Finite Group Databasebeta

← C2×C3⋊(#393⋊C3) #11921365

Subgroups — C2×C3⋊(#393⋊C3) #11921365

Order 1152.

502 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 45 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)C212yesmaximal, direct product factor
3C2×#5231C311yesmaximal, characteristic, complemented
4C2×#393⋊C3—31nomaximal
5C22×D3×A4—41nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic, complemented
7C23×C3×Q8C2×C311yescharacteristic, complemented
8C2×C22⋊SL(2,𝔽3)—21no—
9C2×C22⋊SL(2,𝔽3)D311yescharacteristic, complemented
10C22×(C2×D3)⋊C4—31no—
11#5231—32no—
12#5231C2×C312yescomplemented
13C2×C3⋊#95—31no—
14#393⋊C3—32no—
15C22×C3×A4—41no—
16C2×D3×A4—44no—
17C2×D3×A4—42no—
18C2×C3×SL(2,𝔽3)—82no—
19C2×#393—31noSylow
20C24×D3A411yescharacteristic
21C23×A4—121no—
22C22×C3×Q8—31no—
23C22×C3×Q8—62no—
24C22×C3×Q8—62no—
25C22×C3×Q8C22×C311yescharacteristic, complemented
26C23×C3×C4—31no—
27C23×Dic3—31no—
28C22⋊SL(2,𝔽3)—21no—
29C22⋊SL(2,𝔽3)C2×D311yescharacteristic, complemented
30C2×(C2×D3)⋊C4—32no—
31C2×(C2×D3)⋊C4—32no—
32C2×(C2×D3)⋊C4—62no—
33C2×(C2×D3)⋊C4—62no—
34C3⋊#95—32no—
35C3⋊#95—32no—
36C22×C3×D3—161no—
37C2×C3×A4—42no—
38C2×C3×A4—41no—
39D3×A4—44no—
40C3×SL(2,𝔽3)—82no—
41C23×Q8C3×D311yescharacteristic, complemented
42C22×C22⋊C4—91no—
43#393—32no—
44#393—92no—
45C2×#95—91no—
46C24×C3C2×A411yescharacteristic
47C23×D3—61no—
48C23×D3—61no—
49C23×D3—62no—
50C23×D3SL(2,𝔽3)14yescomplemented
51C23×D3C2×A412yes—
52C22×A4—41no—
53C22×A4—81no—
54C22×A4—122no—
55C22×A4—124no—
56C2×C3×Q8—22no—
57C2×C3×Q8—62no—
58C2×C3×Q8—62no—
59C2×C3×Q8—64no—
60C2×C3×Q8—62no—
61C2×C3×Q8—62no—
62C2×C3×Q8—64no—
63C2×C3×Q8—62no—
64C22×C3×C4—32no—
65C22×C3×C4—32no—
66C22×C3×C4—31no—
67C22×C3×C4—31no—
68C22×C3×C4—62no—
69C22×C3×C4—62no—
70C22×Dic3—32no—
71C22×Dic3—61no—
72C22×Dic3—61no—
73C22×Dic3—64no—
74C2×SL(2,𝔽3)—82no—
75C2×SL(2,𝔽3)—84no—
76(C2×D3)⋊C4—64no—
77(C2×D3)⋊C4—64no—
78C22×C32—161no—
79C2×C3×D3—164no—
80C2×C3×D3—162no—
81C3×A4—41no—
82C25—31no—
83C22×Q8—31no—
84C22×Q8—62no—
85C22×Q8—62no—
86C22×Q8C2×C3×D311yescharacteristic, complemented
87C23×C4—31no—
88C23×C4—91no—
89C2×C22⋊C4—92no—
90C2×C22⋊C4—92no—
91C2×C22⋊C4—182no—
92C2×C22⋊C4—182no—
93#95—92no—
94#95—92no—
95C23×C3—32no—
96C23×C3—31no—
97C23×C3—31no—
98C23×C3—481no—
99C23×C3C2×SL(2,𝔽3)12yes—
100C23×C3C22×A411yescharacteristic
101C22×D3—41no—
102C22×D3—62no—
103C22×D3—64no—
104C22×D3—64no—
105C22×D3—62no—
106C22×D3—122no—
107C22×D3—121no—
108C22×D3—122no—
109C22×D3C2×SL(2,𝔽3)14yescomplemented
110C2×A4—41no—
111C2×A4—42no—
112C2×A4—81no—
113C2×A4—82no—
114C2×A4—124no—
115C3×Q8—22no—
116C3×Q8—62no—
117C3×Q8—62no—
118C3×Q8—62no—
119C3×Q8—64no—
120C2×C3×C4—32no—
121C2×C3×C4—32no—
122C2×C3×C4—62no—
123C2×C3×C4—64no—
124C2×C3×C4—62no—
125C2×C3×C4—62no—
126C2×C3×C4—62no—
127C2×Dic3—62no—
128C2×Dic3—64no—
129C2×Dic3—122no—
130C2×Dic3—122no—
131SL(2,𝔽3)—82no—
132SL(2,𝔽3)—84no—
133C2×C32—162no—
134C2×C32—161no—
135C3×D3—164no—
136C24—32no—
137C24—34no—
138C24—181no—
139C24—181no—
140C24—182no—
141C24D3×A411yescharacteristic
142C2×Q8—22no—
143C2×Q8—62no—
144C2×Q8—62no—
145C2×Q8—64no—
146C2×Q8—62no—
147C2×Q8—62no—
148C2×Q8—64no—
149C2×Q8—62no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—62no—
155C22×C4—62no—
156C22×C4—92no—
157C22×C4—181no—
158C22×C4—181no—
159C22×C4—184no—
160C22⋊C4—184no—
161C22⋊C4—184no—
162C22×C3—31no—
163C22×C3—32no—
164C22×C3—32no—
165C22×C3—32no—
166C22×C3—32no—
167C22×C3—31no—
168C22×C3—31no—
169C22×C3—161no—
170C22×C3—321no—
171C22×C3—484no—
172C22×C3—482no—
173C22×C3#120⋊C311yescharacteristic
174C22×C3C22×SL(2,𝔽3)11yescharacteristic
175C2×D3—44no—
176C2×D3—42no—
177C2×D3—64no—
178C2×D3—122no—
179C2×D3—124no—
180A4—41no—
181A4—81no—
182C3×C4—62no—
183C3×C4—62no—
184Dic3—122no—
185C32—161noSylow
186C23—31no—
187C23—32no—
188C23—31no—
189C23—34no—
190C23—121no—
191C23—182no—
192C23—184no—
193C23—182no—
194C23—184no—
195C23—362no—
196C23—361no—
197C23—362no—
198C23D3×SL(2,𝔽3)12yes—
199C23C2×D3×A411yescharacteristic
200Q8—22no—
201Q8—62no—
202Q8—62no—
203Q8—62no—
204Q8—64no—
205C2×C4—32no—
206C2×C4—32no—
207C2×C4—62no—
208C2×C4—64no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—182no—
213C2×C4—184no—
214C2×C4—362no—
215C2×C4—362no—
216C2×C3—31no—
217C2×C3—31no—
218C2×C3—32no—
219C2×C3—162no—
220C2×C3—161no—
221C2×C3—322no—
222C2×C3—321no—
223C2×C3—484no—
224C2×C3#393⋊C312yescomplemented
225C2×C3C2×#120⋊C311yescharacteristic
226D3—44no—
227C22—31no—
228C22—32no—
229C22—32no—
230C22—32no—
231C22—32no—
232C22—31no—
233C22—31no—
234C22—124no—
235C22—122no—
236C22—184no—
237C22—362no—
238C22—364no—
239C22C3⋊(#120⋊C3)11yescharacteristic
240C22C2×D3×SL(2,𝔽3)11yescharacteristic
241C4—62no—
242C4—62no—
243C4—362no—
244C3—161no—
245C3—321no—
246C3C2×#393⋊C311yescharacteristic, complemented
247C2—31no—
248C2—31no—
249C2—32no—
250C2—124no—
251C2C3⋊(#393⋊C3)12yesdirect product factor
252C2C2×C3⋊(#120⋊C3)11yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊(#120⋊C3)
commutator, cosocle, hypocenterC22×C3×Q8C22×C3
FrattiniC23C2×D3×A4
FittingC23×C3×Q8C2×C3
socleC24×C3C2×A4
LCSG ▷ C22×C3×Q8
UCSC1 ◁ 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)C2direct (N × H)
3. C2×#5231C3semidirect
6. C3×C22⋊SL(2,𝔽3)C22semidirect
7. C23×C3×Q8C2×C3semidirect
9. C2×C22⋊SL(2,𝔽3)D3semidirect
12. #5231C2×C3semidirect
25. C22×C3×Q8C22×C3semidirect
29. C22⋊SL(2,𝔽3)C2×D3semidirect
41. C23×Q8C3×D3semidirect
50. C23×D3SL(2,𝔽3)semidirect
86. C22×Q8C2×C3×D3semidirect
109. C22×D3C2×SL(2,𝔽3)semidirect
224. C2×C3#393⋊C3semidirect
246. C3C2×#393⋊C3semidirect
251. C2C3⋊(#393⋊C3)direct (N × H)

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
C2×#52313
C2×#393⋊C33
C22×D3×A44