Finite Group Databasebeta

← #85292

Subgroups — #85292

Order 384.

371 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 52 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×SL(2,ℤ/4)C211yesmaximal, characteristic
2A4×C22⋊C4C211yesmaximal, characteristic
3#5432C211yesmaximal, characteristic
4SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
5C4×SL(2,ℤ/4)C212yesmaximal, complemented
6#2342—31nomaximal, Sylow
7C23×A4C2211yescharacteristic
8C2×C4×A4C2212yes—
9C2×SL(2,ℤ/4)C414yescomplemented
10C2×SL(2,ℤ/4)C2212yes—
11C2×SL(2,ℤ/4)C2211yescharacteristic
12C2×SL(2,ℤ/4)C2211yescharacteristic
13#877—41nomaximal
14C2×#117—31no—
15C22×C22⋊C4—31no—
16C22×C22⋊C4D311yescharacteristic
17#388—32no—
18#388—32no—
19#385—32no—
20#381—31no—
21#380—31no—
22#379—31no—
23C4×C22⋊C4—31no—
24C4×C22⋊C4—32no—
25C22×A4—22no—
26C22×A4C2×C412yes—
27C22×A4C2311yescharacteristic
28C22×Dic3—41no—
29C4×A4—22no—
30SL(2,ℤ/4)—22no—
31SL(2,ℤ/4)C2×C412yes—
32SL(2,ℤ/4)C2×C412yes—
33C3×C22⋊C4—41no—
34#217—41no—
35Dic3⋊C4—42no—
36C4×Dic3—42no—
37C25C2×D311yescharacteristic
38C23×C4—31no—
39C23×C4—31no—
40C23×C4C2×D312yes—
41#117—61no—
42#117—62no—
43#117—61no—
44C2×C4⋊C4—32no—
45C2×C4⋊C4—32no—
46C2×C22⋊C4—34no—
47C2×C22⋊C4—31no—
48C2×C22⋊C4—31no—
49C2×C22⋊C4—34no—
50C2×C22⋊C4—31no—
51C2×C22⋊C4—31no—
52C2×C22⋊C4—31no—
53C2×C22⋊C4—31no—
54C2×C22⋊C4—32no—
55C2×C22⋊C4—31no—
56C2×C22⋊C4—31no—
57C2×C22⋊C4—32no—
58C2×C22⋊C4—32no—
59C2×C42—32no—
60C2×C42—32no—
61#95—32no—
62#95—32no—
63#95—32no—
64#95—32no—
65C23×C3—41no—
66C2×A4—22no—
67C2×A4C22×C411yescharacteristic
68C2×A4#4112yes—
69C2×C3×C4—42no—
70C2×Dic3—41no—
71C2×Dic3—41no—
72C2×Dic3—44no—
73C2×Dic3—42no—
74C24—22no—
75C24—31no—
76C24—31no—
77C24—32no—
78C24—62no—
79C24C4×D312yes—
80C24C22×D311yescharacteristic
81C22×C4—22no—
82C22×C4—31no—
83C22×C4—31no—
84C22×C4—31no—
85C22×C4—31no—
86C22×C4—34no—
87C22×C4—34no—
88C22×C4—32no—
89C22×C4—32no—
90C22×C4—32no—
91C22×C4—32no—
92C22×C4—61no—
93C22×C4—62no—
94C22×C4—61no—
95C22×C4—62no—
96C22×C4—122no—
97C4⋊C4—62no—
98C4⋊C4—62no—
99C4⋊C4—122no—
100C22⋊C4—32no—
101C22⋊C4—32no—
102C22⋊C4—34no—
103C22⋊C4—34no—
104C22⋊C4—32no—
105C22⋊C4—32no—
106C22⋊C4—31no—
107C22⋊C4—32no—
108C22⋊C4—61no—
109C22⋊C4—62no—
110C22⋊C4—62no—
111C22⋊C4—61no—
112C22⋊C4—62no—
113C22⋊C4—61no—
114C22⋊C4—122no—
115C22⋊C4S411yescharacteristic
116C42—62no—
117C42—62no—
118C42—122no—
119C22×C3—42no—
120C22×C3—41no—
121C22×C3—82no—
122A4#11711yescharacteristic, complemented
123C3×C4—82no—
124Dic3—42no—
125Dic3—42no—
126Dic3—82no—
127C23—22no—
128C23—31no—
129C23—31no—
130C23—32no—
131C23—32no—
132C23—32no—
133C23—32no—
134C23—31no—
135C23—62no—
136C23—62no—
137C23—61no—
138C23—62no—
139C23—62no—
140C23—64no—
141C23—62no—
142C23—122no—
143C23C2×C4×D311yescharacteristic
144C23#23712yes—
145C23C2×S411yescharacteristic
146C2×C4—32no—
147C2×C4—32no—
148C2×C4—32no—
149C2×C4—32no—
150C2×C4—32no—
151C2×C4—62no—
152C2×C4—62no—
153C2×C4—62no—
154C2×C4—62no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—64no—
158C2×C4—64no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—62no—
162C2×C4—62no—
163C2×C4—62no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4—62no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—124no—
170C2×C4C2×S412yes—
171C2×C3—41no—
172C2×C3—42no—
173C2×C3—82no—
174C22—22no—
175C22—31no—
176C22—32no—
177C22—31no—
178C22—31no—
179C22—32no—
180C22—32no—
181C22—32no—
182C22—62no—
183C22—62no—
184C22—64no—
185C22—61no—
186C22—62no—
187C22—62no—
188C22—62no—
189C22—122no—
190C22#87711yescharacteristic, complemented
191C22C4×S412yes—
192C22C22×S411yescharacteristic
193C4—22no—
194C4—62no—
195C4—62no—
196C4—62no—
197C4—62no—
198C4—62no—
199C4—122no—
200C4—122no—
201C3—41noSylow
202C2—22no—
203C2—31no—
204C2—32no—
205C2—31no—
206C2—62no—
207C2C2×C4×S411yescharacteristic
208C2#591412yes—
subgroupG/H
center, FrattiniC22C22×S4
commutatorC2×A4C22×C4
FittingC22×C22⋊C4D3
socleC24C22×D3
prime radicalC23×A4C22
cosocleC22×A4C23
hypercenterC22⋊C4S4
hypocenterA4#117
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. SL(2,ℤ/4)⋊C4C2semidirect
5. C4×SL(2,ℤ/4)C2semidirect
9. C2×SL(2,ℤ/4)C4semidirect
122. A4#117semidirect
190. C22#877semidirect

Maximal subgroups

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

subgroupindex
C22×SL(2,ℤ/4)2
A4×C22⋊C42
#54322
SL(2,ℤ/4)⋊C42
C4×SL(2,ℤ/4)2
#23423
#8774