Finite Group Databasebeta

← #7285

Subgroups — #7285

Order 256.

162 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 41 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#1870C211yesmaximal, characteristic
2#1460C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4C22×M16C411yescharacteristic
5C2×C4×D4C2211yescharacteristic
6#431—21no—
7C2×C22⋊C8—21no—
8C2×C22⋊C8—21no—
9C2×C22⋊C8C411yescharacteristic
10#336—21no—
11#324C412yes—
12#323—21no—
13#323—21no—
14#323C412yes—
15C22×D4Q811yescharacteristic
16C23×C4C2×C411yescharacteristic
17C23×C4D411yescharacteristic
18C2×M16—42no—
19C2×M16—41no—
20C22×C8—21no—
21C22×C8—21no—
22C22×C8—41no—
23C4×D4—41no—
24C4×D4—41no—
25C2×C4⋊C4C2×C411yescharacteristic
26C2×C22⋊C4C2×C411yescharacteristic
27C2×C22⋊C4D411yescharacteristic
28C2×C42D411yescharacteristic
29C4⋊C8—41no—
30C22⋊C8—22no—
31C22⋊C8—22no—
32C22⋊C8—42no—
33C22⋊C8—41no—
34C22⋊C8—41no—
35C4×C8—41no—
36C24C4⋊C411yescharacteristic
37C24C4⋊C411yescharacteristic
38C2×D4—22no—
39C2×D4—22no—
40C2×D4—41no—
41C22×C4—21no—
42C22×C4—21no—
43C22×C4—21no—
44C22×C4—21no—
45C22×C4—41no—
46C22×C4—41no—
47C22×C4—41no—
48C22×C4—41no—
49C22×C4C4211yescharacteristic
50C22×C4C22⋊C411yescharacteristic
51C22×C4C22⋊C411yescharacteristic
52C22×C4C22⋊C411yescharacteristic
53C22×C4C4⋊C411yescharacteristic
54M16—42no—
55M16—42no—
56C2×C8—22no—
57C2×C8—41no—
58C2×C8—41no—
59C2×C8—42no—
60C2×C8—42no—
61C2×C8—41no—
62C2×C8—41no—
63C2×C8—41no—
64C2×C8—81no—
65C2×C8—81no—
66C4⋊C4—41no—
67C22⋊C4—41no—
68C22⋊C4—41no—
69C42—21no—
70C42—21no—
71C23—21no—
72C23—21no—
73C23—21no—
74C23—21no—
75C23—21no—
76C23—41no—
77C23—41no—
78C23—41no—
79C23—41no—
80C23#9511yescharacteristic
81C23#10812yes—
82D4—44no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—42no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—81no—
99C2×C4#9911yescharacteristic
100C2×C4#9911yescharacteristic
101C2×C4#10812yes—
102C8—42no—
103C8—81no—
104C8—82no—
105C8—82no—
106C22—21no—
107C22—21no—
108C22—22no—
109C22—42no—
110C22—41no—
111C22—41no—
112C22—41no—
113C22—41no—
114C22—41no—
115C22—81no—
116C22—81no—
117C22—81no—
118C22#34111yescharacteristic
119C22#34111yescharacteristic
120C22#34211yescharacteristic
121C4—22no—
122C4—21no—
123C4—41no—
124C4—41no—
125C4—41no—
126C4AΣL(1,𝔽16)11yescharacteristic
127C4#35211yescharacteristic
128C2—21no—
129C2—21no—
130C2—42no—
131C2—81no—
132C2#129811yescharacteristic
133C2#139011yescharacteristic
134C2#139511yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC22×C4C42
Frattini, cosocleC2×C4×D4C22
socleC22#342
prime radicalC22×D4Q8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#18702
#14602
#14592