Finite Group Databasebeta

← #11502

Subgroups — #11502

Order 256.

357 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 55 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#2308C211yesmaximal, characteristic
2#1462C212yesmaximal, complemented
3#1317C212yesmaximal, complemented
4#1316C212yesmaximal, complemented
5C23×D4C411yescharacteristic
6C22×C22⋊C4C411yescharacteristic
7C22×C22⋊C4C2211yescharacteristic
8#407—22no—
9#386—22no—
10#386—22no—
11#381C411yescharacteristic
12#381C2211yescharacteristic
13#379C411yescharacteristic
14#379C2211yescharacteristic
15#323—22no—
16#323C2214yes—
17C25C2×C411yescharacteristic
18C25C2×C411yescharacteristic
19C22×D4—21no—
20C22×D4—21no—
21C22×D4—22no—
22C22×D4—22no—
23C22×D4—41no—
24C22×D4—41no—
25C22×D4—42no—
26C23×C4—21no—
27C23×C4C2×C411yescharacteristic
28C23×C4D412yes—
29C2×M16—42no—
30C2×C22⋊C4—21no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—41no—
34C2×C22⋊C4—41no—
35C2×C22⋊C4—42no—
36C2×C22⋊C4C2×C411yescharacteristic
37C2×C22⋊C4C2×C411yescharacteristic
38C2×C22⋊C4C2×C411yescharacteristic
39C2×C22⋊C4D412yes—
40C2×C22⋊C4C2311yescharacteristic
41C22⋊C8—24no—
42C22⋊C8—42no—
43#95—22no—
44#95—22no—
45C24—21no—
46C24—21no—
47C24—22no—
48C24—21no—
49C24—21no—
50C24—21no—
51C24—21no—
52C24—41no—
53C24—41no—
54C24—82no—
55C24—82no—
56C24C22⋊C411yescharacteristic
57C24C22⋊C411yescharacteristic
58C24M1612yescomplemented
59C24C22×C411yescharacteristic
60C2×D4—24no—
61C2×D4—24no—
62C2×D4—44no—
63C2×D4—44no—
64C2×D4—42no—
65C2×D4—42no—
66C2×D4—44no—
67C2×D4—44no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—42no—
71C22×C4—22no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—42no—
80C22×C4—82no—
81C22×C4—82no—
82C22×C4C22⋊C412yes—
83C22×C4M1612yescomplemented
84C22×C4C2×D412yes—
85M16—84no—
86C2×C8—44no—
87C22⋊C4—41no—
88C22⋊C4—42no—
89C22⋊C4—41no—
90C22⋊C4—42no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—82no—
94C22⋊C4—82no—
95C23—22no—
96C23—22no—
97C23—22no—
98C23—22no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23—21no—
105C23—22no—
106C23—42no—
107C23—42no—
108C23—42no—
109C23—42no—
110C23—42no—
111C23—42no—
112C23—42no—
113C23—41no—
114C23—41no—
115C23—82no—
116C23—82no—
117C23—82no—
118C23—84no—
119C23—82no—
120C23—82no—
121C23—82no—
122C23—84no—
123C23—84no—
124C23#9911yescharacteristic
125C23#9911yescharacteristic
126C23#10012yes—
127C23C2×C22⋊C411yescharacteristic
128C23C2×M1611yescharacteristic
129C23C2×M1611yescharacteristic
130D4—44no—
131D4—44no—
132D4—48no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—44no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—82no—
148C2×C4—84no—
149C8—84no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—22no—
158C22—42no—
159C22—41no—
160C22—41no—
161C22—42no—
162C22—42no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—42no—
170C22—82no—
171C22—82no—
172C22—82no—
173C22—82no—
174C22—82no—
175C22—84no—
176C22—84no—
177C22—82no—
178C22#40711yescharacteristic
179C22C2×#9911yescharacteristic
180C22C2×#10011yescharacteristic
181C4—42no—
182C4—42no—
183C4—42no—
184C4—81no—
185C4—81no—
186C2—21no—
187C2—21no—
188C2—41no—
189C2—41no—
190C2—42no—
191C2—41no—
192C2—41no—
193C2—82no—
194C2#146211yescharacteristic
195C2#212112yes—
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC23×D4C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1462C2semidirect
3. #1317C2semidirect
4. #1316C2semidirect
58. C24M16semidirect
83. C22×C4M16semidirect

Maximal subgroups

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

subgroupindex
#23082
#14622
#13172
#13162