Finite Group Databasebeta

← C22×D4×A4 #87526

Subgroups — C22×D4×A4 #87526

Order 384.

2152 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 2152 conjugacy classes of subgroups — too many to draw. Only its 234 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×A4C212yesmaximal, complemented
2C2×D4×A4C2112yesmaximal, direct product factor
3C22×C4×A4C211yesmaximal, characteristic, complemented
4C24×D4C311yesmaximal, characteristic, complemented, Sylow
5C23×A4—28no—
6C23×A4C2211yescharacteristic
7C23×A4C22112yescomplemented
8C22×C3×D4—41nomaximal
9D4×A4C22116yesdirect product factor
10C2×C4×A4C2216yescomplemented
11C26C2×C312yescomplemented
12C23×D4—31no—
13C23×D4—31no—
14C23×D4—32no—
15C23×D4—312no—
16C23×D4C2×C3112yescomplemented
17C24×C4C2×C311yescharacteristic, complemented
18C24×C3—42no—
19C22×A4—224no—
20C22×A4D414yesdirect product factor
21C22×A4C2313yes—
22C22×A4C2318yescomplemented
23C2×C3×D4—412no—
24C22×C3×C4—41no—
25C4×A4C2314yescomplemented
26C25—28no—
27C25—312no—
28C25—32no—
29C25—32no—
30C25—68no—
31C25C22×C311yescharacteristic
32C25C22×C3112yescomplemented
33C22×D4—324no—
34C22×D4—332no—
35C22×D4—32no—
36C22×D4—312no—
37C22×D4—32no—
38C22×D4—312no—
39C22×D4—312no—
40C22×D4—312no—
41C22×D4—324no—
42C22×D4—348no—
43C22×D4—31no—
44C22×D4A411yescharacteristic, direct product factor
45C22×D4C22×C3116yescomplemented
46C23×C4—31no—
47C23×C4—31no—
48C23×C4—36no—
49C23×C4C22×C316yescomplemented
50C23×C3—41no—
51C23×C3—412no—
52C23×C3—88no—
53C2×A4—28no—
54C2×A4C2×D416yesdirect product factor
55C2×A4C2411yescharacteristic
56C3×D4—416no—
57C2×C3×C4—46no—
58C24—224no—
59C24—33no—
60C24—34no—
61C24—31no—
62C24—316no—
63C24—312no—
64C24—324no—
65C24—312no—
66C24—32no—
67C24—312no—
68C24—312no—
69C24—68no—
70C24—68no—
71C24—624no—
72C24—648no—
73C24—68no—
74C24—68no—
75C24—648no—
76C24C3×D414yescomplemented
77C24C2×A412yescomplemented
78C24C23×C313yes—
79C24C23×C318yescomplemented
80C2×D4—396no—
81C2×D4—324no—
82C2×D4—348no—
83C2×D4—324no—
84C2×D4—348no—
85C2×D4—316no—
86C2×D4—316no—
87C2×D4—332no—
88C2×D4—312no—
89C2×D4C2×A4112yesdirect product factor
90C22×C4—36no—
91C22×C4—38no—
92C22×C4—31no—
93C22×C4—36no—
94C22×C4—36no—
95C22×C4—36no—
96C22×C4—312no—
97C22×C4C2×A411yescharacteristic, complemented
98C22×C4C23×C314yescomplemented
99C22×C3—44no—
100C22×C3—43no—
101C22×C3—48no—
102C22×C3—824no—
103A4C22×D411yescharacteristic, direct product factor
104C3×C4—44no—
105C23—28no—
106C23—28no—
107C23—33no—
108C23—312no—
109C23—36no—
110C23—32no—
111C23—312no—
112C23—34no—
113C23—34no—
114C23—33no—
115C23—33no—
116C23—324no—
117C23—38no—
118C23—312no—
119C23—324no—
120C23—38no—
121C23—624no—
122C23—648no—
123C23—624no—
124C23—68no—
125C23—648no—
126C23—624no—
127C23—624no—
128C23—648no—
129C23—664no—
130C23—648no—
131C23—68no—
132C23C2×C3×D416yescomplemented
133C23C22×A411yescharacteristic
134C23C22×A4112yescomplemented
135C23C24×C311yescharacteristic
136D4—332no—
137D4—332no—
138D4—316no—
139D4C22×A4116yesdirect product factor
140C2×C4—312no—
141C2×C4—36no—
142C2×C4—34no—
143C2×C4—312no—
144C2×C4—34no—
145C2×C4C22×A416yescomplemented
146C2×C3—46no—
147C2×C3—41no—
148C2×C3—88no—
149C22—224no—
150C22—36no—
151C22—31no—
152C22—36no—
153C22—38no—
154C22—312no—
155C22—36no—
156C22—36no—
157C22—33no—
158C22—31no—
159C22—38no—
160C22—68no—
161C22—648no—
162C22—68no—
163C22—68no—
164C22—68no—
165C22—648no—
166C22—624no—
167C22D4×A414yesdirect product factor
168C22C22×C3×D411yescharacteristic, complemented
169C22C23×A413yes—
170C22C23×A418yescomplemented
171C4—34no—
172C4C23×A414yescomplemented
173C3—41noSylow
174C2—28no—
175C2—36no—
176C2—31no—
177C2—31no—
178C2—68no—
179C2C2×D4×A416yesdirect product factor
180C2C24×A411yescharacteristic
subgroupG/H
centerC23C22×A4
commutator, cosocleC23C24×C3
FrattiniC2C24×A4
FittingC24×D4C3
socleC25C22×C3
hypercenterC22×D4A4
hypocenterC22C22×C3×D4
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C24×A4C2semidirect
2. C2×D4×A4C2direct (N × H)
3. C22×C4×A4C2semidirect
4. C24×D4C3semidirect
7. C23×A4C22semidirect
9. D4×A4C22direct (N × H)
10. C2×C4×A4C22semidirect
11. C26C2×C3semidirect
16. C23×D4C2×C3semidirect
17. C24×C4C2×C3semidirect
20. C22×A4D4direct (N × H)
22. C22×A4C23semidirect
25. C4×A4C23semidirect
32. C25C22×C3semidirect
44. C22×D4A4direct (N × H)
45. C22×D4C22×C3semidirect
49. C23×C4C22×C3semidirect
54. C2×A4C2×D4direct (N × H)
76. C24C3×D4semidirect
77. C24C2×A4semidirect
79. C24C23×C3semidirect
89. C2×D4C2×A4direct (N × H)
97. C22×C4C2×A4semidirect
98. C22×C4C23×C3semidirect
103. A4C22×D4direct (N × H)
132. C23C2×C3×D4semidirect
134. C23C22×A4semidirect
139. D4C22×A4direct (N × H)
145. C2×C4C22×A4semidirect
167. C22D4×A4direct (N × H)
168. C22C22×C3×D4semidirect
170. C22C23×A4semidirect
172. C4C23×A4semidirect
179. C2C2×D4×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C24×A42
C2×D4×A42
C22×C4×A42
C24×D43
C22×C3×D44