Finite Group Databasebeta

← #85546

Subgroups — #85546

Order 384.

944 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 56 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, complemented
2C2×GL(2,ℤ/4)C213yesmaximal, complemented
3#5432C213yesmaximal, complemented
4#2846—31nomaximal, Sylow
5C23×A4—26no—
6C23×A4C2213yes—
7C22×S4C2211yescharacteristic, complemented
8GL(2,ℤ/4)—26no—
9C2×SL(2,ℤ/4)C2213yescomplemented
10#953—41nomaximal
11C26D311yescharacteristic, complemented
12C2×#120—33no—
13C2×#120—31no—
14C2×#120—33no—
15#379—33no—
16#379—31no—
17#379—33no—
18C24×C3—41no—
19C22×A4—28no—
20C22×A4—26no—
21C22×A4D416yescomplemented
22C22×A4C2311yescharacteristic
23C2×S4—23no—
24C2×#67—43no—
25SL(2,ℤ/4)—23no—
26#217—43no—
27C25—26no—
28C25—33no—
29C25—31no—
30C25—66no—
31C25C2×D313yes—
32C22×D4—33no—
33C22×D4—33no—
34C22×D4—31no—
35#120—66no—
36#120—612no—
37#120—63no—
38#120—66no—
39#120—61no—
40C2×C22⋊C4—36no—
41C2×C22⋊C4—33no—
42C2×C22⋊C4—33no—
43C2×C22⋊C4—33no—
44C2×C22⋊C4—33no—
45C2×C22⋊C4—33no—
46C23×C3—43no—
47C23×C3—86no—
48C22×D3—41no—
49C2×A4—26no—
50C2×A4C2×D413yes—
51S4—41no—
52#67—86no—
53C2×Dic3—43no—
54C24—28no—
55C24—26no—
56C24—33no—
57C24—33no—
58C24—36no—
59C24—312no—
60C24—33no—
61C24—31no—
62C24—66no—
63C24—66no—
64C24—66no—
65C24—61no—
66C24—612no—
67C24—66no—
68C24—612no—
69C24—624no—
70C24—616no—
71C24#6716yescomplemented
72C24S411yescharacteristic, complemented
73C24C22×D311yescharacteristic
74C2×D4—66no—
75C2×D4—63no—
76C2×D4—612no—
77C2×D4—66no—
78C2×D4—66no—
79C2×D4—63no—
80C2×D4—63no—
81C2×D4—63no—
82C22×C4—31no—
83C22×C4—33no—
84C22×C4—33no—
85C22⋊C4—66no—
86C22⋊C4—63no—
87C22⋊C4—612no—
88C22⋊C4—66no—
89C22⋊C4—66no—
90C22⋊C4—63no—
91C22⋊C4—63no—
92C22⋊C4—63no—
93C22×C3—46no—
94C22×C3—41no—
95C22×C3—86no—
96C22×C3—88no—
97C2×D3—83no—
98A4#12011yescharacteristic, complemented
99Dic3—83no—
100C23—26no—
101C23—26no—
102C23—31no—
103C23—36no—
104C23—33no—
105C23—312no—
106C23—36no—
107C23—36no—
108C23—33no—
109C23—66no—
110C23—66no—
111C23—612no—
112C23—68no—
113C23—624no—
114C23—66no—
115C23—63no—
116C23—612no—
117C23—66no—
118C23—612no—
119C23—648no—
120C23—612no—
121C23—63no—
122C23—63no—
123C23—624no—
124C23—624no—
125C23—66no—
126C23—61no—
127C23C2×#6713yes—
128C23C2×S413yes—
129D4—123no—
130D4—1212no—
131D4—126no—
132D4—121no—
133D4—126no—
134C2×C4—66no—
135C2×C4—63no—
136C2×C4—63no—
137C2×C4—63no—
138C2×C4—63no—
139C2×C4—63no—
140C2×C3—43no—
141C2×C3—86no—
142D3—161no—
143C22—28no—
144C22—26no—
145C22—33no—
146C22—33no—
147C22—33no—
148C22—312no—
149C22—36no—
150C22—66no—
151C22—612no—
152C22—66no—
153C22—624no—
154C22—61no—
155C22—66no—
156C22—612no—
157C22—616no—
158C22—66no—
159C22—123no—
160C22—121no—
161C22—123no—
162C22#95311yescharacteristic, complemented
163C22GL(2,ℤ/4)16yescomplemented
164C22C22×S411yescharacteristic
165C4—123no—
166C4—121no—
167C4—123no—
168C3—41noSylow
169C2—26no—
170C2—33no—
171C2—31no—
172C2—66no—
173C2—241no—
174C2C2×GL(2,ℤ/4)13yes—
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC26D3
socleC24C22×D3
hypercenterC24S4
hypocenterA4#120
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×A4C2semidirect
2. C2×GL(2,ℤ/4)C2semidirect
3. #5432C2semidirect
7. C22×S4C22semidirect
9. C2×SL(2,ℤ/4)C22semidirect
11. C26D3semidirect
21. C22×A4D4semidirect
71. C24#67semidirect
72. C24S4semidirect
98. A4#120semidirect
162. C22#953semidirect
163. C22GL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C24×A42
C2×GL(2,ℤ/4)2
#54322
#28463
#9534