Finite Group Databasebeta

← A4×C4⋊(C3⋊Dic3) #13173171

Subgroups — A4×C4⋊(C3⋊Dic3) #13173171

Order 1728.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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×C4×A4C211yesmaximal, characteristic
2C2×A4×C3⋊Dic3C212yesmaximal
3C22×C4⋊(C3⋊Dic3)C311yesmaximal, characteristic, complemented
4A4×C4⋊Dic3—34nomaximal
5C22×C32×A4C2211yescharacteristic
6C32×C4×A4C412yescomplemented
7A4×C3⋊Dic3—22no—
8C3×C4⋊(C3⋊Dic3)—41nomaximal
9C23×C32×C4C2×C311yescharacteristic
10C23×C3⋊Dic3C2×C312yes—
11C2×C3×C4×A4—24no—
12C2×C3×C4×A4D314yes—
13C2×Dic3×A4—38no—
14C2×C4⋊(C3⋊Dic3)—31no—
15C2×C4⋊(C3⋊Dic3)—31no—
16C2×C4⋊(C3⋊Dic3)—32no—
17C2×C32×A4C2×C411yescharacteristic
18C2×C32×A4D411yescharacteristic
19C2×C32×A4Q811yescharacteristic
20C2×C33×C4—41no—
21C2×C3×C3⋊Dic3—42no—
22C22×C4⋊Dic3—34no—
23A4×C4⋊C4—91no—
24C24×C32C22×C311yescharacteristic
25C22×C3×A4—24no—
26C22×C3×A4C2×D314yes—
27C22×C32×C4—31no—
28C22×C32×C4—32no—
29C22×C32×C4—31no—
30C22×C32×C4C3×C412yescomplemented
31C22×C3⋊Dic3—22no—
32C22×C3⋊Dic3—32no—
33C22×C3⋊Dic3—32no—
34C22×C3⋊Dic3—62no—
35C3×C4×A4—28no—
36C3×C4×A4Dic318yescomplemented
37Dic3×A4—68no—
38C4⋊(C3⋊Dic3)—32no—
39C4⋊(C3⋊Dic3)—31no—
40C4⋊(C3⋊Dic3)—32no—
41C4⋊(C3⋊Dic3)A411yescharacteristic, direct product factor
42C3×C4⋊Dic3—124no—
43C22×C33—41no—
44C32×A4C4⋊C411yescharacteristic, complemented
45C33×C4—42no—
46C3×C3⋊Dic3—82no—
47C23×C3×C4C3×D314yes—
48C23×Dic3—38no—
49C2×C4×A4—24no—
50C2×C4×A4—92no—
51C2×C4×A4C3⋊D311yescharacteristic
52C2×C4⋊Dic3—94no—
53C2×C4⋊Dic3—98no—
54C2×C4⋊Dic3—94no—
55C23×C32—31no—
56C23×C32—31no—
57C23×C32—31no—
58C23×C32—31no—
59C23×C32C2×C3×C411yescharacteristic
60C23×C32C3×D411yescharacteristic
61C23×C32C3×Q811yescharacteristic
62C2×C3×A4—24no—
63C2×C3×A4—24no—
64C2×C3×A4—24no—
65C2×C3×A4Dic614yes—
66C2×C3×A4D1214yes—
67C2×C3×A4C2×Dic314yes—
68C2×C32×C4—32no—
69C2×C32×C4—31no—
70C2×C32×C4—32no—
71C2×C32×C4—32no—
72C2×C32×C4—32no—
73C2×C32×C4—44no—
74C2×C32×C4—84no—
75C2×C32×C4C2×A411yescharacteristic
76C2×C3⋊Dic3—32no—
77C2×C3⋊Dic3—62no—
78C2×C3⋊Dic3—62no—
79C2×C3⋊Dic3—62no—
80C2×C3⋊Dic3—62no—
81C2×C3⋊Dic3C2×A412yes—
82C2×C3×Dic3—128no—
83C22×C4⋊C4—91noSylow
84C2×C33—41no—
85C2×C33—41no—
86C2×C33—41no—
87C24×C3C2×C3×D314yes—
88C22×A4—24no—
89C22×A4C2×C3⋊D311yescharacteristic
90C22×C3×C4—34no—
91C22×C3×C4—34no—
92C22×C3×C4—38no—
93C22×C3×C4C3×Dic318yescomplemented
94C22×Dic3—68no—
95C22×Dic3—98no—
96C22×Dic3—98no—
97C22×Dic3—188no—
98C4×A4—28no—
99C4×A4—182no—
100C4×A4C3⋊Dic312yescomplemented
101C3×C4⋊C4—361no—
102C4⋊Dic3—34no—
103C4⋊Dic3—98no—
104C4⋊Dic3—98no—
105C4⋊Dic3—94no—
106C22×C32—31no—
107C22×C32—31no—
108C22×C32—31no—
109C22×C32—32no—
110C22×C32—31no—
111C22×C32—31no—
112C22×C32—31no—
113C22×C32—31no—
114C22×C32—31no—
115C22×C32—31no—
116C22×C32—44no—
117C22×C32—84no—
118C22×C32C3×C4⋊C411yescharacteristic, complemented
119C22×C32C22×A411yescharacteristic
120C3×A4—24no—
121C3×A4C4⋊Dic314yescomplemented
122C32×C4—32no—
123C32×C4—48no—
124C32×C4—88no—
125C32×C4C4×A412yescomplemented
126C3⋊Dic3—22no—
127C3⋊Dic3—62no—
128C3×Dic3—248no—
129C23×C4—92no—
130C23×C4C3×C3⋊D311yescharacteristic
131C2×C4⋊C4—271no—
132C2×C4⋊C4—272no—
133C2×C4⋊C4—271no—
134C33—41noSylow
135C23×C3—34no—
136C23×C3—34no—
137C23×C3—34no—
138C23×C3—34no—
139C23×C3C3×Dic614yes—
140C23×C3C3×D1214yes—
141C23×C3C2×C3×Dic314yes—
142C2×A4—24no—
143C2×A4—24no—
144C2×A4—24no—
145C2×A4#63411yescharacteristic
146C2×A4C4⋊(C3⋊D3)11yescharacteristic
147C2×A4C2×C3⋊Dic311yescharacteristic
148C2×C3×C4—38no—
149C2×C3×C4—34no—
150C2×C3×C4—38no—
151C2×C3×C4—38no—
152C2×C3×C4—38no—
153C2×C3×C4—41no—
154C2×C3×C4—84no—
155C2×C3×C4—362no—
156C2×C3×C4D3×A414yes—
157C2×Dic3—38no—
158C2×Dic3—98no—
159C2×Dic3—188no—
160C2×Dic3—188no—
161C2×Dic3—188no—
162C2×Dic3—188no—
163C2×C32—31no—
164C2×C32—31no—
165C2×C32—31no—
166C2×C32—31no—
167C2×C32—44no—
168C2×C32—44no—
169C2×C32—44no—
170C2×C32—84no—
171C2×C32—84no—
172C2×C32—84no—
173C2×C32C2×C4×A411yescharacteristic
174C2×C32D4×A411yescharacteristic
175C2×C32Q8×A411yescharacteristic
176C24C2×C3×C3⋊D311yescharacteristic
177C22×C4—31no—
178C22×C4—31no—
179C22×C4—32no—
180C22×C4—182no—
181C22×C4—272no—
182C22×C4—272no—
183C22×C4—542no—
184C22×C4C3×C3⋊Dic312yescomplemented
185C4⋊C4—91no—
186C4⋊C4—272no—
187C4⋊C4—272no—
188C4⋊C4—271no—
189C22×C3—34no—
190C22×C3—34no—
191C22×C3—34no—
192C22×C3—38no—
193C22×C3—34no—
194C22×C3—34no—
195C22×C3—34no—
196C22×C3—34no—
197C22×C3—34no—
198C22×C3—34no—
199C22×C3—41no—
200C22×C3—84no—
201C22×C3C3×C4⋊Dic314yescomplemented
202C22×C3C2×D3×A414yes—
203A4—24no—
204A4C4⋊(C3⋊Dic3)11yescharacteristic, direct product factor
205C3×C4—38no—
206C3×C4—42no—
207C3×C4—88no—
208C3×C4—722no—
209C3×C4Dic3×A418yescomplemented
210Dic3—68no—
211Dic3—188no—
212C32—44no—
213C32—84no—
214C32A4×C4⋊C411yescharacteristic, complemented
215C23—31no—
216C23—31no—
217C23—31no—
218C23—31no—
219C23C3×#63411yescharacteristic
220C23C3×C4⋊(C3⋊D3)11yescharacteristic
221C23C2×C3×C3⋊Dic311yescharacteristic
222C2×C4—32no—
223C2×C4—31no—
224C2×C4—32no—
225C2×C4—32no—
226C2×C4—32no—
227C2×C4—92no—
228C2×C4—272no—
229C2×C4—542no—
230C2×C4—542no—
231C2×C4—542no—
232C2×C4—542no—
233C2×C4A4×C3⋊D311yescharacteristic
234C2×C3—34no—
235C2×C3—34no—
236C2×C3—34no—
237C2×C3—34no—
238C2×C3—41no—
239C2×C3—41no—
240C2×C3—41no—
241C2×C3—84no—
242C2×C3—84no—
243C2×C3—84no—
244C2×C3A4×Dic614yes—
245C2×C3A4×D1214yes—
246C2×C3C2×Dic3×A414yes—
247C22—31no—
248C22—31no—
249C22—31no—
250C22—32no—
251C22—31no—
252C22—31no—
253C22—31no—
254C22—31no—
255C22—31no—
256C22—31no—
257C22C3×C4⋊(C3⋊Dic3)11yescharacteristic, complemented
258C22C2×A4×C3⋊D311yescharacteristic
259C4—32no—
260C4—182no—
261C4—542no—
262C4A4×C3⋊Dic312yescomplemented
263C3—41no—
264C3—84no—
265C3A4×C4⋊Dic314yescomplemented
266C2—31no—
267C2—31no—
268C2—31no—
269C2—31no—
270C2A4×#63411yescharacteristic
271C2A4×C4⋊(C3⋊D3)11yescharacteristic
272C2C2×A4×C3⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×C3⋊D3
commutatorC23×C32C2×C3×C4
FittingC23×C32×C4C2×C3
socle, cosocleC24×C32C22×C3
prime radicalC22×C32×A4C22
hypercenterC2×C4A4×C3⋊D3
hypocenterC22×C32C3×C4⋊C4
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊(C3⋊Dic3)C3semidirect
6. C32×C4×A4C4semidirect
30. C22×C32×C4C3×C4semidirect
36. C3×C4×A4Dic3semidirect
41. C4⋊(C3⋊Dic3)A4direct (N × H)
44. C32×A4C4⋊C4semidirect
93. C22×C3×C4C3×Dic3semidirect
100. C4×A4C3⋊Dic3semidirect
118. C22×C32C3×C4⋊C4semidirect
121. C3×A4C4⋊Dic3semidirect
125. C32×C4C4×A4semidirect
184. C22×C4C3×C3⋊Dic3semidirect
201. C22×C3C3×C4⋊Dic3semidirect
204. A4C4⋊(C3⋊Dic3)direct (N × H)
209. C3×C4Dic3×A4semidirect
214. C32A4×C4⋊C4semidirect
257. C22C3×C4⋊(C3⋊Dic3)semidirect
262. C4A4×C3⋊Dic3semidirect
265. C3A4×C4⋊Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×A42
C2×A4×C3⋊Dic32
C22×C4⋊(C3⋊Dic3)3
A4×C4⋊Dic33
C3×C4⋊(C3⋊Dic3)4