3)C211yesmaximal, characteristic, complemented
4#84069C311yesmaximal, characteristic, complemented
5#2614⋊C3—31nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
7A4×#67—41nomaximal
8C23×C3×Q8C2×C311yescharacteristic, complemented
9C2×C22⋊SL(2,𝔽3)—21no—
10C2×C22⋊SL(2,𝔽3)D311yescharacteristic, complemented
11C2×#929—31no—
12#5231—31no—
13#5231C2×C311yescharacteristic, complemented
14#5230—31no—
15#5230C2×C311yescharacteristic, complemented
16#5211—31no—
17#395⋊C3—31no—
18#393⋊C3—31no—
19C22×C3×A4—41no—
20C2×D3×A4—41no—
21C2×C3×SL(2,𝔽3)—82no—
22Dic3×A4—41no—
23#2614—31noSylow
24C22×C3×Q8—31no—
25C22×C3×Q8—62no—
26C22×C3×Q8—62no—
27C22×C3×Q8C22×C311yescharacteristic
28C23×C3×C4—31no—
29C22×#67A411yescharacteristic
30C22⋊SL(2,𝔽3)—21no—
31C22⋊SL(2,𝔽3)C2×D311yescharacteristic
32D4×A4—121no—
33C2×#217—31no—
34#929—62no—
35#929—62no—
36C2×(C2×D3)⋊C4—31no—
37C2×(C2×D3)⋊C4—31no—
38C2×Dic3⋊C4—31no—
39C2×Dic3⋊C4—31no—
40C3⋊#95—32no—
41C3⋊#95—31no—
42C3⋊#95—31no—
43C2×C3×A4—41no—
44C2×C3×A4—81no—
45D3×A4—81no—
46C3×#67—161no—
47C3×SL(2,𝔽3)—82no—
48C23×Q8C3×D311yescharacteristic, complemented
49C2×#123—91no—
50#395—31no—
51#395—91no—
52#393—31no—
53#393—91no—
54#381—91no—
55C24×C3C2×A411yescharacteristic
56C23×D3C2×A411yescharacteristic
57C22×A4—41no—
58C22×A4—81no—
59C22×A4—121no—
60C2×C3×Q8—22no—
61C2×C3×Q8—62no—
62C2×C3×Q8—62no—
63C2×C3×Q8—64no—
64C2×C3×Q8—62no—
65C2×C3×Q8—62no—
66C2×C3×Q8—64no—
67C2×C3×Q8—62no—
68C22×C3×C4—32no—
69C22×C3×C4—32no—
70C22×C3×C4—31no—
71C22×C3×C4—31no—
72C22×C3×C4—62no—
73C22×C3×C4—62no—
74C2×#67—61no—
75C2×#67—61no—
76C22×Dic3—31no—
77C22×Dic3—31no—
78C22×Dic3C2×A411yescharacteristic
79C2×SL(2,𝔽3)—82no—
80C2×SL(2,𝔽3)—84no—
81C4×A4—121no—
82#217—61no—
83#217—61no—
84(C2×D3)⋊C4—62no—
85(C2×D3)⋊C4—62no—
86Dic3⋊C4—62no—
87Dic3⋊C4—62no—
88C22×C32—161no—
89C2×C3×D3—161no—
90C3×A4—41no—
91C3×Dic3—161no—
92C22×Q8—31no—
93C22×Q8—62no—
94C22×Q8—62no—
95C22×Q8C2×C3×D311yescharacteristic
96C22×D4—31no—
97C23×C4—31no—
98#123—182no—
99#123—182no—
100C2×C4⋊C4—91no—
101C2×C4⋊C4—91no—
102C2×C22⋊C4—91no—
103C2×C22⋊C4—91no—
104C2×C22⋊C4—91no—
105#95—91no—
106#95—91no—
107#95—92no—
108C23×C3—21no—
109C23×C3—31no—
110C23×C3—31no—
111C23×C3—61no—
112C23×C3C22×A411yescharacteristic
113C22×D3—21no—
114C22×D3—61no—
115C22×D3—61no—
116C2×A4—41no—
117C2×A4—81no—
118C2×A4—82no—
119C2×A4—81no—
120C2×A4—241no—
121C3×Q8—22no—
122C3×Q8—62no—
123C3×Q8—62no—
124C3×Q8—62no—
125C3×Q8—64no—
126C3×D4—481no—
127C2×C3×C4—32no—
128C2×C3×C4—32no—
129C2×C3×C4—62no—
130C2×C3×C4—64no—
131C2×C3×C4—62no—
132C2×C3×C4—62no—
133C2×C3×C4—62no—
134#67—41no—
135#67—121no—
136C2×Dic3—61no—
137C2×Dic3—61no—
138C2×Dic3—61no—
139C2×Dic3—62no—
140C2×Dic3—61no—
141C2×Dic3—61no—
142SL(2,𝔽3)—82no—
143SL(2,𝔽3)—84no—
144C2×C32—161no—
145C2×C32—321no—
146C3×D3—321no—
147C24—31no—
148C24D3×A411yescharacteristic
149C2×Q8—22no—
150C2×Q8—62no—
151C2×Q8—62no—
152C2×Q8—64no—
153C2×Q8—62no—
154C2×Q8—62no—
155C2×Q8—64no—
156C2×Q8—62no—
157C2×D4—181no—
158C2×D4—181no—
159C22×C4—32no—
160C22×C4—32no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4—91no—
167C22×C4—91no—
168C4⋊C4—182no—
169C4⋊C4—182no—
170C22⋊C4—182no—
171C22⋊C4—182no—
172C22⋊C4—181no—
173C22⋊C4—181no—
174C22×C3—31no—
175C22×C3—31no—
176C22×C3—31no—
177C22×C3—61no—
178C22×C3—61no—
179C22×C3—61no—
180C22×C3—61no—
181C22×C3—161no—
182C22×C3—321no—
183C22×C3—481no—
184C22×C3#120⋊C311yescharacteristic
185C22×C32−1+4⋊C311yescharacteristic
186C2×D3—41no—
187C2×D3—121no—
188C2×D3—121no—
189A4—41no—
190A4—81no—
191C3×C4—62no—
192C3×C4—62no—
193C3×C4—481no—
194Dic3—41no—
195Dic3—121no—
196Dic3—121no—
197C32—161noSylow
198C23—21no—
199C23—31no—
200C23—31no—
201C23—61no—
202C23—61no—
203C23—181no—
204C23—181no—
205C23C2×D3×A411yescharacteristic
206Q8—22no—
207Q8—62no—
208Q8—62no—
209Q8—62no—
210Q8—64no—
211D4—121no—
212D4—361no—
213C2×C4—32no—
214C2×C4—32no—
215C2×C4—62no—
216C2×C4—64no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—181no—
221C2×C4—181no—
222C2×C4—181no—
223C2×C4—181no—
224C2×C4—182no—
225C2×C4—181no—
226C2×C3—21no—
227C2×C3—31no—
228C2×C3—31no—
229C2×C3—61no—
230C2×C3—161no—
231C2×C3—321no—
232C2×C3—322no—
233C2×C3—321no—
234C2×C3—961no—
235C2×C3C2×#120⋊C311yescharacteristic
236D3—81no—
237C22—31no—
238C22—31no—
239C22—31no—
240C22—61no—
241C22—61no—
242C22—61no—
243C22—61no—
244C22—121no—
245C22—361no—
246C22—361no—
247C22C3⋊(#120⋊C3)11yescharacteristic
248C22#1007⋊C311yescharacteristic
249C4—62no—
250C4—62no—
251C4—121no—
252C4—361no—
253C4—361no—
254C3—161no—
255C3—321no—
256C3#2614⋊C311yescharacteristic, complemented
257C2—21no—
258C2—31no—
259C2—31no—
260C2—61no—
261C2—241no—
262C2C2×C3⋊(#120⋊C3)11yescharacteristic
subgroupG/H
centerC2C2×C3⋊(#120⋊C3)
commutator, cosocle, hypocenterC22×C3×Q8C22×C3
FrattiniC23C2×D3×A4
FittingC23×C3×Q8C2×C3
socleC23×C3C22×A4
hypercenterC22C3⋊(#120⋊C3)
LCSG ▷ C22×C3×Q8
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊SL(2,𝔽3)C2semidirect
2. C3⋊(#393⋊C3)C2semidirect
3. C3⋊(#395⋊C3)C2semidirect
4. #84069C3semidirect
8. C23×C3×Q8C2×C3semidirect
10. C2×C22⋊SL(2,𝔽3)D3semidirect
13. #5231C2×C3semidirect
15. #5230C2×C3semidirect
48. C23×Q8C3×D3semidirect
256. C3#2614⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊SL(2,𝔽3)2
C3⋊(#393⋊C3)2
C3⋊(#395⋊C3)2
#840693
#2614⋊C33
A4×#674