Finite Group Databasebeta

← #13048680

Subgroups — #13048680

Order 1296.

671 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 76 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⋊#272C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊((C3⋊D3)⋊C3))C211yesmaximal, characteristic, complemented
4C3×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
5#10619884C212yesmaximal, complemented
6C4⋊#1113—31nomaximal
7C3⋊D3×D12—31nomaximal
8#88655—33nomaximal
9C2×C3⋊#272C2211yescharacteristic
10C2×C3⋊((C3⋊D3)⋊C3)C2212yescomplemented
11C2×C3×(C3⋊D3)⋊C3C2212yescomplemented
12((C3⋊D3)⋊C3)⋊D3—24no—
13C3×C4×H(3,𝔽3)C2211yescharacteristic, complemented
14C3⋊#1084C2211yescharacteristic, complemented
15C2×#1113—32no—
16C2×D3×C3⋊D3—32no—
17C4⋊(C3⋊(C3⋊D3))D311yescharacteristic, complemented
18C3×C4⋊(C3⋊D3)—31no—
19C3×C4⋊(C3⋊D3)D311yescharacteristic, complemented
20C3×C4×C3⋊D3—31no—
21C3×C4×C3⋊D3—31no—
22C3×C4×C3⋊D3—61no—
23C32×D12—31no—
24#6295—32no—
25#6291—32no—
26C2×((C3⋊D3)⋊C3)⋊C2—36no—
27C4×#272—33no—
28C4×#272—63no—
29C4⋊((C3⋊D3)⋊C3)—33no—
30C4⋊((C3⋊D3)⋊C3)D313yescomplemented
31#6198—36no—
32C2×C3×H(3,𝔽3)C2311yescharacteristic
33C3⋊#272D412yescomplemented
34C3⋊((C3⋊D3)⋊C3)—22no—
35C3×(C3⋊D3)⋊C3—22no—
36D4×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—31no—
43C2×C3×C3⋊D3—31no—
44C2×C3×C3⋊D3—32no—
45C2×C3×C3⋊D3—61no—
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—31no—
54C3×C3⋊Dic3—31no—
55C3×C3⋊Dic3—61no—
56C2×#272—33no—
57C2×#272—63no—
58C2×(C3⋊D3)⋊C3—36no—
59C2×(C3⋊D3)⋊C3C2×D316yescomplemented
60((C3⋊D3)⋊C3)⋊C2—612no—
61C4×H(3,𝔽3)—23no—
62C4×H(3,𝔽3)C2×D313yescomplemented
63#1084—33no—
64#1084—63no—
65C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
66C22×C3⋊D3—92no—
67C2×D32—96no—
68C2×D32—92no—
69C2×D32—92no—
70C2×D32—92no—
71C32×D4—91no—
72#638—92no—
73C4⋊(C3⋊D3)—31no—
74C4⋊(C3⋊D3)—33no—
75C4⋊(C3⋊D3)—91no—
76C4⋊(C3⋊D3)—181no—
77C4⋊(C3⋊D3)C3⋊D311yescharacteristic, complemented
78C4×C3⋊D3—31no—
79C4×C3⋊D3—31no—
80C4×C3⋊D3—61no—
81C3×D12—31no—
82C3×D12—33no—
83C3×D12—91no—
84C3×D12—91no—
85C3×D12—91no—
86C3×D12—181no—
87C3×C4×D3—91no—
88C3×C4×D3—93no—
89C3×C4×D3—91no—
90C3×C4×D3—183no—
91C3×C4×D3—181no—
92#626—96no—
93#626—92no—
94#626—92no—
95#625—92no—
96C2×C33—21no—
97C2×C33C22×D311yescharacteristic
98C2×C33C22×D311yescharacteristic
99C3⋊(C3⋊D3)—22no—
100C3×C3⋊D3—22no—
101C3×C3⋊D3—32no—
102C3×C3⋊D3—32no—
103C3×C3⋊D3—62no—
104C3×C3⋊D3—62no—
105C32×D3—62no—
106C2×H(3,𝔽3)—23no—
107C2×H(3,𝔽3)C22×D313yes—
108#272—36no—
109#272—66no—
110(C3⋊D3)⋊C3—26no—
111(C3⋊D3)⋊C3—66no—
112D3×D4—271no—
113D3×D4—271no—
114D3×D4—273no—
115C2×D12—271no—
116C22×C32—92no—
117C2×C3⋊D3—31no—
118C2×C3⋊D3—31no—
119C2×C3⋊D3—32no—
120C2×C3⋊D3—36no—
121C2×C3⋊D3—61no—
122C2×C3⋊D3—92no—
123C2×C3⋊D3—182no—
124C2×C3⋊D3—184no—
125C2×C3⋊D3C2×C3⋊D312yescomplemented
126C2×C3×D3—36no—
127C2×C3×D3—32no—
128C2×C3×D3—92no—
129C2×C3×D3—93no—
130C2×C3×D3—91no—
131C2×C3×D3—91no—
132C2×C3×D3—92no—
133C2×C3×D3—92no—
134C2×C3×D3—183no—
135C2×C3×D3—181no—
136C2×C3×D3—182no—
137D32—1812no—
138D32—184no—
139D32—184no—
140D32—184no—
141C32×C4—21no—
142C32×C4—23no—
143C32×C4—31no—
144C32×C4—31no—
145C32×C4—61no—
146C32×C4—61no—
147C32×C4—61no—
148C32×C4—121no—
149C32×C4D3211yescharacteristic
150C32×C4D3213yescomplemented
151C32×C4C2×C3⋊D311yescharacteristic, complemented
152C3⋊Dic3—31no—
153C3⋊Dic3—31no—
154C3⋊Dic3—61no—
155C3×Dic3—93no—
156C3×Dic3—91no—
157C3×Dic3—91no—
158C3×Dic3—183no—
159C3×Dic3—181no—
160C33—21no—
161C33D3×D411yescharacteristic, complemented
162C33D3×D411yescharacteristic, complemented
163H(3,𝔽3)—23no—
164H(3,𝔽3)D3×D413yescomplemented
165C22×D3—272no—
166C22×D3—272no—
167C22×D3—272no—
168C22×D3—276no—
169C3×D4—93no—
170C3×D4—91no—
171C3×D4—271no—
172C2×C3×C4—271no—
173#67—272no—
174#67—272no—
175#67—276no—
176D12—31no—
177D12—91no—
178D12—91no—
179D12—91no—
180D12—181no—
181D12—273no—
182D12—272no—
183C4×D3—91no—
184C4×D3—93no—
185C4×D3—91no—
186C4×D3—181no—
187C4×D3—183no—
188C2×C32—21no—
189C2×C32—23no—
190C2×C32—31no—
191C2×C32—31no—
192C2×C32—61no—
193C2×C32—61no—
194C2×C32—61no—
195C2×C32—121no—
196C2×C32—182no—
197C2×C32C2×D3211yescharacteristic
198C2×C32C2×D3213yes—
199C2×C32C22×C3⋊D311yescharacteristic
200C3⋊D3—22no—
201C3⋊D3—32no—
202C3⋊D3—32no—
203C3⋊D3—62no—
204C3⋊D3—62no—
205C3⋊D3—66no—
206C3⋊D3—182no—
207C3⋊D3—362no—
208C3×D3—62no—
209C3×D3—66no—
210C3×D3—92no—
211C3×D3—96no—
212C3×D3—92no—
213C3×D3—182no—
214C3×D3—186no—
215C3×D3—182no—
216C3×D3—182no—
217C3×D3—182no—
218C3×D3—362no—
219C2×D4—811noSylow
220C22×C3—96no—
221C22×C3—92no—
222C22×C3—272no—
223C22×C3—271no—
224C2×D3—32no—
225C2×D3—92no—
226C2×D3—91no—
227C2×D3—93no—
228C2×D3—91no—
229C2×D3—92no—
230C2×D3—92no—
231C2×D3—181no—
232C2×D3—183no—
233C2×D3—182no—
234C2×D3—276no—
235C2×D3—544no—
236C2×D3—544no—
237C2×D3—544no—
238C2×D3—5412no—
239C3×C4—21no—
240C3×C4—31no—
241C3×C4—33no—
242C3×C4—61no—
243C3×C4—63no—
244C3×C4—271no—
245C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
246C3×C4D3×C3⋊D311yescharacteristic
247Dic3—91no—
248Dic3—93no—
249Dic3—91no—
250Dic3—181no—
251Dic3—183no—
252C32—21no—
253C32—23no—
254C32—31no—
255C32—31no—
256C32—61no—
257C32—61no—
258C32—61no—
259C32—121no—
260C32#380211yescharacteristic
261C32#380213yescomplemented
262C32D4×C3⋊D311yescharacteristic, complemented
263C23—812no—
264D4—91no—
265D4—271no—
266D4—812no—
267C2×C4—271no—
268C2×C3—21no—
269C2×C3—31no—
270C2×C3—33no—
271C2×C3—61no—
272C2×C3—63no—
273C2×C3—186no—
274C2×C3—182no—
275C2×C3—272no—
276C2×C3—542no—
277C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
278C2×C3C2×D3×C3⋊D311yescharacteristic
279D3—62no—
280D3—92no—
281D3—96no—
282D3—92no—
283D3—182no—
284D3—182no—
285D3—186no—
286D3—182no—
287D3—182no—
288D3—362no—
289D3—546no—
290C22—92no—
291C22—271no—
292C22—272no—
293C22—1624no—
294C4—271no—
295C4((C3⋊D3)⋊C3)⋊D311yescharacteristic, complemented
296C3—21no—
297C3—31no—
298C3—33no—
299C3—61no—
300C3—63no—
301C3#8865511yescharacteristic, complemented
302C3#8902611yescharacteristic
303C2—182no—
304C2—272no—
305C2—542no—
306C2C2×((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⋊#272C2semidirect
3. C4⋊(C3⋊((C3⋊D3)⋊C3))C2semidirect
4. C3×C4⋊((C3⋊D3)⋊C3)C2semidirect
5. #10619884C2semidirect
10. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
11. C2×C3×(C3⋊D3)⋊C3C22semidirect
13. C3×C4×H(3,𝔽3)C22semidirect
14. C3⋊#1084C22semidirect
17. C4⋊(C3⋊(C3⋊D3))D3semidirect
19. C3×C4⋊(C3⋊D3)D3semidirect
30. C4⋊((C3⋊D3)⋊C3)D3semidirect
33. C3⋊#272D4semidirect
41. C2×C3⋊(C3⋊D3)C2×D3semidirect
46. C2×C3×C3⋊D3C2×D3semidirect
51. C33×C4C2×D3semidirect
52. C33×C4C2×D3semidirect
59. C2×(C3⋊D3)⋊C3C2×D3semidirect
62. C4×H(3,𝔽3)C2×D3semidirect
65. C3×H(3,𝔽3)C2×D4semidirect
77. C4⋊(C3⋊D3)C3⋊D3semidirect
125. C2×C3⋊D3C2×C3⋊D3semidirect
150. C32×C4D32semidirect
151. C32×C4C2×C3⋊D3semidirect
161. C33D3×D4semidirect
162. C33D3×D4semidirect
164. H(3,𝔽3)D3×D4semidirect
245. C3×C4((C3⋊D3)⋊C3)⋊C2semidirect
261. C32#3802semidirect
262. C32D4×C3⋊D3semidirect
295. C4((C3⋊D3)⋊C3)⋊D3semidirect
301. C3#88655semidirect

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⋊#2722
C4⋊(C3⋊((C3⋊D3)⋊C3))2
C3×C4⋊((C3⋊D3)⋊C3)2
#106198842
C4⋊#11133
C3⋊D3×D123
#886553