Finite Group Databasebeta

← C2×A4×C22⋊C4 #85570

Subgroups — C2×A4×C22⋊C4 #85570

Order 384.

992 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C24×A4C211yesmaximal, characteristic
2C22×C4×A4C212yesmaximal, complemented
3A4×C22⋊C4C214yesmaximal, direct product factor
4C23×C22⋊C4C311yesmaximal, characteristic, complemented, Sylow
5C23×A4—24no—
6C23×A4C414yescomplemented
7C23×A4C2212yes—
8C23×A4C2211yescharacteristic
9C2×C4×A4—24no—
10C2×C4×A4C2214yescomplemented
11C2×C3×C22⋊C4—41nomaximal
12C26C2×C311yescharacteristic
13C24×C4C2×C312yescomplemented
14C22×C22⋊C4—31no—
15C22×C22⋊C4—31no—
16C22×C22⋊C4—32no—
17C22×C22⋊C4—34no—
18C22×C22⋊C4C2×C314yescomplemented
19C24×C3—41no—
20C22×A4—28no—
21C22×A4—24no—
22C22×A4C2×C412yes—
23C22×A4C2×C414yescomplemented
24C22×A4D414yes—
25C22×A4C2311yescharacteristic
26C22×C3×C4—42no—
27C4×A4—24no—
28C3×C22⋊C4—44no—
29C25—24no—
30C25—32no—
31C25—31no—
32C25—34no—
33C25—31no—
34C25—64no—
35C25C3×C414yescomplemented
36C25C22×C312yes—
37C25C22×C311yescharacteristic
38C23×C4—24no—
39C23×C4—32no—
40C23×C4—32no—
41C23×C4—34no—
42C23×C4—64no—
43C23×C4C22×C314yescomplemented
44C2×C22⋊C4—32no—
45C2×C22⋊C4—34no—
46C2×C22⋊C4—38no—
47C2×C22⋊C4—31no—
48C2×C22⋊C4—32no—
49C2×C22⋊C4—34no—
50C2×C22⋊C4—34no—
51C2×C22⋊C4—34no—
52C2×C22⋊C4—38no—
53C2×C22⋊C4A411yescharacteristic, direct product factor
54C23×C3—44no—
55C23×C3—42no—
56C23×C3—41no—
57C23×C3—84no—
58C2×A4—24no—
59C2×A4C22⋊C414yesdirect product factor
60C2×A4C22×C411yescharacteristic
61C2×A4C2×D412yes—
62C2×C3×C4—44no—
63C2×C3×C4—84no—
64C24—28no—
65C24—24no—
66C24—31no—
67C24—34no—
68C24—32no—
69C24—31no—
70C24—32no—
71C24—38no—
72C24—34no—
73C24—32no—
74C24—38no—
75C24—34no—
76C24—34no—
77C24—34no—
78C24—34no—
79C24—32no—
80C24—32no—
81C24—31no—
82C24—64no—
83C24—64no—
84C24—68no—
85C24—64no—
86C24—68no—
87C24—616no—
88C24—64no—
89C24—64no—
90C24—68no—
91C24—616no—
92C24C2×C3×C412yes—
93C24C2×C3×C414yescomplemented
94C24C3×D414yes—
95C24C2×A411yescharacteristic
96C24C23×C311yescharacteristic
97C22×C4—24no—
98C22×C4—34no—
99C22×C4—32no—
100C22×C4—34no—
101C22×C4—34no—
102C22×C4—34no—
103C22×C4—64no—
104C22×C4—64no—
105C22×C4—68no—
106C22×C4—68no—
107C22×C4—64no—
108C22×C4—64no—
109C22×C4—64no—
110C22×C4C2×A412yescomplemented
111C22⋊C4—38no—
112C22⋊C4—34no—
113C22⋊C4—38no—
114C22⋊C4C2×A414yesdirect product factor
115C22×C3—42no—
116C22×C3—44no—
117C22×C3—44no—
118C22×C3—41no—
119C22×C3—84no—
120C22×C3—88no—
121A4C2×C22⋊C411yescharacteristic, direct product factor
122C3×C4—84no—
123C23—24no—
124C23—24no—
125C23—34no—
126C23—32no—
127C23—32no—
128C23—31no—
129C23—34no—
130C23—38no—
131C23—34no—
132C23—32no—
133C23—38no—
134C23—32no—
135C23—32no—
136C23—34no—
137C23—34no—
138C23—31no—
139C23—31no—
140C23—38no—
141C23—34no—
142C23—34no—
143C23—34no—
144C23—38no—
145C23—34no—
146C23—34no—
147C23—32no—
148C23—64no—
149C23—68no—
150C23—68no—
151C23—64no—
152C23—616no—
153C23—68no—
154C23—64no—
155C23—616no—
156C23—68no—
157C23—64no—
158C23—68no—
159C23—64no—
160C23—68no—
161C23—616no—
162C23—68no—
163C23—632no—
164C23—68no—
165C23—616no—
166C23—64no—
167C23C3×C22⋊C414yescomplemented
168C23C4×A414yescomplemented
169C23C22×C3×C411yescharacteristic
170C23C2×C3×D412yes—
171C23C22×A412yes—
172C23C22×A411yescharacteristic
173C2×C4—24no—
174C2×C4—34no—
175C2×C4—64no—
176C2×C4—68no—
177C2×C4—64no—
178C2×C4—68no—
179C2×C4—64no—
180C2×C4—64no—
181C2×C4—64no—
182C2×C4C22×A414yescomplemented
183C2×C3—41no—
184C2×C3—42no—
185C2×C3—44no—
186C2×C3—84no—
187C22—24no—
188C22—28no—
189C22—34no—
190C22—31no—
191C22—32no—
192C22—38no—
193C22—32no—
194C22—34no—
195C22—32no—
196C22—31no—
197C22—31no—
198C22—34no—
199C22—32no—
200C22—32no—
201C22—34no—
202C22—38no—
203C22—34no—
204C22—34no—
205C22—64no—
206C22—616no—
207C22—68no—
208C22—64no—
209C22—64no—
210C22—64no—
211C22—616no—
212C22—68no—
213C22—64no—
214C22—68no—
215C22C2×C3×C22⋊C411yescharacteristic, complemented
216C22C2×C4×A412yes—
217C22C2×C4×A414yescomplemented
218C22D4×A414yes—
219C22C23×A411yescharacteristic
220C4—24no—
221C4—64no—
222C3—41noSylow
223C2—24no—
224C2—34no—
225C2—31no—
226C2—32no—
227C2—31no—
228C2—64no—
229C2A4×C22⋊C414yesdirect product factor
230C2C22×C4×A411yescharacteristic
231C2C2×D4×A412yes—
subgroupG/H
centerC23C22×A4
commutatorC23C22×C3×C4
FrattiniC22C23×A4
FittingC23×C22⋊C4C3
socleC25C22×C3
prime radicalC24×A4C2
cosocleC24C23×C3
hypercenterC2×C22⋊C4A4
hypocenterC22C2×C3×C22⋊C4
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×A4C2semidirect
3. A4×C22⋊C4C2direct (N × H)
4. C23×C22⋊C4C3semidirect
6. C23×A4C4semidirect
10. C2×C4×A4C22semidirect
13. C24×C4C2×C3semidirect
18. C22×C22⋊C4C2×C3semidirect
23. C22×A4C2×C4semidirect
35. C25C3×C4semidirect
43. C23×C4C22×C3semidirect
53. C2×C22⋊C4A4direct (N × H)
59. C2×A4C22⋊C4direct (N × H)
93. C24C2×C3×C4semidirect
110. C22×C4C2×A4semidirect
114. C22⋊C4C2×A4direct (N × H)
121. A4C2×C22⋊C4direct (N × H)
167. C23C3×C22⋊C4semidirect
168. C23C4×A4semidirect
182. C2×C4C22×A4semidirect
215. C22C2×C3×C22⋊C4semidirect
217. C22C2×C4×A4semidirect
229. C2A4×C22⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C24×A42
C22×C4×A42
A4×C22⋊C42
C23×C22⋊C43
C2×C3×C22⋊C44