Finite Group Databasebeta

← C3×#3828⋊C3 #13048833

Subgroups — C3×#3828⋊C3 #13048833

Order 1296.

544 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 150 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×C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#638⋊C3C212yesmaximal, complemented
3C3×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4C3×C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
5C3×#634⋊C3C211yesmaximal, characteristic, complemented
6C3×#3828C311yesmaximal, characteristic, complemented
7C32×#235—31nomaximal
8#3828⋊C3C313yesmaximal, direct product factor
9C22×C3×H(3,𝔽3)C2211yescharacteristic
10C2×C3×(C3⋊D3)⋊C3C2212yes—
11C3×C4×H(3,𝔽3)C2211yescharacteristic
12C3×C4×H(3,𝔽3)C2211yescharacteristic
13C3×(C3⋊Dic3)⋊C3C2212yes—
14C2×C33×C4—21no—
15C2×C33×C4D311yescharacteristic, complemented
16C2×C33×C4C2×C311yescharacteristic, complemented
17C3×#638C2×C312yescomplemented
18C3×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
19C3×C4×C3⋊D3C2×C312yescomplemented
20C3×#634C2×C311yescharacteristic, complemented
21C32×#67—32no—
22C32×D12—31no—
23C32×C4×D3—32no—
24C32×Dic6—31no—
25C2×C4×H(3,𝔽3)—23no—
26C2×C4×H(3,𝔽3)C2×C313yescomplemented
27#638⋊C3C2×C316yescomplemented
28C4⋊((C3⋊D3)⋊C3)C2×C313yescomplemented
29C4×(C3⋊D3)⋊C3C2×C316yescomplemented
30#634⋊C3C2×C313yescomplemented
31C2×C3×H(3,𝔽3)—21no—
32C2×C3×H(3,𝔽3)C2311yescharacteristic
33C3×(C3⋊D3)⋊C3—22no—
34C32×#41—91no—
35#3828C3211yescharacteristic, complemented
36C3×#235—31no—
37C3×#235—33no—
38C3×#235—91no—
39C22×C33—21no—
40C22×C33C2×D311yescharacteristic
41C22×C33C22×C311yescharacteristic
42C2×C3×C3⋊D3C22×C312yes—
43C2×C32×D3—32no—
44C33×C4—21no—
45C33×C4—21no—
46C33×C4C2×D311yescharacteristic
47C33×C4C2×D311yescharacteristic
48C33×C4C22×C311yescharacteristic
49C33×C4C22×C311yescharacteristic
50C3×C3⋊Dic3C22×C312yes—
51C32×Dic3—32no—
52C22×H(3,𝔽3)—23no—
53C22×H(3,𝔽3)C22×C313yes—
54C2×(C3⋊D3)⋊C3C22×C316yes—
55C4×H(3,𝔽3)—23no—
56C4×H(3,𝔽3)—23no—
57C4×H(3,𝔽3)C22×C313yes—
58C4×H(3,𝔽3)C22×C313yes—
59(C3⋊Dic3)⋊C3C22×C316yes—
60C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
61C32×Q8—91no—
62C32×D4—91no—
63C32×D4—92no—
64C2×C32×C4—21no—
65C2×C32×C4—23no—
66C2×C32×C4—31no—
67C2×C32×C4—31no—
68C2×C32×C4—61no—
69C2×C32×C4—61no—
70C2×C32×C4—62no—
71C2×C32×C4—61no—
72C2×C32×C4—92no—
73C2×C32×C4C3×D311yescharacteristic
74C2×C32×C4C3×D313yescomplemented
75C2×C32×C4C2×C3211yescharacteristic, complemented
76#638C2×C3212yescomplemented
77C4⋊(C3⋊D3)C2×C3211yescharacteristic, complemented
78C4×C3⋊D3C2×C3212yescomplemented
79#634C2×C3211yescharacteristic, complemented
80C3×#67—36no—
81C3×#67—32no—
82C3×#67—92no—
83C3×D12—33no—
84C3×D12—31no—
85C3×D12—91no—
86C3×C4×D3—36no—
87C3×C4×D3—32no—
88C3×C4×D3—92no—
89C3×Dic6—33no—
90C3×Dic6—31no—
91C3×Dic6—91no—
92C2×C33—22no—
93C2×C33—21no—
94C2×C33—21no—
95C2×C33—21no—
96C2×C33C22×D311yescharacteristic
97C2×C33C23×C311yescharacteristic
98C3×C3⋊D3—22no—
99C32×D3—62no—
100C2×H(3,𝔽3)—26no—
101C2×H(3,𝔽3)—23no—
102C2×H(3,𝔽3)—23no—
103C2×H(3,𝔽3)C23×C313yes—
104(C3⋊D3)⋊C3—26no—
105C3×#41—93no—
106C3×#41—91no—
107#235—31no—
108#235—91no—
109C22×C32—21no—
110C22×C32—23no—
111C22×C32—31no—
112C22×C32—31no—
113C22×C32—61no—
114C22×C32—61no—
115C22×C32—62no—
116C22×C32—61no—
117C22×C32—92no—
118C22×C32C2×C3×D311yescharacteristic
119C22×C32C2×C3×D313yes—
120C22×C32C22×C3211yescharacteristic
121C2×C3⋊D3C22×C3212yes—
122C2×C3×D3—36no—
123C2×C3×D3—32no—
124C2×C3×D3—92no—
125C32×C4—21no—
126C32×C4—21no—
127C32×C4—23no—
128C32×C4—23no—
129C32×C4—31no—
130C32×C4—31no—
131C32×C4—31no—
132C32×C4—31no—
133C32×C4—61no—
134C32×C4—61no—
135C32×C4—61no—
136C32×C4—61no—
137C32×C4—62no—
138C32×C4—62no—
139C32×C4—61no—
140C32×C4—61no—
141C32×C4—92no—
142C32×C4C2×C3×D311yescharacteristic
143C32×C4C2×C3×D311yescharacteristic
144C32×C4C2×C3×D313yes—
145C32×C4C2×C3×D313yes—
146C32×C4C22×C3211yescharacteristic
147C32×C4C22×C3211yescharacteristic
148C3⋊Dic3C22×C3212yes—
149C3×Dic3—36no—
150C3×Dic3—32no—
151C3×Dic3—92no—
152C33—21no—
153C33#23511yescharacteristic, complemented
154C33C3×#4111yescharacteristic, complemented
155H(3,𝔽3)—23no—
156H(3,𝔽3)C3×#4113yescomplemented
157C3×Q8—93no—
158C3×Q8—91no—
159C3×D4—93no—
160C3×D4—96no—
161C3×D4—92no—
162C3×D4—91no—
163C2×C3×C4—21no—
164C2×C3×C4—33no—
165C2×C3×C4—31no—
166C2×C3×C4—61no—
167C2×C3×C4—63no—
168C2×C3×C4—96no—
169C2×C3×C4—92no—
170C2×C3×C4(C3⋊D3)⋊C311yescharacteristic, complemented
171C2×C3×C4C32×D311yescharacteristic
172#67—32no—
173#67—92no—
174D12—31no—
175D12—91no—
176C4×D3—32no—
177C4×D3—92no—
178Dic6—31no—
179Dic6—91no—
180C2×C32—21no—
181C2×C32—22no—
182C2×C32—26no—
183C2×C32—23no—
184C2×C32—21no—
185C2×C32—21no—
186C2×C32—23no—
187C2×C32—31no—
188C2×C32—31no—
189C2×C32—61no—
190C2×C32—61no—
191C2×C32—61no—
192C2×C32—61no—
193C2×C32—62no—
194C2×C32—62no—
195C2×C32—62no—
196C2×C32—64no—
197C2×C32—62no—
198C2×C32—61no—
199C2×C32—182no—
200C2×C32C22×C3×D311yescharacteristic
201C2×C32C22×C3×D313yes—
202C2×C32C23×C3211yescharacteristic
203C3⋊D3—22no—
204C3×D3—62no—
205C3×D3—66no—
206C3×D3—182no—
207#41—91noSylow
208C22×C3—21no—
209C22×C3—31no—
210C22×C3—33no—
211C22×C3—61no—
212C22×C3—63no—
213C22×C3—96no—
214C22×C3—92no—
215C22×C3C2×(C3⋊D3)⋊C311yescharacteristic
216C22×C3C2×C32×D311yescharacteristic
217C2×D3—32no—
218C2×D3—92no—
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—31no—
222C3×C4—31no—
223C3×C4—33no—
224C3×C4—33no—
225C3×C4—61no—
226C3×C4—61no—
227C3×C4—63no—
228C3×C4—63no—
229C3×C4—96no—
230C3×C4—92no—
231C3×C4C2×(C3⋊D3)⋊C311yescharacteristic
232C3×C4C2×(C3⋊D3)⋊C311yescharacteristic
233C3×C4C2×C32×D311yescharacteristic
234C3×C4C2×C32×D311yescharacteristic
235Dic3—32no—
236Dic3—92no—
237C32—21no—
238C32—23no—
239C32—31no—
240C32—31no—
241C32—61no—
242C32—61no—
243C32—62no—
244C32—61no—
245C32C3×#23511yescharacteristic
246C32C3×#23513yescomplemented
247C32C32×#4111yescharacteristic, complemented
248Q8—91no—
249D4—92no—
250D4—91no—
251C2×C4—92no—
252C2×C4C3×(C3⋊D3)⋊C311yescharacteristic, complemented
253C2×C3—22no—
254C2×C3—21no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—31no—
258C2×C3—33no—
259C2×C3—61no—
260C2×C3—63no—
261C2×C3—62no—
262C2×C3—66no—
263C2×C3—61no—
264C2×C3—63no—
265C2×C3—186no—
266C2×C3—182no—
267C2×C3C22×(C3⋊D3)⋊C311yescharacteristic
268C2×C3C22×C32×D311yescharacteristic
269D3—62no—
270D3—182no—
271C22—92no—
272C22C2×C3×(C3⋊D3)⋊C311yescharacteristic
273C4—92no—
274C4C2×C3×(C3⋊D3)⋊C311yescharacteristic
275C4C2×C3×(C3⋊D3)⋊C311yescharacteristic
276C3—21no—
277C3—31no—
278C3—33no—
279C3—61no—
280C3—63no—
281C3#3828⋊C311yescharacteristic, direct product factor
282C3C32×#23511yescharacteristic
283C2—21no—
284C2—182no—
285C2C22×C3×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC3×C4C2×(C3⋊D3)⋊C3
commutator, cosocleC2×C32C23×C32
FrattiniC2×C3C22×C32×D3
FittingC2×C3×C4×H(3,𝔽3)C2
socleC2×C32C22×C3×D3
hypercenterC2×C3×C4(C3⋊D3)⋊C3
hypocenterC32C32×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×H(3,𝔽3)C2semidirect
2. C3×#638⋊C3C2semidirect
3. C3×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C3×C4×(C3⋊D3)⋊C3C2semidirect
5. C3×#634⋊C3C2semidirect
6. C3×#3828C3semidirect
8. #3828⋊C3C3direct (N × H)
15. C2×C33×C4D3semidirect
16. C2×C33×C4C2×C3semidirect
17. C3×#638C2×C3semidirect
18. C3×C4⋊(C3⋊D3)C2×C3semidirect
19. C3×C4×C3⋊D3C2×C3semidirect
20. C3×#634C2×C3semidirect
26. C2×C4×H(3,𝔽3)C2×C3semidirect
27. #638⋊C3C2×C3semidirect
28. C4⋊((C3⋊D3)⋊C3)C2×C3semidirect
29. C4×(C3⋊D3)⋊C3C2×C3semidirect
30. #634⋊C3C2×C3semidirect
35. #3828C32semidirect
60. C3×H(3,𝔽3)#41semidirect
74. C2×C32×C4C3×D3semidirect
75. C2×C32×C4C2×C32semidirect
76. #638C2×C32semidirect
77. C4⋊(C3⋊D3)C2×C32semidirect
78. C4×C3⋊D3C2×C32semidirect
79. #634C2×C32semidirect
153. C33#235semidirect
154. C33C3×#41semidirect
156. H(3,𝔽3)C3×#41semidirect
170. C2×C3×C4(C3⋊D3)⋊C3semidirect
246. C32C3×#235semidirect
247. C32C32×#41semidirect
252. C2×C4C3×(C3⋊D3)⋊C3semidirect
281. C3#3828⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×H(3,𝔽3)2
C3×#638⋊C32
C3×C4⋊((C3⋊D3)⋊C3)2
C3×C4×(C3⋊D3)⋊C32
C3×#634⋊C32
C3×#38283
C32×#2353
#3828⋊C33