Finite Group Databasebeta

← C3×#63894 #11723219

Subgroups — C3×#63894 #11723219

Order 864.

502 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 128 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⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×C3×#624C212yesmaximal, complemented
3C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C32×C4×Dic3C211yesmaximal, characteristic, complemented
5C3×#3722C212yesmaximal, complemented
6C3×#893—31nomaximal
7C3×C4×D12—31nomaximal
8#63894C311yesmaximal, characteristic, direct product factor
9C22×C3×C3⋊D3C2212yes—
10C2×C33×C4C2211yescharacteristic
11C3×C4⋊(C3⋊D3)C414yescomplemented
12C2×C32×Dic3C2212yes—
13C2×C32×Dic3C2211yescharacteristic
14C2×C32×Dic3C2211yescharacteristic
15C3×#624—24no—
16C2×C4⋊(C3⋊D3)C2×C311yescharacteristic
17C2×C3×D12—31no—
18C2×C3×D12—31no—
19C2×C3×D12—61no—
20C2×C3×C4×D3—32no—
21C2×C3×C4×D3—32no—
22C2×#624C2×C312yescomplemented
23C32×C4⋊C4—31no—
24C32×C42—31no—
25C3×(C2×D3)⋊C4—32no—
26C3×(C2×D3)⋊C4—32no—
27C3×C4⋊Dic3—21no—
28C3×C4⋊Dic3D311yescharacteristic, complemented
29C3×C4⋊Dic3C2×C311yescharacteristic, complemented
30C3×C4×Dic3—21no—
31C3×C4×Dic3D311yescharacteristic, complemented
32C3×C4×Dic3C2×C311yescharacteristic, complemented
33#3722C2×C312yescomplemented
34C22×C33C2311yescharacteristic
35C2×C3×C3⋊D3—22no—
36C2×C3×C3⋊D3—22no—
37C2×C3×C3⋊D3C2×C414yescomplemented
38C33×C4C2×C412yescomplemented
39C32×Dic3—22no—
40C32×Dic3—21no—
41C32×Dic3D412yescomplemented
42C3×C4×D4—91no—
43#893—31no—
44C4×D12—31no—
45C22×C3⋊D3C22×C312yes—
46C22×C3×D3—32no—
47C22×C3×D3—32no—
48C22×C3×D3—62no—
49C2×C32×C4—21no—
50C2×C32×C4—21no—
51C2×C32×C4—21no—
52C2×C32×C4—31no—
53C2×C32×C4—31no—
54C2×C32×C4—32no—
55C2×C32×C4—41no—
56C2×C32×C4C2×D311yescharacteristic
57C2×C32×C4C2×D311yescharacteristic
58C2×C32×C4C22×C311yescharacteristic
59C4⋊(C3⋊D3)C3×C414yescomplemented
60C2×C3×Dic3—21no—
61C2×C3×Dic3—21no—
62C2×C3×Dic3—22no—
63C2×C3×Dic3C2×D311yescharacteristic
64C2×C3×Dic3C2×D312yes—
65C2×C3×Dic3C2×D311yescharacteristic
66C2×C3×Dic3C22×C311yescharacteristic
67C2×C3×Dic3C22×C312yes—
68C2×C3×Dic3C22×C311yescharacteristic
69C3×D12—34no—
70C3×D12—34no—
71C3×D12—64no—
72C3×C4×D3—64no—
73C3×C4×D3—64no—
74#624—24no—
75C2×C33C22×C411yescharacteristic
76C2×C33C2×D411yescharacteristic
77C2×C33#4111yescharacteristic
78C3×C3⋊D3—24no—
79C2×C3×D4—91no—
80C22×C3×C4—92no—
81C2×D12—31no—
82C2×D12—31no—
83C2×D12—61no—
84C2×C4×D3—32no—
85C2×C4×D3—32no—
86C3×C4⋊C4—31no—
87C3×C4⋊C4—31no—
88C3×C4⋊C4—61no—
89C3×C22⋊C4—92no—
90C3×C42—31no—
91C3×C42—31no—
92C3×C42—61no—
93(C2×D3)⋊C4—32no—
94(C2×D3)⋊C4—32no—
95C4⋊Dic3C3×D311yescharacteristic, complemented
96C4×Dic3C3×D311yescharacteristic, complemented
97C22×C32—21no—
98C22×C32—21no—
99C22×C32—21no—
100C22×C32—41no—
101C22×C32C22×D311yescharacteristic
102C22×C32C22×D311yescharacteristic
103C22×C32C23×C311yescharacteristic
104C2×C3⋊D3—22no—
105C2×C3⋊D3—22no—
106C2×C3⋊D3C2×C3×C414yescomplemented
107C2×C3×D3—34no—
108C2×C3×D3—34no—
109C2×C3×D3—64no—
110C2×C3×D3—62no—
111C2×C3×D3—62no—
112C2×C3×D3—62no—
113C2×C3×D3—62no—
114C2×C3×D3—122no—
115C2×C3×D3—122no—
116C32×C4—22no—
117C32×C4—22no—
118C32×C4—22no—
119C32×C4—32no—
120C32×C4—42no—
121C32×C4—61no—
122C32×C4—62no—
123C32×C4C4×D312yescomplemented
124C32×C4C4×D312yescomplemented
125C32×C4C2×C3×C412yescomplemented
126C3×Dic3—22no—
127C3×Dic3—21no—
128C3×Dic3—22no—
129C3×Dic3—22no—
130C3×Dic3—21no—
131C3×Dic3—22no—
132C3×Dic3—42no—
133C3×Dic3D1212yescomplemented
134C3×Dic3C3×D412yescomplemented
135C4×D4—91noSylow
136C33C4×D411yescharacteristic, complemented, Sylow
137C23×C3—92no—
138C22×D3—32no—
139C22×D3—32no—
140C22×D3—62no—
141C3×D4—94no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—31no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—32no—
149C2×C3×C4—31no—
150C2×C3×C4—32no—
151C2×C3×C4—41no—
152C2×C3×C4—61no—
153C2×C3×C4—61no—
154C2×C3×C4—62no—
155C2×C3×C4—184no—
156C2×C3×C4D3211yescharacteristic
157C2×C3×C4C2×C3×D311yescharacteristic
158C2×C3×C4C2×C3×D311yescharacteristic
159C2×Dic3C2×C3×D311yescharacteristic
160C2×Dic3C2×C3×D312yes—
161C2×Dic3C2×C3×D311yescharacteristic
162D12—34no—
163D12—34no—
164D12—64no—
165C4×D3—64no—
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×C32C2×C4×D311yescharacteristic
180C2×C32C2×C4×D311yescharacteristic
181C2×C32C2×D1211yescharacteristic
182C2×C32#23511yescharacteristic
183C2×C32D3×D411yescharacteristic
184C2×C32#23911yescharacteristic
185C2×C32C22×C3×C411yescharacteristic
186C2×C32C2×C3×D411yescharacteristic
187C2×C32C3×#4111yescharacteristic
188C3⋊D3—24no—
189C3×D3—64no—
190C3×D3—64no—
191C3×D3—124no—
192C2×D4—91no—
193C22×C4—92no—
194C4⋊C4—31no—
195C22⋊C4—92no—
196C42—31no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—41no—
201C22×C3—94no—
202C22×C3—182no—
203C22×C3—182no—
204C22×C3C2×D3211yescharacteristic
205C22×C3C22×C3×D311yescharacteristic
206C22×C3C22×C3×D311yescharacteristic
207C2×D3—34no—
208C2×D3—34no—
209C2×D3—64no—
210C2×D3—62no—
211C2×D3—62no—
212C2×D3—62no—
213C2×D3—62no—
214C2×D3—122no—
215C2×D3—122no—
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—62no—
226C3×C4—62no—
227C3×C4—61no—
228C3×C4—122no—
229C3×C4#62412yescomplemented
230C3×C4C3×C4×D312yescomplemented
231C3×C4C3×C4×D312yescomplemented
232Dic3—21no—
233Dic3—22no—
234Dic3C3×D1212yescomplemented
235C32—21no—
236C32—21no—
237C32—21no—
238C32—41no—
239C32C4×D1211yescharacteristic, complemented
240C32#89311yescharacteristic, complemented
241C32C3×C4×D411yescharacteristic, complemented
242C23—92no—
243D4—94no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—32no—
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—184no—
262C2×C3#379911yescharacteristic
263C2×C3D3×D1211yescharacteristic
264C2×C3C2×#62411yescharacteristic
265C2×C3C2×C3×C4×D311yescharacteristic
266C2×C3C2×C3×C4×D311yescharacteristic
267C2×C3C2×C3×D1211yescharacteristic
268C2×C3C3×#23511yescharacteristic
269C2×C3C3×D3×D411yescharacteristic
270C2×C3C3×#23911yescharacteristic
271D3—64no—
272D3—64no—
273D3—124no—
274C22—94no—
275C22—182no—
276C22—182no—
277C22C2×C3×D3211yescharacteristic
278C4—32no—
279C4—61no—
280C4—62no—
281C4C3×#62412yescomplemented
282C3—21no—
283C3—21no—
284C3—21no—
285C3—41no—
286C3#6389411yescharacteristic, direct product factor
287C3C3×C4×D1211yescharacteristic, complemented
288C3C3×#89311yescharacteristic, complemented
289C2—184no—
290C2C3×#379911yescharacteristic
291C2C3×D3×D1211yescharacteristic
292C2C2×C3×#62411yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutatorC2×C32C22×C3×C4
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socleC22×C33C23
prime radicalC2×C3×C4⋊(C3⋊D3)C2
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×#624C2semidirect
3. C32×C4⋊Dic3C2semidirect
4. C32×C4×Dic3C2semidirect
5. C3×#3722C2semidirect
8. #63894C3direct (N × H)
11. C3×C4⋊(C3⋊D3)C4semidirect
22. C2×#624C2×C3semidirect
28. C3×C4⋊Dic3D3semidirect
29. C3×C4⋊Dic3C2×C3semidirect
31. C3×C4×Dic3D3semidirect
32. C3×C4×Dic3C2×C3semidirect
33. #3722C2×C3semidirect
37. C2×C3×C3⋊D3C2×C4semidirect
38. C33×C4C2×C4semidirect
41. C32×Dic3D4semidirect
59. C4⋊(C3⋊D3)C3×C4semidirect
95. C4⋊Dic3C3×D3semidirect
96. C4×Dic3C3×D3semidirect
106. C2×C3⋊D3C2×C3×C4semidirect
123. C32×C4C4×D3semidirect
124. C32×C4C4×D3semidirect
125. C32×C4C2×C3×C4semidirect
133. C3×Dic3D12semidirect
134. C3×Dic3C3×D4semidirect
136. C33C4×D4semidirect
229. C3×C4#624semidirect
230. C3×C4C3×C4×D3semidirect
231. C3×C4C3×C4×D3semidirect
234. Dic3C3×D12semidirect
239. C32C4×D12semidirect
240. C32#893semidirect
241. C32C3×C4×D4semidirect
281. C4C3×#624semidirect
286. C3#63894direct (N × H)
287. C3C3×C4×D12semidirect
288. C3C3×#893semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊D3)2
C2×C3×#6242
C32×C4⋊Dic32
C32×C4×Dic32
C3×#37222
C3×#8933
C3×C4×D123
#638943