Finite Group Databasebeta

← C2×C3⋊((C3⋊S4)⋊C3) #13049285

Subgroups — C2×C3⋊((C3⋊S4)⋊C3) #13049285

Order 1296.

508 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 54 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)⋊C3)C212yesmaximal, direct product factor
3C2×C3⋊(C3⋊S4)C311yesmaximal, characteristic, complemented
4C2×C3×#638—31nomaximal
5C2×(C3⋊S4)⋊C3—33nomaximal
6C2×C3⋊((C3⋊D3)⋊C3)—41nomaximal
7C3×C22⋊H(3,𝔽3)C2211yescharacteristic, complemented
8C23×C33D311yescharacteristic, complemented
9C22×C3×C3⋊D3—31no—
10C2×C32×A4—21no—
11C2×C32×A4C2×C311yescharacteristic, complemented
12C3⋊(C3⋊S4)C2×C312yescomplemented
13C3×#638—32no—
14C3×#638—32no—
15C2×C3×C3⋊Dic3—31no—
16C2×C22⋊H(3,𝔽3)—23no—
17C2×C22⋊H(3,𝔽3)D313yescomplemented
18(C3⋊S4)⋊C3—36no—
19C2×C3×H(3,𝔽3)—41no—
20C3⋊((C3⋊D3)⋊C3)—42no—
21C2×C3⋊S4—33no—
22C2×C3⋊S4—93no—
23C2×#638—31no—
24C2×C3×#67—93no—
25C2×C3×#67—91no—
26C22×C33—31no—
27C22×C33—31no—
28C22×C33C2×D311yescharacteristic, complemented
29C2×C3⋊(C3⋊D3)—41no—
30C2×C3×C3⋊D3—32no—
31C2×C3×C3⋊D3—61no—
32C2×C3×C3⋊D3—61no—
33C32×A4—21no—
34C32×A4C22×C311yescharacteristic, complemented
35C3×C3⋊Dic3—32no—
36C2×(C3⋊D3)⋊C3—123no—
37C22⋊H(3,𝔽3)—23no—
38C22⋊H(3,𝔽3)C2×D313yescomplemented
39C3×H(3,𝔽3)—41noSylow
40C23×C32—21no—
41C23×C32—33no—
42C23×C32C3×D311yescharacteristic
43C23×C32C3⋊D311yescharacteristic, complemented
44C22×C3⋊D3—31no—
45C22×C3×D3—93no—
46C22×C3×D3—91no—
47C2×C3×A4—23no—
48C2×C3×A4—33no—
49C2×C3×A4—63no—
50C2×C3×A4C3×D313yescomplemented
51C3⋊S4—36no—
52C3⋊S4—96no—
53#638—32no—
54#638—32no—
55C2×C3⋊Dic3—31no—
56C3×#67—92no—
57C3×#67—96no—
58C3×#67—96no—
59C3×#67—92no—
60C2×C3×Dic3—93no—
61C2×C3×Dic3—91no—
62C2×C33—31no—
63C2×C33—31no—
64C2×C33—41no—
65C2×C33—81no—
66C2×C33S411yescharacteristic, complemented
67C3⋊(C3⋊D3)—42no—
68C3×C3⋊D3—62no—
69C2×H(3,𝔽3)—43no—
70C2×H(3,𝔽3)—83no—
71(C3⋊D3)⋊C3—126no—
72C2×S4—273no—
73C2×C3×D4—271no—
74C2×#67—93no—
75C2×#67—91no—
76C22×C32—21no—
77C22×C32—31no—
78C22×C32—31no—
79C22×C32—31no—
80C22×C32—33no—
81C22×C32—31no—
82C22×C32—33no—
83C22×C32—33no—
84C22×C32—61no—
85C22×C32—63no—
86C22×C32—63no—
87C22×C32—61no—
88C22×C32C2×C3×D311yescharacteristic
89C22×C32C2×C3⋊D311yescharacteristic, complemented
90C2×C3⋊D3—32no—
91C2×C3⋊D3—61no—
92C2×C3⋊D3—61no—
93C2×C3⋊D3—123no—
94C2×C3⋊D3—363no—
95C2×C3×D3—96no—
96C2×C3×D3—92no—
97C2×C3×D3—183no—
98C2×C3×D3—181no—
99C2×C3×D3—183no—
100C2×C3×D3—181no—
101C3×A4—23no—
102C3×A4—33no—
103C3×A4—63no—
104C3×A4C2×C3×D313yescomplemented
105C3⋊Dic3—32no—
106C3×Dic3—96no—
107C3×Dic3—92no—
108C33—41no—
109C33—81no—
110C33C2×S411yescharacteristic, complemented
111H(3,𝔽3)—43no—
112H(3,𝔽3)—83no—
113C23×C3—31no—
114C23×C3—61no—
115C23×C3—271no—
116C23×C3(C3⋊D3)⋊C313yescomplemented
117C23×C3C3×C3⋊D311yescharacteristic
118C22×D3—93no—
119C22×D3—91no—
120C2×A4—33no—
121C2×A4—63no—
122S4—276no—
123C3×D4—272no—
124C3×D4—272no—
125C2×C3×C4—271no—
126#67—96no—
127#67—92no—
128#67—96no—
129#67—92no—
130C2×Dic3—93no—
131C2×Dic3—91no—
132C2×C32—21no—
133C2×C32—31no—
134C2×C32—31no—
135C2×C32—33no—
136C2×C32—33no—
137C2×C32—31no—
138C2×C32—31no—
139C2×C32—33no—
140C2×C32—43no—
141C2×C32—61no—
142C2×C32—61no—
143C2×C32—63no—
144C2×C32—63no—
145C2×C32—83no—
146C2×C32—123no—
147C2×C32—243no—
148C2×C32C3×S411yescharacteristic
149C2×C32C3⋊S411yescharacteristic, complemented
150C3⋊D3—62no—
151C3⋊D3—126no—
152C3⋊D3—366no—
153C3×D3—186no—
154C3×D3—182no—
155C2×D4—271noSylow
156C22×C3—33no—
157C22×C3—31no—
158C22×C3—31no—
159C22×C3—33no—
160C22×C3—31no—
161C22×C3—31no—
162C22×C3—31no—
163C22×C3—61no—
164C22×C3—61no—
165C22×C3—63no—
166C22×C3—63no—
167C22×C3—61no—
168C22×C3—272no—
169C22×C3—541no—
170C22×C3—541no—
171C22×C3C2×(C3⋊D3)⋊C313yescomplemented
172C22×C3C2×C3×C3⋊D311yescharacteristic
173C2×D3—92no—
174C2×D3—96no—
175C2×D3—181no—
176C2×D3—183no—
177C2×D3—183no—
178C2×D3—181no—
179C2×D3—1083no—
180A4—33no—
181A4—63no—
182C3×C4—272no—
183Dic3—92no—
184Dic3—96no—
185C32—21no—
186C32—33no—
187C32—43no—
188C32—83no—
189C32—123no—
190C32—243no—
191C32C2×C3×S411yescharacteristic
192C32C2×C3⋊S411yescharacteristic, complemented
193C23—271no—
194C23C3⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
195D4—272no—
196D4—272no—
197C2×C4—271no—
198C2×C3—31no—
199C2×C3—31no—
200C2×C3—31no—
201C2×C3—31no—
202C2×C3—33no—
203C2×C3—33no—
204C2×C3—31no—
205C2×C3—61no—
206C2×C3—61no—
207C2×C3—63no—
208C2×C3—63no—
209C2×C3—61no—
210C2×C3—123no—
211C2×C3—243no—
212C2×C3—542no—
213C2×C3(C3⋊S4)⋊C313yescomplemented
214C2×C3C3×C3⋊S411yescharacteristic
215D3—182no—
216D3—186no—
217D3—1086no—
218C22—31no—
219C22—31no—
220C22—272no—
221C22—541no—
222C22—541no—
223C22C2×C3⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
224C4—272no—
225C3—31no—
226C3—61no—
227C3—123no—
228C3—243no—
229C3C2×(C3⋊S4)⋊C313yescomplemented
230C3C2×C3×C3⋊S411yescharacteristic
231C2—31no—
232C2—31no—
233C2—542no—
234C2C3⋊((C3⋊S4)⋊C3)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊((C3⋊S4)⋊C3)
commutator, cosocle, hypocenterC32×A4C22×C3
FrattiniC3C2×C3×C3⋊S4
FittingC23×C33D3
socleC23×C32C3×D3
LCSG ▷ C32×A4
UCSC1 ◁ C2

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)⋊C3)C2direct (N × H)
3. C2×C3⋊(C3⋊S4)C3semidirect
7. C3×C22⋊H(3,𝔽3)C22semidirect
8. C23×C33D3semidirect
11. C2×C32×A4C2×C3semidirect
12. C3⋊(C3⋊S4)C2×C3semidirect
17. C2×C22⋊H(3,𝔽3)D3semidirect
28. C22×C33C2×D3semidirect
34. C32×A4C22×C3semidirect
38. C22⋊H(3,𝔽3)C2×D3semidirect
43. C23×C32C3⋊D3semidirect
50. C2×C3×A4C3×D3semidirect
66. C2×C33S4semidirect
89. C22×C32C2×C3⋊D3semidirect
104. C3×A4C2×C3×D3semidirect
110. C33C2×S4semidirect
116. C23×C3(C3⋊D3)⋊C3semidirect
149. C2×C32C3⋊S4semidirect
171. C22×C3C2×(C3⋊D3)⋊C3semidirect
192. C32C2×C3⋊S4semidirect
194. C23C3⋊((C3⋊D3)⋊C3)semidirect
213. C2×C3(C3⋊S4)⋊C3semidirect
223. C22C2×C3⋊((C3⋊D3)⋊C3)semidirect
229. C3C2×(C3⋊S4)⋊C3semidirect
234. C2C3⋊((C3⋊S4)⋊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⋊H(3,𝔽3)2
C3⋊((C3⋊S4)⋊C3)2
C2×C3⋊(C3⋊S4)3
C2×C3×#6383
C2×(C3⋊S4)⋊C33
C2×C3⋊((C3⋊D3)⋊C3)4