Finite Group Databasebeta

← C2×C4×D3×A4 #10596608

Subgroups — C2×C4×D3×A4 #10596608

Order 576.

650 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×D3×A4C211yesmaximal, characteristic
2C2×C3×C4×A4C211yesmaximal, characteristic, complemented
3C2×Dic3×A4C211yesmaximal, characteristic, complemented
4C4×D3×A4C214yesmaximal, direct product factor
5C23×C4×D3C311yesmaximal, characteristic, complemented
6C22×C4×A4—31nomaximal
7C22×C3×A4C2211yescharacteristic
8C2×D3×A4C414yesdirect product factor
9C2×D3×A4C2212yes—
10C2×C3×C4×D3—41nomaximal
11C3×C4×A4C2212yescomplemented
12Dic3×A4C2212yescomplemented
13C24×D3C2×C311yescharacteristic
14C23×A4—31no—
15C23×C3×C4C2×C311yescharacteristic, complemented
16C23×Dic3C2×C311yescharacteristic, complemented
17C22×C4×D3—31no—
18C22×C4×D3—31no—
19C22×C4×D3—31no—
20C22×C4×D3—31no—
21C22×C4×D3—34no—
22C22×C4×D3C2×C314yescomplemented
23C2×C4×A4—21no—
24C2×C4×A4—31no—
25C2×C4×A4—34no—
26C2×C4×A4D311yescharacteristic, direct product factor
27C22×C3×D3—41no—
28C2×C3×A4C2×C412yescomplemented
29C2×C3×A4C2311yescharacteristic
30D3×A4C2×C414yesdirect product factor
31C2×C32×C4—41no—
32C2×C3×Dic3—41no—
33C3×C4×D3—44no—
34C24×C4—31noSylow
35C24×C3C22×C311yescharacteristic
36C23×D3—31no—
37C23×D3—31no—
38C23×D3—32no—
39C23×D3—34no—
40C23×D3C3×C414yescomplemented
41C23×D3C22×C312yes—
42C22×A4—21no—
43C22×A4—34no—
44C22×A4—32no—
45C22×A4C2×D311yescharacteristic
46C22×C3×C4—31no—
47C22×C3×C4—31no—
48C22×C3×C4—32no—
49C22×C3×C4—121no—
50C22×C3×C4C22×C312yescomplemented
51C22×Dic3—31no—
52C22×Dic3—31no—
53C22×Dic3—32no—
54C22×Dic3C22×C312yescomplemented
55C2×C4×D3—32no—
56C2×C4×D3—34no—
57C2×C4×D3—34no—
58C2×C4×D3—34no—
59C2×C4×D3—31no—
60C2×C4×D3—31no—
61C2×C4×D3—31no—
62C2×C4×D3—34no—
63C2×C4×D3—34no—
64C2×C4×D3—34no—
65C2×C4×D3—34no—
66C2×C4×D3—34no—
67C2×C4×D3A411yescharacteristic, direct product factor
68C4×A4—22no—
69C4×A4—32no—
70C4×A4C2×D312yesdirect product factor
71C22×C32—41no—
72C2×C3×D3—44no—
73C2×C3×D3—42no—
74C3×A4C22×C411yescharacteristic, complemented
75C32×C4—42no—
76C3×Dic3—42no—
77C25—31no—
78C23×C4—31no—
79C23×C4—34no—
80C23×C4—91no—
81C23×C4—91no—
82C23×C4—94no—
83C23×C4—91no—
84C23×C4—91no—
85C23×C4C3×D311yescharacteristic, complemented
86C23×C3—31no—
87C23×C3—32no—
88C23×C3—31no—
89C23×C3—121no—
90C23×C3C2×C3×C412yescomplemented
91C23×C3C23×C311yescharacteristic
92C22×D3—32no—
93C22×D3—34no—
94C22×D3—38no—
95C22×D3—31no—
96C22×D3—34no—
97C22×D3—34no—
98C22×D3—32no—
99C22×D3—32no—
100C22×D3—32no—
101C22×D3—34no—
102C22×D3—38no—
103C22×D3—34no—
104C22×D3C2×C3×C414yescomplemented
105C22×D3C2×A411yescharacteristic
106C2×A4—22no—
107C2×A4—21no—
108C2×A4—34no—
109C2×A4C4×D312yesdirect product factor
110C2×A4C22×D311yescharacteristic
111C2×C3×C4—32no—
112C2×C3×C4—31no—
113C2×C3×C4—32no—
114C2×C3×C4—32no—
115C2×C3×C4—32no—
116C2×C3×C4—41no—
117C2×C3×C4—81no—
118C2×C3×C4—124no—
119C2×C3×C4—121no—
120C2×C3×C4C2×A411yescharacteristic, complemented
121C2×Dic3—32no—
122C2×Dic3—31no—
123C2×Dic3—32no—
124C2×Dic3—32no—
125C2×Dic3—32no—
126C2×Dic3C2×A411yescharacteristic, complemented
127C4×D3—38no—
128C4×D3—34no—
129C4×D3—34no—
130C4×D3—34no—
131C4×D3C2×A414yesdirect product factor
132C2×C32—42no—
133C2×C32—41no—
134C3×D3—44no—
135C24—34no—
136C24—32no—
137C24—91no—
138C24—92no—
139C24—91no—
140C24—94no—
141C24C2×C3×D311yescharacteristic
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—32no—
146C22×C4—32no—
147C22×C4—94no—
148C22×C4—91no—
149C22×C4—94no—
150C22×C4—91no—
151C22×C4—91no—
152C22×C4—94no—
153C22×C4—94no—
154C22×C4—92no—
155C22×C4—94no—
156C22×C4—94no—
157C22×C4—94no—
158C22×C4—94no—
159C22×C4—91no—
160C22×C4—92no—
161C22×C4—91no—
162C22×C4C2×C3×D312yescomplemented
163C22×C3—32no—
164C22×C3—31no—
165C22×C3—32no—
166C22×C3—32no—
167C22×C3—32no—
168C22×C3—31no—
169C22×C3—31no—
170C22×C3—41no—
171C22×C3—81no—
172C22×C3—122no—
173C22×C3—124no—
174C22×C3C22×C3×C411yescharacteristic, complemented
175C22×C3C22×A411yescharacteristic
176C2×D3—38no—
177C2×D3—34no—
178C2×D3—34no—
179C2×D3—32no—
180C2×D3—34no—
181C2×D3—38no—
182C2×D3—34no—
183C2×D3—34no—
184C2×D3C4×A414yesdirect product factor
185C2×D3C22×A412yes—
186A4—21no—
187A4C2×C4×D311yescharacteristic, direct product factor
188C3×C4—32no—
189C3×C4—42no—
190C3×C4—82no—
191C3×C4—122no—
192C3×C4C22×A412yescomplemented
193Dic3—32no—
194Dic3C22×A412yescomplemented
195C32—41noSylow
196C23—31no—
197C23—32no—
198C23—31no—
199C23—31no—
200C23—34no—
201C23—92no—
202C23—92no—
203C23—94no—
204C23—94no—
205C23—92no—
206C23—98no—
207C23—94no—
208C23—94no—
209C23—92no—
210C23—98no—
211C23—94no—
212C23—91no—
213C23C3×C4×D312yescomplemented
214C23C22×C3×D311yescharacteristic
215C2×C4—32no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—34no—
221C2×C4—32no—
222C2×C4—94no—
223C2×C4—92no—
224C2×C4—94no—
225C2×C4—92no—
226C2×C4—98no—
227C2×C4—92no—
228C2×C4—91no—
229C2×C4—94no—
230C2×C4—92no—
231C2×C4D3×A411yescharacteristic, direct product factor
232C2×C3—32no—
233C2×C3—31no—
234C2×C3—31no—
235C2×C3—42no—
236C2×C3—41no—
237C2×C3—82no—
238C2×C3—81no—
239C2×C3—124no—
240C2×C3C2×C4×A412yescomplemented
241C2×C3C23×A411yescharacteristic
242D3—34no—
243D3C2×C4×A414yesdirect product factor
244C22—32no—
245C22—31no—
246C22—32no—
247C22—32no—
248C22—32no—
249C22—34no—
250C22—31no—
251C22—31no—
252C22—32no—
253C22—94no—
254C22—94no—
255C22—98no—
256C22—94no—
257C22—94no—
258C22—98no—
259C22—94no—
260C22—92no—
261C22C2×C3×C4×D311yescharacteristic, complemented
262C22C2×D3×A411yescharacteristic
263C4—32no—
264C4—32no—
265C4—92no—
266C4C2×D3×A412yesdirect product factor
267C3—41no—
268C3—81no—
269C3C22×C4×A411yescharacteristic, complemented
270C2—31no—
271C2—32no—
272C2—31no—
273C2—34no—
274C2—94no—
275C2C4×D3×A412yesdirect product factor
276C2C22×D3×A411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×A4
commutator, hypocenterC22×C3C22×C3×C4
FrattiniC2C22×D3×A4
FittingC23×C3×C4C2×C3
socleC24×C3C22×C3
prime radicalC22×D3×A4C2
cosocleC23×C3C23×C3
LCSG ▷ C22×C3
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×A4C2semidirect
3. C2×Dic3×A4C2semidirect
4. C4×D3×A4C2direct (N × H)
5. C23×C4×D3C3semidirect
8. C2×D3×A4C4direct (N × H)
11. C3×C4×A4C22semidirect
12. Dic3×A4C22semidirect
15. C23×C3×C4C2×C3semidirect
16. C23×Dic3C2×C3semidirect
22. C22×C4×D3C2×C3semidirect
26. C2×C4×A4D3direct (N × H)
28. C2×C3×A4C2×C4semidirect
30. D3×A4C2×C4direct (N × H)
40. C23×D3C3×C4semidirect
50. C22×C3×C4C22×C3semidirect
54. C22×Dic3C22×C3semidirect
67. C2×C4×D3A4direct (N × H)
70. C4×A4C2×D3direct (N × H)
74. C3×A4C22×C4semidirect
85. C23×C4C3×D3semidirect
90. C23×C3C2×C3×C4semidirect
104. C22×D3C2×C3×C4semidirect
109. C2×A4C4×D3direct (N × H)
120. C2×C3×C4C2×A4semidirect
126. C2×Dic3C2×A4semidirect
131. C4×D3C2×A4direct (N × H)
162. C22×C4C2×C3×D3semidirect
174. C22×C3C22×C3×C4semidirect
184. C2×D3C4×A4direct (N × H)
187. A4C2×C4×D3direct (N × H)
192. C3×C4C22×A4semidirect
194. Dic3C22×A4semidirect
213. C23C3×C4×D3semidirect
231. C2×C4D3×A4direct (N × H)
240. C2×C3C2×C4×A4semidirect
243. D3C2×C4×A4direct (N × H)
261. C22C2×C3×C4×D3semidirect
266. C4C2×D3×A4direct (N × H)
269. C3C22×C4×A4semidirect
275. C2C4×D3×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×A42
C2×C3×C4×A42
C2×Dic3×A42
C4×D3×A42
C23×C4×D33
C22×C4×A43
C2×C3×C4×D34