Finite Group Databasebeta

← C2×A4×M16 #85625

Subgroups — C2×A4×M16 #85625

Order 384.

400 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 90 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×A4C211yesmaximal, characteristic
2A4×M16C214yesmaximal, direct product factor
3C2×C8×A4C212yesmaximal, complemented
4C23×M16C311yesmaximal, characteristic, complemented, Sylow
5C23×A4C411yescharacteristic
6C2×C4×A4C412yes—
7C2×C4×A4C411yescharacteristic
8C2×C4×A4C2211yescharacteristic
9C2×C4×A4C2212yes—
10C2×C3×M16—41nomaximal
11C8×A4C2214yescomplemented
12C24×C4C2×C311yescharacteristic
13C22×M16—31no—
14C22×M16—32no—
15C22×M16—31no—
16C22×M16—34no—
17C22×M16C2×C314yescomplemented
18C23×C8C2×C312yescomplemented
19C22×A4—22no—
20C22×A4C2×C412yes—
21C22×A4C2×C411yescharacteristic
22C22×C3×C4—41no—
23C4×A4C2×C411yescharacteristic
24C4×A4C2×C412yes—
25C4×A4C2311yescharacteristic
26C3×M16—44no—
27C2×C3×C8—42no—
28C25C3×C411yescharacteristic
29C23×C4—31no—
30C23×C4—32no—
31C23×C4—31no—
32C23×C4—31no—
33C23×C4—31no—
34C23×C4—32no—
35C23×C4C3×C412yes—
36C23×C4C3×C411yescharacteristic
37C23×C4C22×C311yescharacteristic
38C23×C4C22×C312yes—
39C2×M16—34no—
40C2×M16—38no—
41C2×M16—32no—
42C2×M16—31no—
43C2×M16—34no—
44C2×M16—34no—
45C2×M16—38no—
46C2×M16—34no—
47C2×M16—32no—
48C2×M16A411yescharacteristic, direct product factor
49C22×C8—32no—
50C22×C8—32no—
51C22×C8—34no—
52C22×C8C22×C314yescomplemented
53C23×C3—41no—
54C2×A4—22no—
55C2×A4M1612yesdirect product factor
56C2×A4C22×C411yescharacteristic
57C2×C3×C4—41no—
58C2×C3×C4—42no—
59C2×C3×C4—42no—
60C2×C3×C4—41no—
61C3×C8—44no—
62C24—22no—
63C24—32no—
64C24—31no—
65C24—31no—
66C24—62no—
67C24C2×C3×C412yes—
68C24C2×C3×C411yescharacteristic
69C22×C4—32no—
70C22×C4—34no—
71C22×C4—32no—
72C22×C4—32no—
73C22×C4—34no—
74C22×C4—31no—
75C22×C4—32no—
76C22×C4—31no—
77C22×C4—31no—
78C22×C4—32no—
79C22×C4—32no—
80C22×C4—32no—
81C22×C4—31no—
82C22×C4—31no—
83C22×C4—31no—
84C22×C4—32no—
85C22×C4—31no—
86C22×C4—32no—
87C22×C4—32no—
88C22×C4—32no—
89C22×C4—32no—
90C22×C4—31no—
91C22×C4—32no—
92C22×C4—31no—
93C22×C4—32no—
94C22×C4C2×C3×C411yescharacteristic
95C22×C4C2×C3×C412yes—
96C22×C4C2×A411yescharacteristic
97C22×C4C23×C311yescharacteristic
98M16—38no—
99M16—34no—
100M16—38no—
101M16C2×A414yesdirect product factor
102C2×C8—34no—
103C2×C8—32no—
104C2×C8—34no—
105C2×C8—34no—
106C2×C8—34no—
107C2×C8C2×A412yescomplemented
108C22×C3—41no—
109C22×C3—42no—
110C22×C3—82no—
111A4C2×M1611yescharacteristic, direct product factor
112C3×C4—42no—
113C3×C4—41no—
114C3×C4—41no—
115C23—22no—
116C23—31no—
117C23—32no—
118C23—31no—
119C23—32no—
120C23—32no—
121C23—32no—
122C23—31no—
123C23—32no—
124C23—62no—
125C23—62no—
126C23—62no—
127C23—64no—
128C23—62no—
129C23—62no—
130C23—64no—
131C23C3×M1612yescomplemented
132C23C4×A411yescharacteristic
133C23C22×C3×C411yescharacteristic
134C2×C4—32no—
135C2×C4—32no—
136C2×C4—34no—
137C2×C4—34no—
138C2×C4—32no—
139C2×C4—32no—
140C2×C4—31no—
141C2×C4—31no—
142C2×C4—32no—
143C2×C4—32no—
144C2×C4—32no—
145C2×C4—31no—
146C2×C4—32no—
147C2×C4—32no—
148C2×C4—31no—
149C2×C4—31no—
150C2×C4—31no—
151C2×C4—32no—
152C2×C4—31no—
153C2×C4—32no—
154C2×C4—31no—
155C2×C4C4×A411yescharacteristic
156C2×C4C4×A412yes—
157C2×C4C22×A412yes—
158C2×C4C22×A411yescharacteristic
159C8—34no—
160C8C22×A414yescomplemented
161C2×C3—41no—
162C2×C3—42no—
163C2×C3—82no—
164C22—22no—
165C22—32no—
166C22—31no—
167C22—32no—
168C22—31no—
169C22—31no—
170C22—32no—
171C22—32no—
172C22—32no—
173C22—62no—
174C22—64no—
175C22—62no—
176C22—62no—
177C22—62no—
178C22—64no—
179C22—62no—
180C22C2×C3×M1611yescharacteristic, complemented
181C22C2×C4×A412yes—
182C22C2×C4×A411yescharacteristic
183C4—31no—
184C4—31no—
185C4—32no—
186C4C2×C4×A412yes—
187C4C2×C4×A411yescharacteristic
188C4C23×A411yescharacteristic
189C3—41noSylow
190C2—22no—
191C2—32no—
192C2—31no—
193C2—31no—
194C2—62no—
195C2A4×M1612yesdirect product factor
196C2C22×C4×A411yescharacteristic
subgroupG/H
centerC2×C4C22×A4
commutatorC23C22×C3×C4
FrattiniC4C23×A4
FittingC23×M16C3
socleC24C2×C3×C4
prime radicalC23×A4C4
cosocleC22×C4C23×C3
hypercenterC2×M16A4
hypocenterC22C2×C3×M16
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. A4×M16C2direct (N × H)
3. C2×C8×A4C2semidirect
4. C23×M16C3semidirect
11. C8×A4C22semidirect
17. C22×M16C2×C3semidirect
18. C23×C8C2×C3semidirect
48. C2×M16A4direct (N × H)
52. C22×C8C22×C3semidirect
55. C2×A4M16direct (N × H)
101. M16C2×A4direct (N × H)
107. C2×C8C2×A4semidirect
111. A4C2×M16direct (N × H)
131. C23C3×M16semidirect
160. C8C22×A4semidirect
180. C22C2×C3×M16semidirect
195. C2A4×M16direct (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×M162
C2×C8×A42
C23×M163
C2×C3×M164