Finite Group Databasebeta

← C2×C3×(C3⋊S4)⋊C3 #13049273

Subgroups — C2×C3×(C3⋊S4)⋊C3 #13049273

Order 1296.

460 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 62 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⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C3×(C3⋊S4)⋊C3C212yesmaximal, direct product factor
3C2×C3×C3⋊S4C311yesmaximal, characteristic, complemented
4C2×C32×#67—31nomaximal
5C2×(C3⋊S4)⋊C3C313yesmaximal, direct product factor
6C2×C3×(C3⋊D3)⋊C3—41nomaximal
7C3×C22⋊H(3,𝔽3)C2211yescharacteristic, complemented
8C23×C33D311yescharacteristic, complemented
9C22×C32×D3—31no—
10C2×C32×A4—21no—
11C2×C32×A4C2×C311yescharacteristic, complemented
12C3×C3⋊S4C2×C312yescomplemented
13C32×#67—32no—
14C32×#67—32no—
15C2×C32×Dic3—31no—
16C2×C22⋊H(3,𝔽3)—23no—
17C2×C22⋊H(3,𝔽3)C2×C313yescomplemented
18(C3⋊S4)⋊C3C2×C316yesdirect product factor
19C2×C3×H(3,𝔽3)—41no—
20C3×(C3⋊D3)⋊C3—42no—
21C2×C3⋊S4C3211yescharacteristic, complemented
22C2×C3×S4—91no—
23C2×C32×D4—91no—
24C2×C3×#67—31no—
25C2×C3×#67—33no—
26C22×C33—31no—
27C22×C33—31no—
28C22×C33C2×D311yescharacteristic, complemented
29C2×C3×C3⋊D3—41no—
30C2×C32×D3—32no—
31C2×C32×D3—61no—
32C2×C32×D3—61no—
33C32×A4—21no—
34C32×A4C22×C311yescharacteristic, complemented
35C32×Dic3—32no—
36C2×(C3⋊D3)⋊C3—43no—
37C22⋊H(3,𝔽3)—23no—
38C22⋊H(3,𝔽3)C22×C313yescomplemented
39C3×H(3,𝔽3)—41noSylow
40C23×C32—31no—
41C23×C32—61no—
42C23×C32—91no—
43C23×C32C3×D311yescharacteristic
44C23×C32C3×D313yescomplemented
45C22×C3×D3—31no—
46C22×C3×D3—33no—
47C2×C3×A4—21no—
48C2×C3×A4—23no—
49C2×C3×A4—31no—
50C2×C3×A4—61no—
51C2×C3×A4—62no—
52C2×C3×A4—61no—
53C2×C3×A4C2×C3211yescharacteristic, complemented
54C3⋊S4C2×C3212yescomplemented
55C3×S4—92no—
56C32×D4—92no—
57C32×D4—92no—
58C2×C32×C4—91no—
59C3×#67—32no—
60C3×#67—36no—
61C3×#67—32no—
62C3×#67—36no—
63C2×C3×Dic3—31no—
64C2×C3×Dic3—33no—
65C2×C33—31no—
66C2×C33—31no—
67C2×C33—41no—
68C2×C33—81no—
69C2×C33S411yescharacteristic, complemented
70C3×C3⋊D3—42no—
71C32×D3—62no—
72C2×H(3,𝔽3)—43no—
73C2×H(3,𝔽3)—83no—
74(C3⋊D3)⋊C3—46no—
75C2×S4—91no—
76C2×C3×D4—93no—
77C2×C3×D4—91no—
78C2×#67—31no—
79C22×C32—31no—
80C22×C32—31no—
81C22×C32—31no—
82C22×C32—31no—
83C22×C32—33no—
84C22×C32—31no—
85C22×C32—33no—
86C22×C32—61no—
87C22×C32—61no—
88C22×C32—61no—
89C22×C32—63no—
90C22×C32—63no—
91C22×C32—92no—
92C22×C32—181no—
93C22×C32—181no—
94C22×C32C2×C3×D311yescharacteristic
95C22×C32C2×C3×D313yescomplemented
96C2×C3⋊D3—41no—
97C2×C3×D3—32no—
98C2×C3×D3—36no—
99C2×C3×D3—61no—
100C2×C3×D3—63no—
101C2×C3×D3—61no—
102C2×C3×D3—63no—
103C2×C3×D3—361no—
104C3×A4—21no—
105C3×A4—23no—
106C3×A4—31no—
107C3×A4—61no—
108C3×A4—62no—
109C3×A4—61no—
110C3×A4C22×C3211yescharacteristic, complemented
111C32×C4—92no—
112C3×Dic3—32no—
113C3×Dic3—36no—
114C33—41no—
115C33—81no—
116C33C2×S411yescharacteristic, complemented
117H(3,𝔽3)—43no—
118H(3,𝔽3)—83no—
119C23×C3—21no—
120C23×C3—33no—
121C23×C3—93no—
122C23×C3—91no—
123C23×C3(C3⋊D3)⋊C311yescharacteristic, complemented
124C23×C3C32×D311yescharacteristic
125C22×D3—31no—
126C2×A4—31no—
127C2×A4—61no—
128C2×A4—63no—
129S4—92no—
130C3×D4—92no—
131C3×D4—96no—
132C3×D4—96no—
133C3×D4—92no—
134C2×C3×C4—93no—
135C2×C3×C4—91no—
136#67—32no—
137#67—32no—
138C2×Dic3—31no—
139C2×C32—31no—
140C2×C32—31no—
141C2×C32—31no—
142C2×C32—31no—
143C2×C32—31no—
144C2×C32—33no—
145C2×C32—33no—
146C2×C32—41no—
147C2×C32—61no—
148C2×C32—61no—
149C2×C32—61no—
150C2×C32—63no—
151C2×C32—63no—
152C2×C32—83no—
153C2×C32—81no—
154C2×C32—121no—
155C2×C32—182no—
156C2×C32—241no—
157C2×C32—242no—
158C2×C32—241no—
159C2×C32C3×S411yescharacteristic
160C2×C32C3×S413yescomplemented
161C3⋊D3—42no—
162C3×D3—62no—
163C3×D3—66no—
164C3×D3—362no—
165C2×D4—91noSylow
166C22×C3—21no—
167C22×C3—31no—
168C22×C3—31no—
169C22×C3—33no—
170C22×C3—31no—
171C22×C3—31no—
172C22×C3—33no—
173C22×C3—33no—
174C22×C3—61no—
175C22×C3—63no—
176C22×C3—63no—
177C22×C3—61no—
178C22×C3—96no—
179C22×C3—92no—
180C22×C3—183no—
181C22×C3—181no—
182C22×C3—183no—
183C22×C3—181no—
184C22×C3C2×(C3⋊D3)⋊C311yescharacteristic, complemented
185C22×C3C2×C32×D311yescharacteristic
186C2×D3—32no—
187C2×D3—61no—
188C2×D3—61no—
189C2×D3—361no—
190A4—31no—
191A4—61no—
192A4—63no—
193C3×C4—96no—
194C3×C4—92no—
195Dic3—32no—
196C32—31no—
197C32—41no—
198C32—61no—
199C32—83no—
200C32—81no—
201C32—121no—
202C32—241no—
203C32—242no—
204C32—241no—
205C32C2×C3×S411yescharacteristic
206C32C2×C3×S413yescomplemented
207C23—91no—
208C23C3×(C3⋊D3)⋊C311yescharacteristic, complemented
209D4—92no—
210D4—92no—
211C2×C4—91no—
212C2×C3—21no—
213C2×C3—33no—
214C2×C3—33no—
215C2×C3—31no—
216C2×C3—31no—
217C2×C3—31no—
218C2×C3—31no—
219C2×C3—33no—
220C2×C3—61no—
221C2×C3—61no—
222C2×C3—63no—
223C2×C3—63no—
224C2×C3—121no—
225C2×C3—186no—
226C2×C3—182no—
227C2×C3—241no—
228C2×C3—243no—
229C2×C3(C3⋊S4)⋊C311yescharacteristic, direct product factor
230C2×C3C32×S411yescharacteristic
231D3—62no—
232D3—362no—
233C22—31no—
234C22—31no—
235C22—92no—
236C22—181no—
237C22—181no—
238C22C2×C3×(C3⋊D3)⋊C311yescharacteristic, complemented
239C4—92no—
240C3—21no—
241C3—33no—
242C3—121no—
243C3—241no—
244C3—243no—
245C3C2×(C3⋊S4)⋊C311yescharacteristic, direct product factor
246C3C2×C32×S411yescharacteristic
247C2—31no—
248C2—31no—
249C2—182no—
250C2C3×(C3⋊S4)⋊C311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×A4C22×C32
FrattiniC3C2×C32×S4
FittingC23×C33D3
socleC23×C32C3×D3
LCSG ▷ C3×A4
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊H(3,𝔽3)C2semidirect
2. C3×(C3⋊S4)⋊C3C2direct (N × H)
3. C2×C3×C3⋊S4C3semidirect
5. C2×(C3⋊S4)⋊C3C3direct (N × H)
7. C3×C22⋊H(3,𝔽3)C22semidirect
8. C23×C33D3semidirect
11. C2×C32×A4C2×C3semidirect
12. C3×C3⋊S4C2×C3semidirect
17. C2×C22⋊H(3,𝔽3)C2×C3semidirect
18. (C3⋊S4)⋊C3C2×C3direct (N × H)
21. C2×C3⋊S4C32semidirect
28. C22×C33C2×D3semidirect
34. C32×A4C22×C3semidirect
38. C22⋊H(3,𝔽3)C22×C3semidirect
44. C23×C32C3×D3semidirect
53. C2×C3×A4C2×C32semidirect
54. C3⋊S4C2×C32semidirect
69. C2×C33S4semidirect
95. C22×C32C2×C3×D3semidirect
110. C3×A4C22×C32semidirect
116. C33C2×S4semidirect
123. C23×C3(C3⋊D3)⋊C3semidirect
160. C2×C32C3×S4semidirect
184. C22×C3C2×(C3⋊D3)⋊C3semidirect
206. C32C2×C3×S4semidirect
208. C23C3×(C3⋊D3)⋊C3semidirect
229. C2×C3(C3⋊S4)⋊C3direct (N × H)
238. C22C2×C3×(C3⋊D3)⋊C3semidirect
245. C3C2×(C3⋊S4)⋊C3direct (N × H)
250. C2C3×(C3⋊S4)⋊C3direct (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⋊H(3,𝔽3)2
C3×(C3⋊S4)⋊C32
C2×C3×C3⋊S43
C2×C32×#673
C2×(C3⋊S4)⋊C33
C2×C3×(C3⋊D3)⋊C34