Finite Group Databasebeta

← C22×C4×S4 #87472

Subgroups — C22×C4×S4 #87472

Order 384.

1291 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1291 conjugacy classes of subgroups — too many to draw. Only its 172 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
1C23×S4C211yesmaximal, characteristic
2C22×C4×A4C211yesmaximal, characteristic, complemented
3C22×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C2×C4×S4C2112yesmaximal, direct product factor
5C22×C4×D4—31nomaximal, Sylow
6C23×A4C2211yescharacteristic
7C22×S4C418yesdirect product factor
8C22×S4C2216yes—
9C22×C4×D3—41nomaximal
10C2×C4×A4C2216yescomplemented
11C2×SL(2,ℤ/4)C2216yescomplemented
12C4×S4C22116yesdirect product factor
13C23×D4—31no—
14C24×C4—31no—
15C24×C4D311yescharacteristic, complemented
16C2×C4×D4—312no—
17C2×C4×D4—312no—
18C22×C4⋊C4—31no—
19C22×C22⋊C4—31no—
20C22×C22⋊C4—31no—
21C22×C42—31no—
22C23×D3—41no—
23C22×A4C2×C414yescomplemented
24C22×A4C2313yes—
25C2×S4C2×C4124yesdirect product factor
26C2×S4C2314yes—
27C22×C3×C4—41no—
28C22×Dic3—41no—
29C2×C4×D3—412no—
30C4×A4C2314yescomplemented
31SL(2,ℤ/4)C2314yescomplemented
32C25—31no—
33C25C2×D311yescharacteristic
34C22×D4—36no—
35C22×D4—38no—
36C22×D4—38no—
37C22×D4—36no—
38C23×C4—31no—
39C23×C4—31no—
40C23×C4—31no—
41C23×C4—36no—
42C23×C4—31no—
43C23×C4—31no—
44C23×C4—312no—
45C23×C4—61no—
46C23×C4—66no—
47C23×C4—61no—
48C23×C4C2×D316yescomplemented
49C4×D4—348no—
50C4×D4—316no—
51C2×C4⋊C4—312no—
52C2×C22⋊C4—312no—
53C2×C22⋊C4—36no—
54C2×C22⋊C4—36no—
55C2×C42—312no—
56C23×C3—41no—
57C22×D3—46no—
58C22×D3—48no—
59C2×A4C22×C416yescomplemented
60C2×A4C2411yescharacteristic
61S4C22×C418yesdirect product factor
62C2×C3×C4—46no—
63C2×Dic3—46no—
64C4×D3—416no—
65C24—31no—
66C24—33no—
67C24—34no—
68C24—36no—
69C24—38no—
70C24—63no—
71C24—64no—
72C24—61no—
73C24C4×D314yescomplemented
74C24C22×D313yes—
75C2×D4—312no—
76C2×D4—348no—
77C2×D4—324no—
78C2×D4—324no—
79C2×D4—34no—
80C22×C4—36no—
81C22×C4—38no—
82C22×C4—31no—
83C22×C4—36no—
84C22×C4—36no—
85C22×C4—36no—
86C22×C4—312no—
87C22×C4—36no—
88C22×C4—36no—
89C22×C4—316no—
90C22×C4—64no—
91C22×C4—63no—
92C22×C4—612no—
93C22×C4—624no—
94C22×C4—64no—
95C22×C4—63no—
96C22×C4—61no—
97C22×C4—612no—
98C22×C4S411yescharacteristic, direct product factor
99C22×C4C22×D314yescomplemented
100C4⋊C4—316no—
101C22⋊C4—316no—
102C22⋊C4—312no—
103C22⋊C4—34no—
104C42—316no—
105C22×C3—44no—
106C22×C3—43no—
107C2×D3—44no—
108C2×D3—424no—
109A4C23×C411yescharacteristic, complemented
110C3×C4—44no—
111Dic3—44no—
112C23—34no—
113C23—34no—
114C23—33no—
115C23—33no—
116C23—33no—
117C23—312no—
118C23—36no—
119C23—34no—
120C23—324no—
121C23—61no—
122C23—66no—
123C23—66no—
124C23—612no—
125C23—624no—
126C23—68no—
127C23—66no—
128C23C2×C4×D316yescomplemented
129C23C2×S411yescharacteristic
130C23C23×D311yescharacteristic
131D4—348no—
132D4—38no—
133D4—38no—
134C2×C4—34no—
135C2×C4—324no—
136C2×C4—36no—
137C2×C4—34no—
138C2×C4—312no—
139C2×C4—34no—
140C2×C4—34no—
141C2×C4—34no—
142C2×C4—66no—
143C2×C4—616no—
144C2×C4—612no—
145C2×C4—66no—
146C2×C4—66no—
147C2×C4—616no—
148C2×C4C2×S416yesdirect product factor
149C2×C3—46no—
150C2×C3—41no—
151D3—48no—
152C22—36no—
153C22—31no—
154C22—36no—
155C22—36no—
156C22—312no—
157C22—31no—
158C22—33no—
159C22—38no—
160C22—63no—
161C22—64no—
162C22—64no—
163C22—624no—
164C22—624no—
165C22—64no—
166C22C4×S414yesdirect product factor
167C22C22×C4×D311yescharacteristic, complemented
168C22C22×S413yes—
169C4—38no—
170C4—34no—
171C4—64no—
172C4—64no—
173C4C22×S414yesdirect product factor
174C3—41noSylow
175C2—31no—
176C2—31no—
177C2—36no—
178C2—68no—
179C2C2×C4×S416yesdirect product factor
180C2C23×S411yescharacteristic
subgroupG/H
center, hypercenterC22×C4S4
commutator, hypocenterA4C23×C4
FrattiniC2C23×S4
FittingC24×C4D3
socleC25C2×D3
prime radicalC23×S4C2
cosocleC2×A4C24
LCSG ▷ A4
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×A4C2semidirect
3. C22×SL(2,ℤ/4)C2semidirect
4. C2×C4×S4C2direct (N × H)
7. C22×S4C4direct (N × H)
10. C2×C4×A4C22semidirect
11. C2×SL(2,ℤ/4)C22semidirect
12. C4×S4C22direct (N × H)
15. C24×C4D3semidirect
23. C22×A4C2×C4semidirect
25. C2×S4C2×C4direct (N × H)
30. C4×A4C23semidirect
31. SL(2,ℤ/4)C23semidirect
48. C23×C4C2×D3semidirect
59. C2×A4C22×C4semidirect
61. S4C22×C4direct (N × H)
73. C24C4×D3semidirect
98. C22×C4S4direct (N × H)
99. C22×C4C22×D3semidirect
109. A4C23×C4semidirect
128. C23C2×C4×D3semidirect
148. C2×C4C2×S4direct (N × H)
166. C22C4×S4direct (N × H)
167. C22C22×C4×D3semidirect
173. C4C22×S4direct (N × H)
179. C2C2×C4×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×S42
C22×C4×A42
C22×SL(2,ℤ/4)2
C2×C4×S42
C22×C4×D43
C22×C4×D34