Finite Group Databasebeta

← C2×A4×C4⋊C4 #85571

Subgroups — C2×A4×C4⋊C4 #85571

Order 384.

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

normal subgroups only

This group has 568 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
1C22×C4×A4C212yesmaximal
2C22×C4×A4C211yesmaximal, characteristic
3A4×C4⋊C4C214yesmaximal, direct product factor
4C23×C4⋊C4C311yesmaximal, characteristic, complemented, Sylow
5C23×A4C2211yescharacteristic
6C2×C4×A4—24no—
7C2×C4×A4C414yescomplemented
8C2×C4×A4C2214yes—
9C2×C4×A4C2212yes—
10C2×C3×C4⋊C4—41nomaximal
11C24×C4C2×C312yes—
12C24×C4C2×C311yescharacteristic
13C22×C4⋊C4—31no—
14C22×C4⋊C4—31no—
15C22×C4⋊C4—32no—
16C22×C4⋊C4—34no—
17C22×C4⋊C4C2×C314yescomplemented
18C22×A4C2×C412yes—
19C22×A4D412yes—
20C22×A4Q812yes—
21C22×A4C2311yescharacteristic
22C22×C3×C4—41no—
23C22×C3×C4—42no—
24C4×A4—24no—
25C4×A4C2×C414yescomplemented
26C3×C4⋊C4—44no—
27C25C22×C311yescharacteristic
28C23×C4—24no—
29C23×C4—31no—
30C23×C4—31no—
31C23×C4—32no—
32C23×C4—34no—
33C23×C4—32no—
34C23×C4—32no—
35C23×C4—34no—
36C23×C4—64no—
37C23×C4C3×C414yescomplemented
38C23×C4C22×C314yes—
39C23×C4C22×C312yes—
40C2×C4⋊C4—32no—
41C2×C4⋊C4—38no—
42C2×C4⋊C4—34no—
43C2×C4⋊C4—32no—
44C2×C4⋊C4—31no—
45C2×C4⋊C4—34no—
46C2×C4⋊C4—34no—
47C2×C4⋊C4—34no—
48C2×C4⋊C4—38no—
49C2×C4⋊C4A411yescharacteristic, direct product factor
50C23×C3—41no—
51C2×A4C4⋊C414yesdirect product factor
52C2×A4C22×C411yescharacteristic
53C2×A4C2×D411yescharacteristic
54C2×A4C2×Q811yescharacteristic
55C2×C3×C4—44no—
56C2×C3×C4—42no—
57C2×C3×C4—44no—
58C2×C3×C4—84no—
59C24—31no—
60C24—32no—
61C24—32no—
62C24—32no—
63C24—31no—
64C24C2×C3×C412yes—
65C24C3×D412yes—
66C24C3×Q812yes—
67C24C23×C311yescharacteristic
68C22×C4—24no—
69C22×C4—32no—
70C22×C4—34no—
71C22×C4—38no—
72C22×C4—31no—
73C22×C4—34no—
74C22×C4—34no—
75C22×C4—32no—
76C22×C4—32no—
77C22×C4—32no—
78C22×C4—34no—
79C22×C4—38no—
80C22×C4—34no—
81C22×C4—34no—
82C22×C4—32no—
83C22×C4—34no—
84C22×C4—34no—
85C22×C4—34no—
86C22×C4—64no—
87C22×C4—64no—
88C22×C4—68no—
89C22×C4—68no—
90C22×C4—64no—
91C22×C4—64no—
92C22×C4—64no—
93C22×C4C2×C3×C414yescomplemented
94C22×C4C2×A412yes—
95C22×C4C2×A411yescharacteristic
96C4⋊C4—38no—
97C4⋊C4—38no—
98C4⋊C4—34no—
99C4⋊C4C2×A414yesdirect product factor
100C22×C3—42no—
101C22×C3—42no—
102C22×C3—42no—
103C22×C3—41no—
104A4C2×C4⋊C411yescharacteristic, direct product factor
105C3×C4—44no—
106C3×C4—84no—
107C23—32no—
108C23—31no—
109C23—32no—
110C23—31no—
111C23—34no—
112C23—34no—
113C23—32no—
114C23—34no—
115C23—31no—
116C23—32no—
117C23—32no—
118C23—32no—
119C23—38no—
120C23—32no—
121C23—32no—
122C23—32no—
123C23—32no—
124C23—32no—
125C23—32no—
126C23—31no—
127C23—31no—
128C23C3×C4⋊C414yescomplemented
129C23C22×C3×C411yescharacteristic
130C23C2×C3×D411yescharacteristic
131C23C2×C3×Q811yescharacteristic
132C23C22×A411yescharacteristic
133C2×C4—24no—
134C2×C4—38no—
135C2×C4—34no—
136C2×C4—34no—
137C2×C4—32no—
138C2×C4—34no—
139C2×C4—38no—
140C2×C4—34no—
141C2×C4—34no—
142C2×C4—34no—
143C2×C4—64no—
144C2×C4—68no—
145C2×C4—64no—
146C2×C4—68no—
147C2×C4—64no—
148C2×C4—64no—
149C2×C4—64no—
150C2×C4C4×A414yescomplemented
151C2×C4C22×A414yes—
152C2×C4C22×A412yes—
153C2×C3—41no—
154C2×C3—41no—
155C2×C3—41no—
156C2×C3—44no—
157C22—34no—
158C22—31no—
159C22—31no—
160C22—31no—
161C22—34no—
162C22—32no—
163C22—34no—
164C22—34no—
165C22—32no—
166C22—31no—
167C22—31no—
168C22—32no—
169C22—31no—
170C22—31no—
171C22—32no—
172C22—31no—
173C22—31no—
174C22—34no—
175C22—34no—
176C22—34no—
177C22—34no—
178C22C2×C3×C4⋊C411yescharacteristic, complemented
179C22C2×C4×A412yes—
180C22D4×A412yes—
181C22Q8×A412yes—
182C22C23×A411yescharacteristic
183C4—24no—
184C4—34no—
185C4—64no—
186C4C2×C4×A414yescomplemented
187C3—41noSylow
188C2—34no—
189C2—31no—
190C2—31no—
191C2—31no—
192C2—31no—
193C2A4×C4⋊C414yesdirect product factor
194C2C22×C4×A411yescharacteristic
195C2C2×D4×A411yescharacteristic
196C2C2×Q8×A411yescharacteristic
subgroupG/H
centerC23C22×A4
commutatorC23C22×C3×C4
FrattiniC22C23×A4
FittingC23×C4⋊C4C3
socleC25C22×C3
prime radicalC23×A4C22
cosocleC24C23×C3
hypercenterC2×C4⋊C4A4
hypocenterC22C2×C3×C4⋊C4
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. A4×C4⋊C4C2direct (N × H)
4. C23×C4⋊C4C3semidirect
7. C2×C4×A4C4semidirect
17. C22×C4⋊C4C2×C3semidirect
25. C4×A4C2×C4semidirect
37. C23×C4C3×C4semidirect
49. C2×C4⋊C4A4direct (N × H)
51. C2×A4C4⋊C4direct (N × H)
93. C22×C4C2×C3×C4semidirect
99. C4⋊C4C2×A4direct (N × H)
104. A4C2×C4⋊C4direct (N × H)
128. C23C3×C4⋊C4semidirect
150. C2×C4C4×A4semidirect
178. C22C2×C3×C4⋊C4semidirect
186. C4C2×C4×A4semidirect
193. C2A4×C4⋊C4direct (N × H)

Maximal subgroups

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

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