Finite Group Databasebeta

← #13048678

Subgroups — #13048678

Order 1296.

671 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 78 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)⋊C3)C211yesmaximal, characteristic, complemented
5#10619886C212yesmaximal, complemented
6C4⋊#1113—31nomaximal
7#89026—31nomaximal
8#88654—33nomaximal
9C2×C3⋊#272C2212yescomplemented
10C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C3×(C3⋊D3)⋊C3C2212yescomplemented
12((C3⋊D3)⋊C3)⋊D3—24no—
13C3×C4×H(3,𝔽3)C2211yescharacteristic, complemented
14C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, 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)—61no—
21C3×C4⋊(C3⋊D3)D311yescharacteristic, complemented
22C3×C4×C3⋊D3—31no—
23C32×D12—31no—
24#6295—32no—
25#6290—32no—
26C2×((C3⋊D3)⋊C3)⋊C2—36no—
27C4⋊#272—33no—
28C4⋊#272—63no—
29C4⋊((C3⋊D3)⋊C3)D313yescomplemented
30C4×(C3⋊D3)⋊C3—33no—
31#6200—36no—
32C2×C3×H(3,𝔽3)C2311yescharacteristic
33C3⋊#272—22no—
34C3⋊((C3⋊D3)⋊C3)D412yescomplemented
35C3×(C3⋊D3)⋊C3—22no—
36D4×C3⋊D3—91no—
37#3802—93no—
38#3802—91no—
39D3×D12—91no—
40D3×D12—91no—
41C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
42C2×C3×C3⋊D3—31no—
43C2×C3×C3⋊D3—32no—
44C2×C3×C3⋊D3—32no—
45C2×C3×C3⋊D3—62no—
46C2×C3×C3⋊D3C2×D312yescomplemented
47C2×C32×D3—32no—
48#1113—64no—
49D3×C3⋊D3—64no—
50C33×C4—21no—
51C33×C4C2×D311yescharacteristic, complemented
52C33×C4C2×D311yescharacteristic, complemented
53C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
54C3×C3⋊Dic3—31no—
55C2×#272—36no—
56C2×#272—66no—
57C2×(C3⋊D3)⋊C3—33no—
58C2×(C3⋊D3)⋊C3C2×D316yescomplemented
59((C3⋊D3)⋊C3)⋊C2—612no—
60C4×H(3,𝔽3)—23no—
61C4×H(3,𝔽3)C2×D313yescomplemented
62(C3⋊Dic3)⋊C3—33no—
63C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
64C22×C3⋊D3—92no—
65C2×D32—96no—
66C2×D32—92no—
67C2×D32—92no—
68C2×D32—92no—
69C32×D4—91no—
70#638—92no—
71C4⋊(C3⋊D3)—31no—
72C4⋊(C3⋊D3)—31no—
73C4⋊(C3⋊D3)—61no—
74C4⋊(C3⋊D3)C3⋊D311yescharacteristic, complemented
75C4×C3⋊D3—31no—
76C4×C3⋊D3—33no—
77C4×C3⋊D3—91no—
78C4×C3⋊D3—181no—
79C3×D12—31no—
80C3×D12—33no—
81C3×D12—91no—
82C3×D12—91no—
83C3×D12—93no—
84C3×D12—91no—
85C3×D12—183no—
86C3×D12—181no—
87C3×C4×D3—91no—
88C3×C4×D3—91no—
89C3×C4×D3—181no—
90#626—92no—
91#626—92no—
92#625—96no—
93#625—92no—
94C2×C33—21no—
95C2×C33C22×D311yescharacteristic
96C2×C33C22×D311yescharacteristic
97C3⋊(C3⋊D3)D1212yescomplemented
98C3×C3⋊D3—22no—
99C3×C3⋊D3—32no—
100C3×C3⋊D3—62no—
101C3×C3⋊D3—62no—
102C3×C3⋊D3—122no—
103C32×D3—62no—
104C2×H(3,𝔽3)—23no—
105C2×H(3,𝔽3)C22×D313yes—
106#272—66no—
107#272—126no—
108(C3⋊D3)⋊C3—26no—
109(C3⋊D3)⋊C3—36no—
110D3×D4—271no—
111D3×D4—271no—
112D3×D4—273no—
113C2×D12—271no—
114C22×C32—92no—
115C2×C3⋊D3—32no—
116C2×C3⋊D3—31no—
117C2×C3⋊D3—32no—
118C2×C3⋊D3—33no—
119C2×C3⋊D3—62no—
120C2×C3⋊D3—91no—
121C2×C3⋊D3—181no—
122C2×C3⋊D3—184no—
123C2×C3⋊D3C2×C3⋊D312yescomplemented
124C2×C3×D3—36no—
125C2×C3×D3—32no—
126C2×C3×D3—92no—
127C2×C3×D3—91no—
128C2×C3×D3—91no—
129C2×C3×D3—96no—
130C2×C3×D3—92no—
131C2×C3×D3—92no—
132C2×C3×D3—181no—
133C2×C3×D3—186no—
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—121no—
147C32×C4D3211yescharacteristic
148C32×C4D3213yescomplemented
149C32×C4C2×C3⋊D311yescharacteristic, complemented
150C3⋊Dic3—31no—
151C3⋊Dic3—33no—
152C3⋊Dic3—91no—
153C3⋊Dic3—181no—
154C3×Dic3—91no—
155C3×Dic3—91no—
156C3×Dic3—181no—
157C33—21no—
158C33C2×D1211yescharacteristic, complemented
159C33D3×D411yescharacteristic, complemented
160H(3,𝔽3)—23no—
161H(3,𝔽3)D3×D413yescomplemented
162C22×D3—272no—
163C22×D3—272no—
164C22×D3—272no—
165C22×D3—276no—
166C3×D4—93no—
167C3×D4—91no—
168C3×D4—271no—
169C2×C3×C4—271no—
170#67—272no—
171#67—272no—
172#67—276no—
173D12—31no—
174D12—91no—
175D12—91no—
176D12—93no—
177D12—91no—
178D12—181no—
179D12—183no—
180D12—272no—
181C4×D3—91no—
182C4×D3—91no—
183C4×D3—181no—
184C4×D3—273no—
185C2×C32—21no—
186C2×C32—23no—
187C2×C32—31no—
188C2×C32—31no—
189C2×C32—61no—
190C2×C32—61no—
191C2×C32—61no—
192C2×C32—121no—
193C2×C32—182no—
194C2×C32C2×D3211yescharacteristic
195C2×C32C2×D3213yes—
196C2×C32C22×C3⋊D311yescharacteristic
197C3⋊D3—22no—
198C3⋊D3—32no—
199C3⋊D3—36no—
200C3⋊D3—62no—
201C3⋊D3—62no—
202C3⋊D3—92no—
203C3⋊D3—122no—
204C3⋊D3—182no—
205C3×D3—62no—
206C3×D3—66no—
207C3×D3—92no—
208C3×D3—92no—
209C3×D3—182no—
210C3×D3—182no—
211C3×D3—182no—
212C3×D3—186no—
213C3×D3—182no—
214C3×D3—366no—
215C3×D3—362no—
216C2×D4—811noSylow
217C22×C3—96no—
218C22×C3—92no—
219C22×C3—271no—
220C22×C3—272no—
221C2×D3—32no—
222C2×D3—92no—
223C2×D3—91no—
224C2×D3—91no—
225C2×D3—92no—
226C2×D3—96no—
227C2×D3—92no—
228C2×D3—181no—
229C2×D3—182no—
230C2×D3—186no—
231C2×D3—273no—
232C2×D3—544no—
233C2×D3—544no—
234C2×D3—544no—
235C2×D3—5412no—
236C3×C4—21no—
237C3×C4—33no—
238C3×C4—31no—
239C3×C4—61no—
240C3×C4—63no—
241C3×C4—271no—
242C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
243C3×C4D3×C3⋊D311yescharacteristic
244Dic3—91no—
245Dic3—91no—
246Dic3—181no—
247Dic3—273no—
248C32—21no—
249C32—23no—
250C32—31no—
251C32—31no—
252C32—61no—
253C32—61no—
254C32—61no—
255C32—121no—
256C32D3×D1211yescharacteristic
257C32D3×D1213yescomplemented
258C32D4×C3⋊D311yescharacteristic, complemented
259C23—812no—
260D4—91no—
261D4—271no—
262D4—812no—
263C2×C4—271no—
264C2×C3—21no—
265C2×C3—33no—
266C2×C3—31no—
267C2×C3—61no—
268C2×C3—63no—
269C2×C3—186no—
270C2×C3—182no—
271C2×C3—272no—
272C2×C3—542no—
273C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
274C2×C3C2×D3×C3⋊D311yescharacteristic
275D3—62no—
276D3—92no—
277D3—92no—
278D3—182no—
279D3—182no—
280D3—182no—
281D3—186no—
282D3—182no—
283D3—276no—
284D3—362no—
285D3—366no—
286C22—92no—
287C22—271no—
288C22—272no—
289C22—1624no—
290C4—271no—
291C4((C3⋊D3)⋊C3)⋊D311yescharacteristic, complemented
292C3—21no—
293C3—31no—
294C3—33no—
295C3—61no—
296C3—63no—
297C3#8865411yescharacteristic, complemented
298C3C3⋊D3×D1211yescharacteristic
299C2—182no—
300C2—272no—
301C2—542no—
302C2C2×((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)⋊C3)C2semidirect
5. #10619886C2semidirect
9. C2×C3⋊#272C22semidirect
11. C2×C3×(C3⋊D3)⋊C3C22semidirect
13. C3×C4×H(3,𝔽3)C22semidirect
14. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
17. C4×C3⋊(C3⋊D3)D3semidirect
21. C3×C4⋊(C3⋊D3)D3semidirect
29. C4⋊((C3⋊D3)⋊C3)D3semidirect
34. C3⋊((C3⋊D3)⋊C3)D4semidirect
46. C2×C3×C3⋊D3C2×D3semidirect
51. C33×C4C2×D3semidirect
52. C33×C4C2×D3semidirect
53. C3⋊(C3⋊Dic3)C2×D3semidirect
58. C2×(C3⋊D3)⋊C3C2×D3semidirect
61. C4×H(3,𝔽3)C2×D3semidirect
63. C3×H(3,𝔽3)C2×D4semidirect
74. C4⋊(C3⋊D3)C3⋊D3semidirect
97. C3⋊(C3⋊D3)D12semidirect
123. C2×C3⋊D3C2×C3⋊D3semidirect
148. C32×C4D32semidirect
149. C32×C4C2×C3⋊D3semidirect
158. C33C2×D12semidirect
159. C33D3×D4semidirect
161. H(3,𝔽3)D3×D4semidirect
242. C3×C4((C3⋊D3)⋊C3)⋊C2semidirect
257. C32D3×D12semidirect
258. C32D4×C3⋊D3semidirect
291. C4((C3⋊D3)⋊C3)⋊D3semidirect
297. 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)⋊C3)2
#106198862
C4⋊#11133
#890263
#886543