Finite Group Databasebeta

← C2×C3⋊((C3⋊D3)⋊A4) #13049306

Subgroups — C2×C3⋊((C3⋊D3)⋊A4) #13049306

Order 1296.

852 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 61 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)⋊A4)C212yesmaximal, direct product factor
3C23×C3⋊(C3⋊D3)C311yesmaximal, characteristic, complemented
4C2×A4×C3⋊D3—31nomaximal
5C2×(C3⋊D3)⋊A4—33nomaximal
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⋊D3)C2×C312yescomplemented
13C2×C32×A4—21no—
14C2×C32×A4D311yescharacteristic, complemented
15A4×C3⋊D3—32no—
16C2×C22⋊H(3,𝔽3)—23no—
17C2×C22⋊H(3,𝔽3)D313yescomplemented
18(C3⋊D3)⋊A4—36no—
19C2×C3×H(3,𝔽3)—41no—
20C3⋊((C3⋊D3)⋊C3)—42no—
21C23×C3⋊D3—31no—
22C23×C3⋊D3—33no—
23C23×C3⋊D3—93no—
24C2×D3×A4—93no—
25C2×D3×A4—91no—
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⋊D3)A411yescharacteristic, complemented
35C2×C3×C3⋊D3—121no—
36C32×A4—21no—
37C32×A4C2×D311yescharacteristic, complemented
38C2×(C3⋊D3)⋊C3—123no—
39C22⋊H(3,𝔽3)—23no—
40C22⋊H(3,𝔽3)C2×D313yescomplemented
41C3×H(3,𝔽3)—41noSylow
42C23×C32—33no—
43C23×C32C3×D311yescharacteristic
44C23×C32C3×D313yescomplemented
45C22×C3⋊D3—32no—
46C22×C3⋊D3—36no—
47C22×C3⋊D3—96no—
48C22×C3⋊D3—91no—
49C22×C3⋊D3—93no—
50C22×C3⋊D3—99no—
51C22×C3⋊D3—918no—
52C22×C3⋊D3—99no—
53C22×C3⋊D3—96no—
54C22×C3⋊D3—93no—
55C22×C3⋊D3—92no—
56C22×C3⋊D3—91no—
57C2×C3×A4—21no—
58C2×C3×A4—23no—
59C2×C3×A4—33no—
60C2×C3×A4—63no—
61C2×C3×A4C3⋊D311yescharacteristic, complemented
62D3×A4—96no—
63D3×A4—92no—
64C2×C33—31no—
65C2×C33—31no—
66C2×C33—41no—
67C2×C33—81no—
68C2×C33C2×A411yescharacteristic, complemented
69C3⋊(C3⋊D3)—32no—
70C3⋊(C3⋊D3)C2×A412yescomplemented
71C3×C3⋊D3—122no—
72C2×H(3,𝔽3)—43no—
73C2×H(3,𝔽3)—83no—
74(C3⋊D3)⋊C3—126no—
75C23×D3—93no—
76C23×D3—91no—
77C23×D3—273no—
78C22×A4—271no—
79C22×C32—31no—
80C22×C32—33no—
81C22×C32—39no—
82C22×C32—39no—
83C22×C32—31no—
84C22×C32—33no—
85C22×C32—33no—
86C22×C32C2×C3×D311yescharacteristic
87C22×C32C2×C3×D313yescomplemented
88C2×C3⋊D3—31no—
89C2×C3⋊D3—33no—
90C2×C3⋊D3—93no—
91C2×C3⋊D3—92no—
92C2×C3⋊D3—96no—
93C2×C3⋊D3—92no—
94C2×C3⋊D3—96no—
95C2×C3⋊D3—918no—
96C2×C3⋊D3—918no—
97C2×C3⋊D3—918no—
98C2×C3⋊D3—92no—
99C2×C3⋊D3—96no—
100C2×C3⋊D3—99no—
101C2×C3⋊D3—918no—
102C2×C3⋊D3—91no—
103C2×C3⋊D3—93no—
104C2×C3⋊D3—96no—
105C2×C3⋊D3—92no—
106C2×C3×D3—361no—
107C2×C3×D3—363no—
108C3×A4—21no—
109C3×A4—23no—
110C3×A4—33no—
111C3×A4—63no—
112C3×A4C2×C3⋊D311yescharacteristic, complemented
113C33—41no—
114C33—81no—
115C33C22×A411yescharacteristic, complemented
116H(3,𝔽3)—43no—
117H(3,𝔽3)—83no—
118C23×C3—33no—
119C23×C3(C3⋊D3)⋊C313yescomplemented
120C23×C3C3×C3⋊D311yescharacteristic
121C22×D3—96no—
122C22×D3—92no—
123C22×D3—276no—
124C22×D3—273no—
125C22×D3—271no—
126C22×D3—279no—
127C22×D3—2718no—
128C22×D3—279no—
129C22×D3—276no—
130C22×D3—272no—
131C22×D3—273no—
132C22×D3—271no—
133C2×A4—31no—
134C2×A4—61no—
135C2×A4—63no—
136C2×A4—272no—
137C2×C32—39no—
138C2×C32—39no—
139C2×C32—31no—
140C2×C32—31no—
141C2×C32—33no—
142C2×C32—33no—
143C2×C32—33no—
144C2×C32—41no—
145C2×C32—83no—
146C2×C32—81no—
147C2×C32—123no—
148C2×C32—243no—
149C2×C32D3×A411yescharacteristic
150C2×C32D3×A413yescomplemented
151C3⋊D3—32no—
152C3⋊D3—36no—
153C3⋊D3—96no—
154C3⋊D3—918no—
155C3⋊D3—92no—
156C3⋊D3—96no—
157C3×D3—362no—
158C3×D3—366no—
159C24—271noSylow
160C22×C3—33no—
161C22×C3—31no—
162C22×C3—39no—
163C22×C3—39no—
164C22×C3—33no—
165C22×C3—31no—
166C22×C3—33no—
167C22×C3—1081no—
168C22×C3C2×(C3⋊D3)⋊C313yescomplemented
169C22×C3C2×C3×C3⋊D311yescharacteristic
170C2×D3—93no—
171C2×D3—91no—
172C2×D3—273no—
173C2×D3—272no—
174C2×D3—276no—
175C2×D3—272no—
176C2×D3—276no—
177C2×D3—2718no—
178C2×D3—2718no—
179C2×D3—2718no—
180C2×D3—2718no—
181C2×D3—272no—
182C2×D3—276no—
183C2×D3—279no—
184C2×D3—273no—
185C2×D3—271no—
186C2×D3—272no—
187C2×D3—276no—
188A4—31no—
189A4—61no—
190A4—63no—
191C32—33no—
192C32—41no—
193C32—83no—
194C32—81no—
195C32—123no—
196C32—243no—
197C32C2×D3×A411yescharacteristic
198C32C2×D3×A413yescomplemented
199C23—272no—
200C23—811no—
201C23—812no—
202C23—811no—
203C23C3⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
204C2×C3—39no—
205C2×C3—39no—
206C2×C3—31no—
207C2×C3—31no—
208C2×C3—33no—
209C2×C3—33no—
210C2×C3—33no—
211C2×C3—121no—
212C2×C3—241no—
213C2×C3—243no—
214C2×C3—1082no—
215C2×C3(C3⋊D3)⋊A413yescomplemented
216C2×C3A4×C3⋊D311yescharacteristic
217D3—92no—
218D3—96no—
219D3—276no—
220D3—2718no—
221D3—272no—
222D3—276no—
223C22—31no—
224C22—31no—
225C22—271no—
226C22—812no—
227C22—812no—
228C22—812no—
229C22—812no—
230C22—811no—
231C22C2×C3⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
232C3—33no—
233C3—121no—
234C3—241no—
235C3—243no—
236C3C2×(C3⋊D3)⋊A413yescomplemented
237C3C2×A4×C3⋊D311yescharacteristic
238C2—31no—
239C2—31no—
240C2—272no—
241C2—812no—
242C2C3⋊((C3⋊D3)⋊A4)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊((C3⋊D3)⋊A4)
commutator, cosocle, hypocenterC22×C33C22×C3
FrattiniC3C2×A4×C3⋊D3
FittingC23×C33C2×C3
socleC23×C32C3×D3
LCSG ▷ C22×C33
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⋊D3)⋊A4)C2direct (N × H)
3. C23×C3⋊(C3⋊D3)C3semidirect
7. C3×C22⋊H(3,𝔽3)C22semidirect
8. C23×C33C2×C3semidirect
12. C22×C3⋊(C3⋊D3)C2×C3semidirect
14. C2×C32×A4D3semidirect
17. C2×C22⋊H(3,𝔽3)D3semidirect
28. C22×C33C22×C3semidirect
34. C2×C3⋊(C3⋊D3)A4semidirect
37. C32×A4C2×D3semidirect
40. C22⋊H(3,𝔽3)C2×D3semidirect
44. C23×C32C3×D3semidirect
61. C2×C3×A4C3⋊D3semidirect
68. C2×C33C2×A4semidirect
70. C3⋊(C3⋊D3)C2×A4semidirect
87. C22×C32C2×C3×D3semidirect
112. C3×A4C2×C3⋊D3semidirect
115. C33C22×A4semidirect
119. C23×C3(C3⋊D3)⋊C3semidirect
150. C2×C32D3×A4semidirect
168. C22×C3C2×(C3⋊D3)⋊C3semidirect
198. C32C2×D3×A4semidirect
203. C23C3⋊((C3⋊D3)⋊C3)semidirect
215. C2×C3(C3⋊D3)⋊A4semidirect
231. C22C2×C3⋊((C3⋊D3)⋊C3)semidirect
236. C3C2×(C3⋊D3)⋊A4semidirect
242. C2C3⋊((C3⋊D3)⋊A4)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⋊D3)⋊A4)2
C23×C3⋊(C3⋊D3)3
C2×A4×C3⋊D33
C2×(C3⋊D3)⋊A43
C2×C3⋊((C3⋊D3)⋊C3)4