Finite Group Databasebeta

← #7253

Subgroups — #7253

Order 256.

164 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 51 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#1798C211yesmaximal, characteristic
2#1742C211yesmaximal, characteristic
3#1458C211yesmaximal, characteristic
4C22×M16—21no—
5C2×#117C2211yescharacteristic
6#431—21no—
7C2×C4⋊C8C411yescharacteristic
8C2×C4⋊C8C411yescharacteristic
9C2×C22⋊C8—21no—
10#343—21no—
11#343C412yes—
12#336—21no—
13#324—21no—
14#323C412yes—
15C23×C4D411yescharacteristic
16C2×M16—22no—
17C2×M16—22no—
18C2×M16—22no—
19C2×M16—41no—
20C2×M16—41no—
21C2×M16—41no—
22C22×C8—21no—
23C22×C8—21no—
24C22×C8—41no—
25#117—41no—
26#117—41no—
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C4⋊C4D411yescharacteristic
29C2×C22⋊C4C2×C411yescharacteristic
30C2×C22⋊C4Q811yescharacteristic
31C2×C42C2×C411yescharacteristic
32C2×C42D411yescharacteristic
33C4⋊C8—41no—
34C4⋊C8—41no—
35C4⋊C8—41no—
36C22⋊C8—22no—
37C22⋊C8—41no—
38C22⋊C8—41no—
39C4×C8—41no—
40C24C4⋊C411yescharacteristic
41C22×C4—21no—
42C22×C4—21no—
43C22×C4—21no—
44C22×C4—21no—
45C22×C4C4211yescharacteristic
46C22×C4C22⋊C411yescharacteristic
47C22×C4C22⋊C411yescharacteristic
48C22×C4C22⋊C411yescharacteristic
49C22×C4C4⋊C411yescharacteristic
50C22×C4C4⋊C411yescharacteristic
51C22×C4D811yescharacteristic
52C22×C4SD1611yescharacteristic
53C22×C4SD1611yescharacteristic
54C22×C4Q1611yescharacteristic
55M16—24no—
56M16—42no—
57M16—42no—
58M16—42no—
59C2×C8—42no—
60C2×C8—42no—
61C2×C8—42no—
62C2×C8—42no—
63C2×C8—41no—
64C2×C8—41no—
65C2×C8—41no—
66C2×C8—41no—
67C2×C8—41no—
68C2×C8—41no—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C22⋊C4—41no—
72C22⋊C4—41no—
73C42—21no—
74C42—21no—
75C42—21no—
76C42—21no—
77C23—21no—
78C23—21no—
79C23—41no—
80C23—41no—
81C23#9511yescharacteristic
82C23#10611yescharacteristic
83C23#10711yescharacteristic
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4#9911yescharacteristic
103C2×C4#9911yescharacteristic
104C2×C4#10211yescharacteristic
105C2×C4#10211yescharacteristic
106C2×C4#10311yescharacteristic
107C2×C4#10311yescharacteristic
108C2×C4#10411yescharacteristic
109C2×C4#10411yescharacteristic
110C8—44no—
111C8—82no—
112C8—81no—
113C8—81no—
114C22—21no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—41no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—81no—
123C22#33911yescharacteristic
124C22#34011yescharacteristic
125C22#34211yescharacteristic
126C4—21no—
127C4—21no—
128C4—41no—
129C4—41no—
130C4—41no—
131C4—41no—
132C4#32711yescharacteristic
133C4#32711yescharacteristic
134C4#32811yescharacteristic
135C4#32811yescharacteristic
136C2—21no—
137C2—21no—
138C2—41no—
139C2—41no—
140C2#129411yescharacteristic
141C2#139211yescharacteristic
142C2#139511yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC22×C4C42
Frattini, cosocleC2×#117C22
socleC22#342
prime radicalC24C4⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#17982
#17422
#14582