Finite Group Databasebeta

← C3×Dic3×D12 #11723217

Subgroups — C3×Dic3×D12 #11723217

Order 864.

502 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 144 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×C32×D12C211yesmaximal, characteristic
2C2×C3×D3×Dic3C212yesmaximal, complemented
3C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C32×C4×Dic3C211yesmaximal, characteristic, complemented
5C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
6C3×D4×Dic3—31nomaximal
7C3×C4×D12—31nomaximal
8Dic3×D12C311yesmaximal, characteristic, direct product factor
9C22×C32×D3C2212yes—
10C2×C33×C4C2211yescharacteristic
11C2×C3×C3⋊Dic3C2212yes—
12C2×C32×Dic3C2211yescharacteristic
13C2×C32×Dic3C2211yescharacteristic
14C32×D12C414yescomplemented
15C3×D3×Dic3—24no—
16C2×C32×D4—31no—
17C22×C3×Dic3—32no—
18C2×C3×D12—21no—
19C2×C3×D12D311yescharacteristic
20C2×C3×D12C2×C311yescharacteristic
21C2×C3×C4×D3—32no—
22C2×D3×Dic3C2×C312yescomplemented
23C32×C42—31no—
24C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
25C3×#217—32no—
26C3×(C2×D3)⋊C4—32no—
27C3×C4⋊Dic3—31no—
28C3×C4⋊Dic3—31no—
29C3×C4⋊Dic3—61no—
30C3×C4×Dic3—21no—
31C3×C4×Dic3D311yescharacteristic, complemented
32C3×C4×Dic3C2×C311yescharacteristic, complemented
33(C2×D3)⋊Dic3C2×C312yescomplemented
34C22×C33C2311yescharacteristic
35C2×C32×D3—22no—
36C2×C32×D3—22no—
37C2×C32×D3C2×C414yescomplemented
38C33×C4C2×C412yescomplemented
39C3×C3⋊Dic3—22no—
40C32×Dic3—21no—
41C32×Dic3D412yescomplemented
42C3×C4×D4—91no—
43D4×Dic3—31no—
44C4×D12—31no—
45C23×C32—32no—
46C22×C3×D3—22no—
47C22×C3×D3C2×D312yes—
48C22×C3×D3C22×C312yes—
49C32×D4—34no—
50C2×C32×C4—21no—
51C2×C32×C4—21no—
52C2×C32×C4—21no—
53C2×C32×C4—31no—
54C2×C32×C4—31no—
55C2×C32×C4—41no—
56C2×C32×C4C2×D311yescharacteristic
57C2×C32×C4C2×D311yescharacteristic
58C2×C32×C4C22×C311yescharacteristic
59C2×C3⋊Dic3C22×C312yes—
60C2×C3×Dic3—21no—
61C2×C3×Dic3—21no—
62C2×C3×Dic3—32no—
63C2×C3×Dic3—32no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3—64no—
66C2×C3×Dic3C2×D311yescharacteristic
67C2×C3×Dic3C2×D311yescharacteristic
68C2×C3×Dic3C22×C311yescharacteristic
69C2×C3×Dic3C22×C311yescharacteristic
70C3×D12—24no—
71C3×D12Dic314yesdirect product factor
72C3×D12C3×C414yescomplemented
73C3×C4×D3—64no—
74D3×Dic3—24no—
75C2×C33C22×C411yescharacteristic
76C2×C33C2×D411yescharacteristic
77C2×C33#4111yescharacteristic
78C32×D3—24no—
79C2×C3×D4—31no—
80C2×C3×D4—31no—
81C2×C3×D4—61no—
82C22×C3×C4—92no—
83C22×Dic3—32no—
84C2×D12C3×D311yescharacteristic
85C2×C4×D3—32no—
86C3×C4⋊C4—91no—
87C3×C22⋊C4—92no—
88C3×C42—31no—
89C3×C42—31no—
90C3×C42—61no—
91#217—32no—
92(C2×D3)⋊C4—32no—
93C4⋊Dic3—31no—
94C4⋊Dic3—31no—
95C4⋊Dic3—61no—
96C4×Dic3C3×D311yescharacteristic, complemented
97C22×C32—21no—
98C22×C32—21no—
99C22×C32—21no—
100C22×C32—34no—
101C22×C32—41no—
102C22×C32—62no—
103C22×C32—62no—
104C22×C32C22×D311yescharacteristic
105C22×C32C22×D311yescharacteristic
106C22×C32C23×C311yescharacteristic
107C2×C3×D3—22no—
108C2×C3×D3—22no—
109C2×C3×D3—22no—
110C2×C3×D3—22no—
111C2×C3×D3—24no—
112C2×C3×D3—42no—
113C2×C3×D3—42no—
114C2×C3×D3C2×Dic314yescomplemented
115C2×C3×D3C2×C3×C414yescomplemented
116C32×C4—22no—
117C32×C4—22no—
118C32×C4—22no—
119C32×C4—32no—
120C32×C4—42no—
121C32×C4—61no—
122C32×C4C4×D312yescomplemented
123C32×C4C2×Dic312yescomplemented
124C32×C4C2×C3×C412yescomplemented
125C3⋊Dic3—22no—
126C3×Dic3—22no—
127C3×Dic3—21no—
128C3×Dic3—21no—
129C3×Dic3—22no—
130C3×Dic3—62no—
131C3×Dic3—62no—
132C3×Dic3—122no—
133C3×Dic3D1212yesdirect product factor
134C3×Dic3C3×D412yescomplemented
135C4×D4—91noSylow
136C33C4×D411yescharacteristic, complemented, Sylow
137C23×C3—32no—
138C23×C3—32no—
139C23×C3—62no—
140C22×D3C2×C3×D312yes—
141C3×D4—34no—
142C3×D4—34no—
143C3×D4—64no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—41no—
152C2×C3×C4—61no—
153C2×C3×C4—61no—
154C2×C3×C4—92no—
155C2×C3×C4—184no—
156C2×C3×C4D3211yescharacteristic
157C2×C3×C4C2×C3×D311yescharacteristic
158C2×C3×C4C2×C3×D311yescharacteristic
159C2×Dic3—32no—
160C2×Dic3—32no—
161C2×Dic3—62no—
162C2×Dic3—64no—
163C2×Dic3C2×C3×D311yescharacteristic
164C2×Dic3C2×C3×D311yescharacteristic
165D12C3×Dic314yesdirect product factor
166C4×D3—64no—
167C2×C32—21no—
168C2×C32—21no—
169C2×C32—21no—
170C2×C32—21no—
171C2×C32—21no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32—41no—
177C2×C32—41no—
178C2×C32—41no—
179C2×C32—64no—
180C2×C32C2×C4×D311yescharacteristic
181C2×C32C2×D1211yescharacteristic
182C2×C32#23511yescharacteristic
183C2×C32D3×D411yescharacteristic
184C2×C32#23711yescharacteristic
185C2×C32C22×Dic311yescharacteristic
186C2×C32C22×C3×C411yescharacteristic
187C2×C32C2×C3×D411yescharacteristic
188C2×C32C3×#4111yescharacteristic
189C3×D3—24no—
190C3×D3—24no—
191C3×D3—44no—
192C2×D4—31no—
193C22×C4—92no—
194C4⋊C4—91no—
195C22⋊C4—92no—
196C42—31no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—34no—
201C22×C3—34no—
202C22×C3—41no—
203C22×C3—64no—
204C22×C3—62no—
205C22×C3—62no—
206C22×C3—62no—
207C22×C3—62no—
208C22×C3—122no—
209C22×C3—122no—
210C22×C3C2×D3211yescharacteristic
211C22×C3C22×C3×D311yescharacteristic
212C22×C3C22×C3×D311yescharacteristic
213C2×D3—22no—
214C2×D3—22no—
215C2×D3C2×C3×Dic314yescomplemented
216C3×C4—22no—
217C3×C4—22no—
218C3×C4—22no—
219C3×C4—32no—
220C3×C4—32no—
221C3×C4—42no—
222C3×C4—62no—
223C3×C4—62no—
224C3×C4—61no—
225C3×C4—61no—
226C3×C4—182no—
227C3×C4D3×Dic312yescomplemented
228C3×C4C3×C4×D312yescomplemented
229C3×C4C2×C3×Dic312yescomplemented
230Dic3—21no—
231Dic3—62no—
232Dic3—62no—
233Dic3—122no—
234Dic3C3×D1212yesdirect product factor
235C32—21no—
236C32—21no—
237C32—21no—
238C32—41no—
239C32C4×D1211yescharacteristic, complemented
240C32D4×Dic311yescharacteristic, complemented
241C32C3×C4×D411yescharacteristic, complemented
242C23—32no—
243D4—34no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—92no—
247C2×C4—184no—
248C2×C4C3×D3211yescharacteristic
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—21no—
254C2×C3—21no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—21no—
258C2×C3—41no—
259C2×C3—41no—
260C2×C3—41no—
261C2×C3—64no—
262C2×C3—64no—
263C2×C3—124no—
264C2×C3#379511yescharacteristic
265C2×C3D3×D1211yescharacteristic
266C2×C3C2×D3×Dic311yescharacteristic
267C2×C3C2×C3×C4×D311yescharacteristic
268C2×C3C2×C3×D1211yescharacteristic
269C2×C3C3×#23511yescharacteristic
270C2×C3C3×D3×D411yescharacteristic
271C2×C3C3×#23711yescharacteristic
272C2×C3C22×C3×Dic311yescharacteristic
273D3—24no—
274C22—34no—
275C22—62no—
276C22—62no—
277C22C2×C3×D3211yescharacteristic
278C4—32no—
279C4—61no—
280C4—182no—
281C4C3×D3×Dic312yescomplemented
282C3—21no—
283C3—21no—
284C3—21no—
285C3—41no—
286C3Dic3×D1211yescharacteristic, direct product factor
287C3C3×C4×D1211yescharacteristic, complemented
288C3C3×D4×Dic311yescharacteristic, complemented
289C2—64no—
290C2C3×#379511yescharacteristic
291C2C3×D3×D1211yescharacteristic
292C2C2×C3×D3×Dic311yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutatorC2×C32C22×C3×C4
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socleC22×C33C23
prime radicalC2×C32×D12C2
cosocleC22×C32C23×C3
hypercenterC2×C3×C4D32
hypocenterC32C3×C4×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×D3×Dic3C2semidirect
3. C3×C4⋊(C3⋊Dic3)C2semidirect
4. C32×C4×Dic3C2semidirect
5. C3×(C2×D3)⋊Dic3C2semidirect
8. Dic3×D12C3direct (N × H)
14. C32×D12C4semidirect
22. C2×D3×Dic3C2×C3semidirect
24. C4⋊(C3⋊Dic3)C2×C3semidirect
31. C3×C4×Dic3D3semidirect
32. C3×C4×Dic3C2×C3semidirect
33. (C2×D3)⋊Dic3C2×C3semidirect
37. C2×C32×D3C2×C4semidirect
38. C33×C4C2×C4semidirect
41. C32×Dic3D4semidirect
71. C3×D12Dic3direct (N × H)
72. C3×D12C3×C4semidirect
96. C4×Dic3C3×D3semidirect
114. C2×C3×D3C2×Dic3semidirect
115. C2×C3×D3C2×C3×C4semidirect
122. C32×C4C4×D3semidirect
123. C32×C4C2×Dic3semidirect
124. C32×C4C2×C3×C4semidirect
133. C3×Dic3D12direct (N × H)
134. C3×Dic3C3×D4semidirect
136. C33C4×D4semidirect
165. D12C3×Dic3direct (N × H)
215. C2×D3C2×C3×Dic3semidirect
227. C3×C4D3×Dic3semidirect
228. C3×C4C3×C4×D3semidirect
229. C3×C4C2×C3×Dic3semidirect
234. Dic3C3×D12direct (N × H)
239. C32C4×D12semidirect
240. C32D4×Dic3semidirect
241. C32C3×C4×D4semidirect
281. C4C3×D3×Dic3semidirect
286. C3Dic3×D12direct (N × H)
287. C3C3×C4×D12semidirect
288. C3C3×D4×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D122
C2×C3×D3×Dic32
C3×C4⋊(C3⋊Dic3)2
C32×C4×Dic32
C3×(C2×D3)⋊Dic32
C3×D4×Dic33
C3×C4×D123
Dic3×D123