Finite Group Databasebeta

← #85492

Subgroups — #85492

Order 384.

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

normal subgroups only

This group has 411 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
1C2×D4×A4C211yesmaximal, characteristic
2C2×A4⋊Q8C211yesmaximal, characteristic, complemented
3#5432C214yesmaximal, complemented
4C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5#2435—31nomaximal, Sylow
6C23×A4C2212yes—
7D4×A4—22no—
8C2×C4×A4C2211yescharacteristic
9C2×SL(2,ℤ/4)C2212yes—
10C2×SL(2,ℤ/4)C2212yes—
11A4⋊Q8—22no—
12#936—41nomaximal
13C23×D4D311yescharacteristic
14C2×#124—31no—
15C2×#124—31no—
16C2×C22⋊Q8—31no—
17#394—34no—
18#390—31no—
19#390—31no—
20#379—34no—
21C4×C22⋊C4—31no—
22C22×A4—22no—
23C22×A4—24no—
24C22×A4C2311yescharacteristic
25C2×C3×D4—41no—
26C2×Dic6—41no—
27C4×A4D412yes—
28SL(2,ℤ/4)—22no—
29SL(2,ℤ/4)—22no—
30#217—44no—
31C4×Dic3—41no—
32C25C2×D312yes—
33C22×Q8—31no—
34C22×D4—22no—
35C22×D4—31no—
36C22×D4—31no—
37C22×D4—32no—
38C22×D4—62no—
39C23×C4C2×D311yescharacteristic
40#124—64no—
41#124—62no—
42#124—61no—
43#124—61no—
44C22⋊Q8—62no—
45C22⋊Q8—62no—
46C2×C4⋊C4—31no—
47C2×C4⋊C4—32no—
48C2×C22⋊C4—34no—
49C2×C22⋊C4—34no—
50C2×C22⋊C4—32no—
51C2×C22⋊C4—32no—
52C2×C22⋊C4—34no—
53C2×C22⋊C4—34no—
54C2×C42—31no—
55C2×C42—31no—
56#95—32no—
57C23×C3—42no—
58C2×A4—42no—
59C2×A4C2×D411yescharacteristic
60C2×A4#4112yes—
61C3×D4—82no—
62C2×C3×C4—41no—
63C2×Dic3—42no—
64C2×Dic3—42no—
65Dic6—82no—
66C24—24no—
67C24—22no—
68C24—32no—
69C24—32no—
70C24—62no—
71C24—64no—
72C24C22×D311yescharacteristic
73C2×Q8—62no—
74C2×Q8—61no—
75C2×Q8—62no—
76C2×Q8—61no—
77C2×D4—34no—
78C2×D4—32no—
79C2×D4—31no—
80C2×D4—62no—
81C2×D4—62no—
82C2×D4—61no—
83C2×D4—62no—
84C2×D4—64no—
85C2×D4—64no—
86C2×D4S411yescharacteristic
87C22×C4—31no—
88C22×C4—31no—
89C22×C4—32no—
90C22×C4—32no—
91C22×C4—32no—
92C22×C4—32no—
93C22×C4—32no—
94C22×C4#6712yes—
95C4⋊C4—62no—
96C4⋊C4—64no—
97C22⋊C4—62no—
98C22⋊C4—62no—
99C22⋊C4—68no—
100C22⋊C4—64no—
101C22⋊C4—62no—
102C22⋊C4—62no—
103C22⋊C4—64no—
104C22⋊C4—124no—
105C22⋊C4—124no—
106C42—61no—
107C42—62no—
108C42—61no—
109C22×C3—41no—
110C22×C3—82no—
111C22×C3—84no—
112A4#12411yescharacteristic, complemented
113C3×C4—42no—
114Dic3—82no—
115Dic3—82no—
116C23—31no—
117C23—31no—
118C23—32no—
119C23—34no—
120C23—32no—
121C23—42no—
122C23—64no—
123C23—62no—
124C23—62no—
125C23—64no—
126C23—64no—
127C23—64no—
128C23—62no—
129C23—124no—
130C23—122no—
131C23—124no—
132C23#23712yes—
133C23C2×#6711yescharacteristic
134C23C2×S412yes—
135Q8—122no—
136Q8—122no—
137D4—22no—
138D4—64no—
139D4—64no—
140D4—62no—
141C2×C4—31no—
142C2×C4—32no—
143C2×C4—32no—
144C2×C4—32no—
145C2×C4—61no—
146C2×C4—62no—
147C2×C4—62no—
148C2×C4—62no—
149C2×C4—62no—
150C2×C4—62no—
151C2×C4—62no—
152C2×C4—62no—
153C2×C4—64no—
154C2×C4—62no—
155C2×C4—62no—
156C2×C4—62no—
157C2×C4C2×S411yescharacteristic
158C2×C3—42no—
159C2×C3—41no—
160C2×C3—162no—
161C22—24no—
162C22—22no—
163C22—32no—
164C22—31no—
165C22—32no—
166C22—32no—
167C22—31no—
168C22—31no—
169C22—61no—
170C22—68no—
171C22—64no—
172C22—62no—
173C22—122no—
174C22—122no—
175C22—124no—
176C22—122no—
177C22—122no—
178C22#93611yescharacteristic, complemented
179C22C22×S411yescharacteristic
180C4—32no—
181C4—122no—
182C4—122no—
183C4—122no—
184C4—122no—
185C4GL(2,ℤ/4)12yes—
186C3—41noSylow
187C2—31no—
188C2—32no—
189C2—31no—
190C2—42no—
191C2—122no—
192C2#591412yes—
193C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC23×D4D3
socleC24C22×D3
prime radicalC2×D4×A4C2
hypercenterC2×D4S4
hypocenterA4#124
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×A4⋊Q8C2semidirect
3. #5432C2semidirect
4. C4×SL(2,ℤ/4)C2semidirect
112. A4#124semidirect
178. C22#936semidirect

Maximal subgroups

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

subgroupindex
C2×D4×A42
C2×A4⋊Q82
#54322
C4×SL(2,ℤ/4)2
#24353
#9364