Finite Group Databasebeta

← #9504

Subgroups — #9504

Order 256.

439 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#2975C211yesmaximal, characteristic, complemented
2#2408C211yesmaximal, characteristic
3#1783C212yesmaximal, complemented
4#1465C211yesmaximal, characteristic, complemented
5#1305C212yesmaximal, complemented
6C23×D4C2211yescharacteristic
7C2×#121—21no—
8C2×#120C412yes—
9C2×C4×D4C2211yescharacteristic
10#436—24no—
11#431—21no—
12C2×#100—24no—
13#407—21no—
14#407C2212yes—
15#407C2212yes—
16#394—21no—
17#392—21no—
18#390—21no—
19#380C2211yescharacteristic
20#379C412yes—
21#324—21no—
22#323—21no—
23C25C2×C412yes—
24C22×D4—22no—
25C22×D4—21no—
26C22×D4—22no—
27C22×D4—21no—
28C22×D4—22no—
29C22×D4—44no—
30C22×D4C2×C411yescharacteristic
31C23×C4D411yescharacteristic
32C23×C4C2311yescharacteristic
33C2×M16—22no—
34C2×M16—22no—
35C2×M16D414yescomplemented
36#121—44no—
37#120—44no—
38C4×D4—22no—
39C4×D4—22no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4D411yescharacteristic
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4C2×C411yescharacteristic
46C2×C22⋊C4C2×C412yes—
47C2×C22⋊C4D411yescharacteristic
48C2×C42D411yescharacteristic
49C4⋊C8—22no—
50#100—48no—
51C22⋊C8—22no—
52C22⋊C8—22no—
53C22⋊C8—22no—
54C22⋊C8—22no—
55C22⋊C8—22no—
56C4×C8—22no—
57#95—21no—
58C24—22no—
59C24—24no—
60C24—44no—
61C24—42no—
62C24—42no—
63C24—82no—
64C24C22⋊C411yescharacteristic
65C24C22×C411yescharacteristic
66C2×D4—28no—
67C2×D4—24no—
68C2×D4—22no—
69C2×D4—22no—
70C2×D4—41no—
71C2×D4—42no—
72C2×D4—44no—
73C2×D4—44no—
74C2×D4—48no—
75C2×D4—44no—
76C2×D4—44no—
77C2×D4—41no—
78C2×D4—42no—
79C2×D4—41no—
80C2×D4—82no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22⋊C411yescharacteristic
88C22×C4C22⋊C411yescharacteristic
89C22×C4C2×D412yes—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4#4112yes—
93M16—44no—
94M16—44no—
95M16—82no—
96C2×C8—24no—
97C2×C8—42no—
98C4⋊C4—22no—
99C4⋊C4—42no—
100C22⋊C4—22no—
101C22⋊C4—22no—
102C22⋊C4—44no—
103C22⋊C4—44no—
104C22⋊C4—42no—
105C22⋊C4—82no—
106C42—21no—
107C42—21no—
108C23—21no—
109C23—21no—
110C23—24no—
111C23—28no—
112C23—21no—
113C23—21no—
114C23—21no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—42no—
120C23—42no—
121C23—44no—
122C23—42no—
123C23—42no—
124C23—42no—
125C23—42no—
126C23—84no—
127C23—84no—
128C23—88no—
129C23—88no—
130C23C2×C22⋊C411yescharacteristic
131C23C4×D412yes—
132D4—44no—
133D4—48no—
134D4—44no—
135D4—44no—
136D4—88no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—81no—
152C2×C4#10012yescomplemented
153C2×C4#12011yescharacteristic
154C2×C4#12112yes—
155C2×C4#12311yescharacteristic
156C8—44no—
157C8—82no—
158C22—21no—
159C22—21no—
160C22—22no—
161C22—22no—
162C22—22no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—41no—
171C22—41no—
172C22—81no—
173C22—82no—
174C22—82no—
175C22—88no—
176C22—88no—
177C22—82no—
178C22—82no—
179C22#38611yescharacteristic
180C22C2×#10011yescharacteristic
181C22#41311yescharacteristic
182C4—22no—
183C4—21no—
184C4—22no—
185C4—41no—
186C4—41no—
187C4—82no—
188C4B(4,𝔽2)12yes—
189C2—21no—
190C2—21no—
191C2—42no—
192C2—81no—
193C2—84no—
194C2#188911yescharacteristic
195C2#190011yescharacteristic
196C2#191411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
prime radical#2408C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2975C2semidirect
3. #1783C2semidirect
4. #1465C2semidirect
5. #1305C2semidirect
35. C2×M16D4semidirect
152. C2×C4#100semidirect

Maximal subgroups

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

subgroupindex
#29752
#24082
#17832
#14652
#13052