Finite Group Databasebeta

← C2×C3×(C3⋊D3)⋊A4 #13049298

Subgroups — C2×C3×(C3⋊D3)⋊A4 #13049298

Order 1296.

532 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 64 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⋊D3)⋊A4C212yesmaximal, direct product factor
3C23×C3×C3⋊D3C311yesmaximal, characteristic, complemented
4C2×C3×D3×A4—31nomaximal
5C2×(C3⋊D3)⋊A4C313yesmaximal, direct product factor
6C2×C3×(C3⋊D3)⋊C3—41nomaximal
7C3×C22⋊H(3,𝔽3)C2211yescharacteristic, complemented
8C23×C33C2×C311yescharacteristic, complemented
9C22×C3×C3⋊D3—31no—
10C22×C3×C3⋊D3—31no—
11C22×C3×C3⋊D3—32no—
12C22×C3×C3⋊D3C2×C312yescomplemented
13C2×C32×A4—21no—
14C2×C32×A4D311yescharacteristic, complemented
15C3×D3×A4—32no—
16C2×C22⋊H(3,𝔽3)—23no—
17C2×C22⋊H(3,𝔽3)C2×C313yescomplemented
18(C3⋊D3)⋊A4C2×C316yesdirect product factor
19C2×C3×H(3,𝔽3)—41no—
20C3×(C3⋊D3)⋊C3—42no—
21C23×C3⋊D3C3211yescharacteristic, complemented
22C23×C3×D3—31no—
23C23×C3×D3—91no—
24C22×C3×A4—91no—
25C2×D3×A4—33no—
26C22×C33—31no—
27C22×C33—31no—
28C22×C33C22×C311yescharacteristic, complemented
29C2×C3×C3⋊D3—32no—
30C2×C3×C3⋊D3—31no—
31C2×C3×C3⋊D3—32no—
32C2×C3×C3⋊D3—32no—
33C2×C3×C3⋊D3—32no—
34C2×C3×C3⋊D3A411yescharacteristic, complemented
35C2×C32×D3—121no—
36C32×A4—21no—
37C32×A4C2×D311yescharacteristic, complemented
38C2×(C3⋊D3)⋊C3—43no—
39C22⋊H(3,𝔽3)—23no—
40C22⋊H(3,𝔽3)C22×C313yescomplemented
41C3×H(3,𝔽3)—41noSylow
42C23×C32—21no—
43C23×C32—31no—
44C23×C32—61no—
45C23×C32C3×D311yescharacteristic
46C23×C32C2×C3211yescharacteristic, complemented
47C22×C3⋊D3—31no—
48C22×C3⋊D3—31no—
49C22×C3⋊D3—32no—
50C22×C3⋊D3C2×C3212yescomplemented
51C22×C3×D3—32no—
52C22×C3×D3—92no—
53C22×C3×D3—91no—
54C22×C3×D3—93no—
55C22×C3×D3—96no—
56C22×C3×D3—93no—
57C22×C3×D3—92no—
58C22×C3×D3—91no—
59C2×C3×A4—23no—
60C2×C3×A4—31no—
61C2×C3×A4—61no—
62C2×C3×A4—62no—
63C2×C3×A4—61no—
64C2×C3×A4—92no—
65C2×C3×A4C3×D313yescomplemented
66D3×A4—36no—
67C2×C33—31no—
68C2×C33—31no—
69C2×C33—41no—
70C2×C33—81no—
71C2×C33C2×A411yescharacteristic, complemented
72C3×C3⋊D3—32no—
73C3×C3⋊D3C2×A412yescomplemented
74C32×D3—122no—
75C2×H(3,𝔽3)—43no—
76C2×H(3,𝔽3)—83no—
77(C3⋊D3)⋊C3—46no—
78C24×C3—91no—
79C23×D3—31no—
80C23×D3—91no—
81C22×A4—93no—
82C22×C32—21no—
83C22×C32—31no—
84C22×C32—31no—
85C22×C32—31no—
86C22×C32—33no—
87C22×C32—33no—
88C22×C32—31no—
89C22×C32—31no—
90C22×C32—61no—
91C22×C32—61no—
92C22×C32—61no—
93C22×C32—63no—
94C22×C32—63no—
95C22×C32—361no—
96C22×C32C2×C3×D311yescharacteristic
97C22×C32C22×C3211yescharacteristic, complemented
98C2×C3⋊D3—32no—
99C2×C3⋊D3—31no—
100C2×C3⋊D3—32no—
101C2×C3⋊D3—32no—
102C2×C3⋊D3—32no—
103C2×C3⋊D3C3×A411yescharacteristic, complemented
104C2×C3×D3—31no—
105C2×C3×D3—91no—
106C2×C3×D3—92no—
107C2×C3×D3—92no—
108C2×C3×D3—96no—
109C2×C3×D3—96no—
110C2×C3×D3—96no—
111C2×C3×D3—96no—
112C2×C3×D3—92no—
113C2×C3×D3—93no—
114C2×C3×D3—91no—
115C2×C3×D3—92no—
116C2×C3×D3—123no—
117C3×A4—23no—
118C3×A4—31no—
119C3×A4—61no—
120C3×A4—62no—
121C3×A4—61no—
122C3×A4C2×C3×D313yescomplemented
123C33—41no—
124C33—81no—
125C33C22×A411yescharacteristic, complemented
126H(3,𝔽3)—43no—
127H(3,𝔽3)—83no—
128C23×C3—21no—
129C23×C3—31no—
130C23×C3—61no—
131C23×C3—92no—
132C23×C3—271no—
133C23×C3—272no—
134C23×C3—271no—
135C23×C3(C3⋊D3)⋊C311yescharacteristic, complemented
136C23×C3C32×D311yescharacteristic
137C22×D3—32no—
138C22×D3—92no—
139C22×D3—91no—
140C22×D3—93no—
141C22×D3—96no—
142C22×D3—93no—
143C22×D3—92no—
144C22×D3—91no—
145C2×A4—33no—
146C2×A4—63no—
147C2×A4—96no—
148C2×C32—21no—
149C2×C32—31no—
150C2×C32—31no—
151C2×C32—33no—
152C2×C32—33no—
153C2×C32—31no—
154C2×C32—31no—
155C2×C32—31no—
156C2×C32—43no—
157C2×C32—61no—
158C2×C32—61no—
159C2×C32—61no—
160C2×C32—63no—
161C2×C32—63no—
162C2×C32—83no—
163C2×C32—121no—
164C2×C32—241no—
165C2×C32—242no—
166C2×C32—241no—
167C2×C32—362no—
168C2×C32D3×A411yescharacteristic
169C2×C32C2×C3×A411yescharacteristic, complemented
170C3⋊D3—32no—
171C3⋊D3C2×C3×A412yescomplemented
172C3×D3—32no—
173C3×D3—92no—
174C3×D3—96no—
175C3×D3—92no—
176C3×D3—126no—
177C24—91noSylow
178C22×C3—21no—
179C22×C3—31no—
180C22×C3—31no—
181C22×C3—33no—
182C22×C3—33no—
183C22×C3—31no—
184C22×C3—31no—
185C22×C3—31no—
186C22×C3—61no—
187C22×C3—61no—
188C22×C3—63no—
189C22×C3—63no—
190C22×C3—61no—
191C22×C3—91no—
192C22×C3—272no—
193C22×C3—272no—
194C22×C3—272no—
195C22×C3—272no—
196C22×C3—271no—
197C22×C3—363no—
198C22×C3C2×(C3⋊D3)⋊C311yescharacteristic, complemented
199C22×C3C2×C32×D311yescharacteristic
200C2×D3—31no—
201C2×D3—91no—
202C2×D3—92no—
203C2×D3—92no—
204C2×D3—96no—
205C2×D3—96no—
206C2×D3—96no—
207C2×D3—96no—
208C2×D3—92no—
209C2×D3—93no—
210C2×D3—91no—
211C2×D3—92no—
212A4—33no—
213A4—63no—
214C32—21no—
215C32—31no—
216C32—43no—
217C32—61no—
218C32—83no—
219C32—121no—
220C32—241no—
221C32—242no—
222C32—241no—
223C32C2×D3×A411yescharacteristic
224C32C22×C3×A411yescharacteristic, complemented
225C23—92no—
226C23—271no—
227C23—272no—
228C23—271no—
229C23C3×(C3⋊D3)⋊C311yescharacteristic, complemented
230C2×C3—21no—
231C2×C3—33no—
232C2×C3—33no—
233C2×C3—31no—
234C2×C3—31no—
235C2×C3—31no—
236C2×C3—31no—
237C2×C3—31no—
238C2×C3—61no—
239C2×C3—61no—
240C2×C3—63no—
241C2×C3—63no—
242C2×C3—61no—
243C2×C3—92no—
244C2×C3—123no—
245C2×C3—243no—
246C2×C3—272no—
247C2×C3—366no—
248C2×C3(C3⋊D3)⋊A411yescharacteristic, direct product factor
249C2×C3C3×D3×A411yescharacteristic
250D3—32no—
251D3—92no—
252D3—96no—
253D3—92no—
254C22—31no—
255C22—31no—
256C22—91no—
257C22—272no—
258C22—272no—
259C22—272no—
260C22—272no—
261C22—271no—
262C22C2×C3×(C3⋊D3)⋊C311yescharacteristic, complemented
263C3—21no—
264C3—31no—
265C3—61no—
266C3—123no—
267C3—243no—
268C3C2×(C3⋊D3)⋊A411yescharacteristic, direct product factor
269C3C2×C3×D3×A411yescharacteristic
270C2—31no—
271C2—31no—
272C2—92no—
273C2—272no—
274C2C3×(C3⋊D3)⋊A411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3(C3⋊D3)⋊A4
commutator, cosocle, hypocenterC22×C32C22×C32
FrattiniC3C2×C3×D3×A4
FittingC23×C33C2×C3
socleC23×C32C3×D3
LCSG ▷ C22×C32
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⋊D3)⋊A4C2direct (N × H)
3. C23×C3×C3⋊D3C3semidirect
5. C2×(C3⋊D3)⋊A4C3direct (N × H)
7. C3×C22⋊H(3,𝔽3)C22semidirect
8. C23×C33C2×C3semidirect
12. C22×C3×C3⋊D3C2×C3semidirect
14. C2×C32×A4D3semidirect
17. C2×C22⋊H(3,𝔽3)C2×C3semidirect
18. (C3⋊D3)⋊A4C2×C3direct (N × H)
21. C23×C3⋊D3C32semidirect
28. C22×C33C22×C3semidirect
34. C2×C3×C3⋊D3A4semidirect
37. C32×A4C2×D3semidirect
40. C22⋊H(3,𝔽3)C22×C3semidirect
46. C23×C32C2×C32semidirect
50. C22×C3⋊D3C2×C32semidirect
65. C2×C3×A4C3×D3semidirect
71. C2×C33C2×A4semidirect
73. C3×C3⋊D3C2×A4semidirect
97. C22×C32C22×C32semidirect
103. C2×C3⋊D3C3×A4semidirect
122. C3×A4C2×C3×D3semidirect
125. C33C22×A4semidirect
135. C23×C3(C3⋊D3)⋊C3semidirect
169. C2×C32C2×C3×A4semidirect
171. C3⋊D3C2×C3×A4semidirect
198. C22×C3C2×(C3⋊D3)⋊C3semidirect
224. C32C22×C3×A4semidirect
229. C23C3×(C3⋊D3)⋊C3semidirect
248. C2×C3(C3⋊D3)⋊A4direct (N × H)
262. C22C2×C3×(C3⋊D3)⋊C3semidirect
268. C3C2×(C3⋊D3)⋊A4direct (N × H)
274. C2C3×(C3⋊D3)⋊A4direct (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⋊D3)⋊A42
C23×C3×C3⋊D33
C2×C3×D3×A43
C2×(C3⋊D3)⋊A43
C2×C3×(C3⋊D3)⋊C34