Finite Group Databasebeta

← #85303

Subgroups — #85303

Order 384.

311 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C2×A4⋊Q8C212yesmaximal
2A4×C4⋊C4C211yesmaximal, characteristic
3C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4SL(2,ℤ/4)⋊C4C212yesmaximal
5C4×SL(2,ℤ/4)C211yesmaximal, characteristic
6#2440—31nomaximal, Sylow
7C2×C4×A4C2212yes—
8C2×C4×A4C2211yescharacteristic
9C2×SL(2,ℤ/4)C2212yes—
10C2×SL(2,ℤ/4)C2211yescharacteristic
11C2×SL(2,ℤ/4)C2211yescharacteristic
12A4⋊Q8—24no—
13#888—41nomaximal
14C2×C4⋊Q8—31no—
15C2×C22⋊Q8—32no—
16C22×C4⋊C4D311yescharacteristic
17#398—32no—
18#395—32no—
19#389—31no—
20#389—31no—
21#385—32no—
22#384—31no—
23#380—31no—
24C4×C22⋊C4—31no—
25C22×A4C2311yescharacteristic
26C2×Dic6—42no—
27C4×A4—22no—
28C4×A4Q812yescomplemented
29SL(2,ℤ/4)—22no—
30SL(2,ℤ/4)D412yes—
31SL(2,ℤ/4)Q812yescomplemented
32C3×C4⋊C4—41no—
33C4⋊Dic3—41no—
34Dic3⋊C4—42no—
35C4×Dic3—41no—
36C22×Q8—32no—
37C23×C4C2×D312yes—
38C23×C4C2×D311yescharacteristic
39C4⋊Q8—62no—
40C4⋊Q8—61no—
41C4⋊Q8—61no—
42C22⋊Q8—64no—
43C22⋊Q8—64no—
44C2×C4⋊C4—31no—
45C2×C4⋊C4—31no—
46C2×C4⋊C4—32no—
47C2×C4⋊C4—32no—
48C2×C4⋊C4—32no—
49C2×C4⋊C4—32no—
50C2×C4⋊C4—31no—
51C2×C4⋊C4—32no—
52C2×C4⋊C4—31no—
53C2×C4⋊C4—32no—
54C2×C22⋊C4—31no—
55C2×C22⋊C4—31no—
56C2×C22⋊C4—32no—
57C2×C42—31no—
58C2×C42—31no—
59#95—31no—
60#95—31no—
61#95—32no—
62#95—32no—
63#95—32no—
64C2×A4C2×D411yescharacteristic
65C2×A4C2×Q811yescharacteristic
66C2×A4C2×Q811yescharacteristic
67C2×C3×C4—41no—
68C2×C3×C4—42no—
69C2×Dic3—41no—
70C2×Dic3—41no—
71C2×Dic3—42no—
72Dic6—84no—
73C24C22×D311yescharacteristic
74C2×Q8—64no—
75C2×Q8—62no—
76C2×Q8—62no—
77C2×Q8—122no—
78C22×C4—22no—
79C22×C4—31no—
80C22×C4—31no—
81C22×C4—32no—
82C22×C4—31no—
83C22×C4—31no—
84C22×C4—31no—
85C22×C4—31no—
86C22×C4—32no—
87C22×C4—32no—
88C22×C4—32no—
89C22×C4—32no—
90C22×C4—62no—
91C22×C4Dic612yescomplemented
92C4⋊C4—32no—
93C4⋊C4—31no—
94C4⋊C4—61no—
95C4⋊C4—62no—
96C4⋊C4—64no—
97C4⋊C4—64no—
98C4⋊C4—64no—
99C4⋊C4—61no—
100C4⋊C4—62no—
101C4⋊C4—62no—
102C4⋊C4—61no—
103C4⋊C4—122no—
104C4⋊C4S411yescharacteristic
105C22⋊C4—32no—
106C22⋊C4—32no—
107C22⋊C4—32no—
108C22⋊C4—32no—
109C22⋊C4—62no—
110C22⋊C4—62no—
111C42—62no—
112C42—61no—
113C42—61no—
114C22×C3—41no—
115A4C4⋊Q811yescharacteristic, complemented
116C3×C4—42no—
117C3×C4—82no—
118Dic3—42no—
119Dic3—42no—
120Dic3—82no—
121C23—31no—
122C23—31no—
123C23—31no—
124C23—31no—
125C23C2×Dic611yescharacteristic
126C23D3×D411yescharacteristic
127C23D3×Q811yescharacteristic
128Q8—124no—
129Q8—124no—
130C2×C4—31no—
131C2×C4—32no—
132C2×C4—32no—
133C2×C4—32no—
134C2×C4—32no—
135C2×C4—32no—
136C2×C4—32no—
137C2×C4—32no—
138C2×C4—32no—
139C2×C4—62no—
140C2×C4—62no—
141C2×C4—61no—
142C2×C4—61no—
143C2×C4—61no—
144C2×C4—61no—
145C2×C4—61no—
146C2×C4—61no—
147C2×C4—61no—
148C2×C4—61no—
149C2×C4—61no—
150C2×C4—62no—
151C2×C4—62no—
152C2×C4—62no—
153C2×C4—62no—
154C2×C4—62no—
155C2×C4—62no—
156C2×C4—62no—
157C2×C4—62no—
158C2×C4C2×S412yes—
159C2×C4C2×S411yescharacteristic
160C2×C3—41no—
161C2×C3—41no—
162C2×C3—41no—
163C22—31no—
164C22—31no—
165C22—31no—
166C22—31no—
167C22—31no—
168C22—31no—
169C22—31no—
170C22—31no—
171C22—31no—
172C22—61no—
173C22#88811yescharacteristic, complemented
174C22C22×S411yescharacteristic
175C4—22no—
176C4—32no—
177C4—62no—
178C4—62no—
179C4—62no—
180C4—62no—
181C4—62no—
182C4—122no—
183C4—122no—
184C4A4⋊Q812yescomplemented
185C3—41noSylow
186C2—31no—
187C2—31no—
188C2—31no—
189C2—31no—
190C2C2×A4⋊Q811yescharacteristic
191C2D4×S411yescharacteristic
192C2Q8×S411yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, prime radical, cosocleC22×A4C23
FittingC22×C4⋊C4D3
socleC24C22×D3
hypercenterC4⋊C4S4
hypocenterA4C4⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
28. C4×A4Q8semidirect
31. SL(2,ℤ/4)Q8semidirect
91. C22×C4Dic6semidirect
115. A4C4⋊Q8semidirect
173. C22#888semidirect
184. C4A4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×A4⋊Q82
A4×C4⋊C42
C4⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊C42
C4×SL(2,ℤ/4)2
#24403
#8884