Finite Group Databasebeta

← #85305

Subgroups — #85305

Order 384.

255 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 44 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
1A4×C4⋊C4C211yesmaximal, characteristic
2C4⋊SL(2,ℤ/4)C212yesmaximal
3C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4SL(2,ℤ/4)⋊C4C212yesmaximal
5C4×SL(2,ℤ/4)C211yesmaximal, characteristic
6#2442—31nomaximal, Sylow
7C2×C4×A4C2212yes—
8C2×C4×A4C2211yescharacteristic
9C2×SL(2,ℤ/4)C2211yescharacteristic
10C2×SL(2,ℤ/4)C2211yescharacteristic
11C2×SL(2,ℤ/4)C2212yes—
12#890—41nomaximal
13C2×#125—31no—
14C22×C4⋊C4D311yescharacteristic
15#400—32no—
16#398—32no—
17#389—31no—
18#389—31no—
19#385—32no—
20#382—31no—
21#380—32no—
22#380—31no—
23C4×C22⋊C4—31no—
24C22×A4C2311yescharacteristic
25C4×A4—22no—
26C4×A4Q812yes—
27SL(2,ℤ/4)—21no—
28SL(2,ℤ/4)—21no—
29SL(2,ℤ/4)—22no—
30C3×C4⋊C4—41no—
31C4⋊Dic3—42no—
32C4⋊Dic3—41no—
33Dic3⋊C4—42no—
34C4×Dic3—41no—
35C23×C4C2×D312yes—
36C23×C4C2×D311yescharacteristic
37#125—62no—
38#125—61no—
39#125—61no—
40C2×C4⋊C4—31no—
41C2×C4⋊C4—31no—
42C2×C4⋊C4—32no—
43C2×C4⋊C4—31no—
44C2×C4⋊C4—32no—
45C2×C4⋊C4—32no—
46C2×C4⋊C4—31no—
47C2×C4⋊C4—32no—
48C2×C4⋊C4—32no—
49C2×C22⋊C4—31no—
50C2×C22⋊C4—31no—
51C2×C22⋊C4—32no—
52C2×C42—31no—
53C2×C42—31no—
54#95—31no—
55#95—31no—
56#95—32no—
57#95—32no—
58#95—32no—
59#95—32no—
60#95—32no—
61C2×A4C2×Q811yescharacteristic
62C2×A4#4111yescharacteristic
63C2×A4#4111yescharacteristic
64C2×C3×C4—41no—
65C2×C3×C4—42no—
66C2×Dic3—41no—
67C2×Dic3—41no—
68C2×Dic3—42no—
69C24C22×D311yescharacteristic
70C22×C4—22no—
71C22×C4—31no—
72C22×C4—31no—
73C22×C4—32no—
74C22×C4—32no—
75C22×C4—32no—
76C22×C4—31no—
77C22×C4—31no—
78C22×C4—31no—
79C22×C4—32no—
80C22×C4—32no—
81C22×C4—31no—
82C22×C4—62no—
83C22×C4Dic612yes—
84C4⋊C4—32no—
85C4⋊C4—31no—
86C4⋊C4—61no—
87C4⋊C4—62no—
88C4⋊C4—61no—
89C4⋊C4—62no—
90C4⋊C4—62no—
91C4⋊C4—62no—
92C4⋊C4—62no—
93C4⋊C4—61no—
94C4⋊C4—122no—
95C4⋊C4—122no—
96C4⋊C4S411yescharacteristic
97C22⋊C4—61no—
98C22⋊C4—62no—
99C22⋊C4—61no—
100C22⋊C4—61no—
101C22⋊C4—61no—
102C22⋊C4—62no—
103C42—62no—
104C42—61no—
105C42—61no—
106C22×C3—41no—
107A4#12511yescharacteristic, complemented
108C3×C4—42no—
109C3×C4—82no—
110Dic3—81no—
111Dic3—81no—
112Dic3—82no—
113C23—31no—
114C23—31no—
115C23—31no—
116C23—31no—
117C23C2×Dic611yescharacteristic
118C23#23711yescharacteristic
119C23#23911yescharacteristic
120C2×C4—31no—
121C2×C4—32no—
122C2×C4—32no—
123C2×C4—32no—
124C2×C4—32no—
125C2×C4—62no—
126C2×C4—62no—
127C2×C4—61no—
128C2×C4—62no—
129C2×C4—62no—
130C2×C4—61no—
131C2×C4—61no—
132C2×C4—61no—
133C2×C4—61no—
134C2×C4—62no—
135C2×C4—61no—
136C2×C4—61no—
137C2×C4—62no—
138C2×C4—62no—
139C2×C4—61no—
140C2×C4—62no—
141C2×C4—61no—
142C2×C4—61no—
143C2×C4—61no—
144C2×C4—62no—
145C2×C4—61no—
146C2×C4—61no—
147C2×C4—62no—
148C2×C4C2×S412yes—
149C2×C4C2×S411yescharacteristic
150C2×C3—41no—
151C2×C3—41no—
152C2×C3—41no—
153C22—31no—
154C22—31no—
155C22—31no—
156C22—31no—
157C22—31no—
158C22—31no—
159C22—31no—
160C22—31no—
161C22—31no—
162C22—61no—
163C22#89011yescharacteristic, complemented
164C22C22×S411yescharacteristic
165C4—22no—
166C4—32no—
167C4—62no—
168C4—121no—
169C4—122no—
170C4—121no—
171C4—121no—
172C4—121no—
173C4—122no—
174C4A4⋊Q812yes—
175C3—41noSylow
176C2—31no—
177C2—31no—
178C2—31no—
179C2—31no—
180C2C2×A4⋊Q811yescharacteristic
181C2#591411yescharacteristic
182C2#591911yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, prime radical, cosocleC22×A4C23
FittingC22×C4⋊C4D3
socleC24C22×D3
hypercenterC4⋊C4S4
hypocenterA4#125
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
107. A4#125semidirect
163. C22#890semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊C42
C4⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊C42
C4×SL(2,ℤ/4)2
#24423
#8904