Finite Group Databasebeta

← #13048679

Subgroups — #13048679

Order 1296.

671 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 86 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⋊D3)⋊C3)⋊D3C212yesmaximal, complemented
2C4⋊(C3⋊#272)C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊((C3⋊D3)⋊C3))C211yesmaximal, characteristic, complemented
4C3×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5#10619887C212yesmaximal, complemented
6C4⋊#1113—31nomaximal
7D3×C4⋊(C3⋊D3)—31nomaximal
8#88654—33nomaximal
9C2×C3⋊#272C2212yescomplemented
10C2×C3⋊((C3⋊D3)⋊C3)C2212yescomplemented
11C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
12((C3⋊D3)⋊C3)⋊D3—24no—
13C3×C4×H(3,𝔽3)C2211yescharacteristic, complemented
14C3×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
15C2×#1113—32no—
16C2×D3×C3⋊D3—32no—
17C4⋊(C3⋊(C3⋊D3))D311yescharacteristic, complemented
18C3×C4⋊(C3⋊D3)—31no—
19C3×C4⋊(C3⋊D3)—31no—
20C3×C4⋊(C3⋊D3)—31no—
21C3×C4⋊(C3⋊D3)—61no—
22C3×C4×C3⋊D3D311yescharacteristic, complemented
23C32×C4×D3—31no—
24#6295—32no—
25#6292—32no—
26C2×((C3⋊D3)⋊C3)⋊C2—36no—
27C4⋊#272—33no—
28C4⋊#272—63no—
29C4⋊((C3⋊D3)⋊C3)—33no—
30C4×(C3⋊D3)⋊C3D313yescomplemented
31#6200—36no—
32C2×C3×H(3,𝔽3)C2311yescharacteristic
33C3⋊#272—22no—
34C3⋊((C3⋊D3)⋊C3)—22no—
35C3×(C3⋊D3)⋊C3D412yescomplemented
36C2×C4⋊(C3⋊D3)—91no—
37#3802—91no—
38D3×D12—93no—
39D3×D12—91no—
40D3×D12—91no—
41C2×C3⋊(C3⋊D3)C2×D312yescomplemented
42C2×C3×C3⋊D3—32no—
43C2×C3×C3⋊D3—32no—
44C2×C3×C3⋊D3—32no—
45C2×C3×C3⋊D3—62no—
46C2×C3×C3⋊D3C2×D311yescharacteristic
47C2×C32×D3—31no—
48#1113—64no—
49D3×C3⋊D3—64no—
50C33×C4—21no—
51C33×C4C2×D311yescharacteristic, complemented
52C33×C4C2×D311yescharacteristic, complemented
53C3×C3⋊Dic3C2×D311yescharacteristic, complemented
54C32×Dic3—31no—
55C2×#272—36no—
56C2×#272—66no—
57C2×(C3⋊D3)⋊C3—36no—
58C2×(C3⋊D3)⋊C3C2×D313yes—
59((C3⋊D3)⋊C3)⋊C2—612no—
60C4×H(3,𝔽3)—23no—
61C4×H(3,𝔽3)C2×D313yescomplemented
62(C3⋊Dic3)⋊C3C2×D313yescomplemented
63C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
64C22×C3⋊D3—92no—
65C2×D32—96no—
66C2×D32—92no—
67C2×D32—92no—
68C2×D32—92no—
69C2×C32×C4—91no—
70C4⋊(C3⋊D3)—31no—
71C4⋊(C3⋊D3)—31no—
72C4⋊(C3⋊D3)—31no—
73C4⋊(C3⋊D3)—33no—
74C4⋊(C3⋊D3)—61no—
75C4⋊(C3⋊D3)—91no—
76C4⋊(C3⋊D3)—92no—
77C4⋊(C3⋊D3)—181no—
78C4×C3⋊D3C3⋊D311yescharacteristic, complemented
79C3×D12—91no—
80C3×D12—91no—
81C3×D12—93no—
82C3×D12—91no—
83C3×D12—91no—
84C3×D12—183no—
85C3×D12—181no—
86C3×D12—181no—
87C3×C4×D3—31no—
88C3×C4×D3—33no—
89C3×C4×D3—91no—
90#626—96no—
91#626—92no—
92#626—92no—
93#625—92no—
94C2×C33—21no—
95C2×C33C22×D311yescharacteristic
96C2×C33C22×D311yescharacteristic
97C3⋊(C3⋊D3)—22no—
98C3×C3⋊D3—62no—
99C3×C3⋊D3—62no—
100C3×C3⋊D3—62no—
101C3×C3⋊D3—122no—
102C3×C3⋊D3D1212yescomplemented
103C32×D3—32no—
104C2×H(3,𝔽3)—23no—
105C2×H(3,𝔽3)C22×D313yes—
106#272—66no—
107#272—126no—
108(C3⋊D3)⋊C3—66no—
109(C3⋊D3)⋊C3D1216yescomplemented
110D3×D4—271no—
111D3×D4—271no—
112C2×D12—271no—
113C2×D12—273no—
114C22×C32—91no—
115C2×C3⋊D3—32no—
116C2×C3⋊D3—32no—
117C2×C3⋊D3—32no—
118C2×C3⋊D3—36no—
119C2×C3⋊D3—62no—
120C2×C3⋊D3—92no—
121C2×C3⋊D3—182no—
122C2×C3⋊D3—184no—
123C2×C3⋊D3C2×C3⋊D311yescharacteristic
124C2×C3×D3—33no—
125C2×C3×D3—31no—
126C2×C3×D3—91no—
127C2×C3×D3—92no—
128C2×C3×D3—96no—
129C2×C3×D3—92no—
130C2×C3×D3—92no—
131C2×C3×D3—92no—
132C2×C3×D3—186no—
133C2×C3×D3—182no—
134C2×C3×D3—182no—
135D32—1812no—
136D32—184no—
137D32—184no—
138D32—184no—
139C32×C4—21no—
140C32×C4—23no—
141C32×C4—31no—
142C32×C4—31no—
143C32×C4—61no—
144C32×C4—61no—
145C32×C4—61no—
146C32×C4—91no—
147C32×C4—121no—
148C32×C4D3211yescharacteristic
149C32×C4D3213yescomplemented
150C32×C4C2×C3⋊D311yescharacteristic, complemented
151C3⋊Dic3C2×C3⋊D311yescharacteristic, complemented
152C3×Dic3—33no—
153C3×Dic3—31no—
154C3×Dic3—91no—
155C33—21no—
156C33C2×D1211yescharacteristic, complemented
157C33D3×D411yescharacteristic, complemented
158H(3,𝔽3)—23no—
159H(3,𝔽3)C2×D1213yescomplemented
160C22×D3—272no—
161C22×D3—272no—
162C22×D3—272no—
163C22×D3—276no—
164C3×D4—271no—
165C3×D4—271no—
166C2×C3×C4—93no—
167C2×C3×C4—91no—
168#67—272no—
169#67—272no—
170D12—93no—
171D12—91no—
172D12—91no—
173D12—91no—
174D12—91no—
175D12—181no—
176D12—181no—
177D12—183no—
178D12—273no—
179D12—272no—
180D12—276no—
181C4×D3—31no—
182C4×D3—91no—
183C2×C32—21no—
184C2×C32—23no—
185C2×C32—31no—
186C2×C32—31no—
187C2×C32—61no—
188C2×C32—61no—
189C2×C32—61no—
190C2×C32—92no—
191C2×C32—121no—
192C2×C32C2×D3211yescharacteristic
193C2×C32C2×D3213yes—
194C2×C32C22×C3⋊D311yescharacteristic
195C3⋊D3—62no—
196C3⋊D3—62no—
197C3⋊D3—62no—
198C3⋊D3—66no—
199C3⋊D3—122no—
200C3⋊D3—182no—
201C3⋊D3—362no—
202C3⋊D3C4⋊(C3⋊D3)12yescomplemented
203C3×D3—32no—
204C3×D3—36no—
205C3×D3—92no—
206C3×D3—182no—
207C3×D3—182no—
208C3×D3—186no—
209C3×D3—182no—
210C3×D3—182no—
211C3×D3—366no—
212C3×D3—362no—
213C3×D3—362no—
214C2×D4—811noSylow
215C22×C3—93no—
216C22×C3—91no—
217C22×C3—272no—
218C22×C3—272no—
219C2×D3—31no—
220C2×D3—91no—
221C2×D3—96no—
222C2×D3—92no—
223C2×D3—92no—
224C2×D3—92no—
225C2×D3—92no—
226C2×D3—182no—
227C2×D3—182no—
228C2×D3—186no—
229C2×D3—276no—
230C2×D3—544no—
231C2×D3—544no—
232C2×D3—544no—
233C2×D3—5412no—
234C3×C4—21no—
235C3×C4—31no—
236C3×C4—33no—
237C3×C4—61no—
238C3×C4—63no—
239C3×C4—93no—
240C3×C4—91no—
241C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
242C3×C4D3×C3⋊D311yescharacteristic
243Dic3—31no—
244Dic3—91no—
245C32—21no—
246C32—23no—
247C32—31no—
248C32—31no—
249C32—61no—
250C32—61no—
251C32—61no—
252C32—121no—
253C32D3×D1211yescharacteristic
254C32D3×D1213yescomplemented
255C32C2×C4⋊(C3⋊D3)11yescharacteristic, complemented
256C23—812no—
257D4—271no—
258D4—271no—
259D4—812no—
260C2×C4—91no—
261C2×C3—21no—
262C2×C3—31no—
263C2×C3—33no—
264C2×C3—61no—
265C2×C3—63no—
266C2×C3—96no—
267C2×C3—92no—
268C2×C3—542no—
269C2×C3—542no—
270C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
271C2×C3C2×D3×C3⋊D311yescharacteristic
272D3—32no—
273D3—92no—
274D3—182no—
275D3—186no—
276D3—182no—
277D3—182no—
278D3—182no—
279D3—362no—
280D3—362no—
281D3—366no—
282D3—546no—
283C22—91no—
284C22—272no—
285C22—272no—
286C22—1624no—
287C4—91no—
288C4((C3⋊D3)⋊C3)⋊D311yescharacteristic, complemented
289C3—21no—
290C3—31no—
291C3—33no—
292C3—61no—
293C3—63no—
294C3#8865411yescharacteristic, complemented
295C3D3×C4⋊(C3⋊D3)11yescharacteristic
296C2—92no—
297C2—542no—
298C2—542no—
299C2C2×((C3⋊D3)⋊C3)⋊D311yescharacteristic
subgroupG/H
centerC2C2×((C3⋊D3)⋊C3)⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC4((C3⋊D3)⋊C3)⋊D3
hypocenterC3×H(3,𝔽3)C2×D4
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×((C3⋊D3)⋊C3)⋊D3C2semidirect
2. C4⋊(C3⋊#272)C2semidirect
3. C4⋊(C3⋊((C3⋊D3)⋊C3))C2semidirect
4. C3×C4×(C3⋊D3)⋊C3C2semidirect
5. #10619887C2semidirect
9. C2×C3⋊#272C22semidirect
10. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
13. C3×C4×H(3,𝔽3)C22semidirect
14. C3×(C3⋊Dic3)⋊C3C22semidirect
17. C4⋊(C3⋊(C3⋊D3))D3semidirect
22. C3×C4×C3⋊D3D3semidirect
30. C4×(C3⋊D3)⋊C3D3semidirect
35. C3×(C3⋊D3)⋊C3D4semidirect
41. C2×C3⋊(C3⋊D3)C2×D3semidirect
51. C33×C4C2×D3semidirect
52. C33×C4C2×D3semidirect
53. C3×C3⋊Dic3C2×D3semidirect
61. C4×H(3,𝔽3)C2×D3semidirect
62. (C3⋊Dic3)⋊C3C2×D3semidirect
63. C3×H(3,𝔽3)C2×D4semidirect
78. C4×C3⋊D3C3⋊D3semidirect
102. C3×C3⋊D3D12semidirect
109. (C3⋊D3)⋊C3D12semidirect
149. C32×C4D32semidirect
150. C32×C4C2×C3⋊D3semidirect
151. C3⋊Dic3C2×C3⋊D3semidirect
156. C33C2×D12semidirect
157. C33D3×D4semidirect
159. H(3,𝔽3)C2×D12semidirect
202. C3⋊D3C4⋊(C3⋊D3)semidirect
241. C3×C4((C3⋊D3)⋊C3)⋊C2semidirect
254. C32D3×D12semidirect
255. C32C2×C4⋊(C3⋊D3)semidirect
288. C4((C3⋊D3)⋊C3)⋊D3semidirect
294. C3#88654semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×((C3⋊D3)⋊C3)⋊D32
C4⋊(C3⋊#272)2
C4⋊(C3⋊((C3⋊D3)⋊C3))2
C3×C4×(C3⋊D3)⋊C32
#106198872
C4⋊#11133
D3×C4⋊(C3⋊D3)3
#886543