Finite Group Databasebeta

← #7321

Subgroups — #7321

Order 256.

164 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 39 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#1800C211yesmaximal, characteristic
3#1465C211yesmaximal, characteristic
4C22×M16C411yescharacteristic
5C2×C4×D4C2211yescharacteristic
6#423—21no—
7C2×#100C412yes—
8#407—21no—
9C2×C22⋊C8—21no—
10C2×C22⋊C8C411yescharacteristic
11#343—21no—
12#336—21no—
13#324—21no—
14#323C412yes—
15C22×D4C2×C411yescharacteristic
16C23×C4C2×C411yescharacteristic
17C23×C4D411yescharacteristic
18C2×M16—22no—
19C2×M16—42no—
20C2×M16—41no—
21C2×M16—41no—
22C22×C8—21no—
23C22×C8—21no—
24C4×D4—41no—
25C4×D4—41no—
26C2×C4⋊C4D411yescharacteristic
27C2×C22⋊C4C2×C411yescharacteristic
28C2×C22⋊C4Q811yescharacteristic
29C2×C42D411yescharacteristic
30C4⋊C8—41no—
31C4⋊C8—41no—
32#100—42no—
33C22⋊C8—22no—
34C22⋊C8—42no—
35C22⋊C8—41no—
36C22⋊C8—41no—
37C24C4211yescharacteristic
38C24C4⋊C411yescharacteristic
39C2×D4—21no—
40C2×D4—21no—
41C2×D4—41no—
42C2×D4—41no—
43C22×C4—21no—
44C22×C4—21no—
45C22×C4—21no—
46C22×C4—21no—
47C22×C4—41no—
48C22×C4—41no—
49C22×C4—41no—
50C22×C4—41no—
51C22×C4C22⋊C411yescharacteristic
52C22×C4C22⋊C411yescharacteristic
53C22×C4C22⋊C411yescharacteristic
54C22×C4C4⋊C411yescharacteristic
55C22×C4C4⋊C411yescharacteristic
56M16—42no—
57M16—42no—
58M16—42no—
59M16—42no—
60M16—81no—
61M16—81no—
62C2×C8—22no—
63C2×C8—41no—
64C2×C8—41no—
65C2×C8—42no—
66C2×C8—41no—
67C2×C8—42no—
68C2×C8—41no—
69C2×C8—41no—
70C4⋊C4—41no—
71C22⋊C4—41no—
72C22⋊C4—41no—
73C42—21no—
74C42—21no—
75C23—21no—
76C23—21no—
77C23—21no—
78C23—21no—
79C23—21no—
80C23—21no—
81C23—41no—
82C23—41no—
83C23—41no—
84C23—41no—
85C23#9511yescharacteristic
86D4—42no—
87D4—42no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—42no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—81no—
104C2×C4#9911yescharacteristic
105C2×C4#9911yescharacteristic
106C2×C4#10011yescharacteristic
107C2×C4#10111yescharacteristic
108C8—42no—
109C8—81no—
110C8—82no—
111C8—82no—
112C22—21no—
113C22—21no—
114C22—42no—
115C22—41no—
116C22—41no—
117C22—41no—
118C22—41no—
119C22—41no—
120C22—41no—
121C22—81no—
122C22—81no—
123C22—81no—
124C22#34211yescharacteristic
125C22#34311yescharacteristic
126C22#34411yescharacteristic
127C4—21no—
128C4—21no—
129C4—21no—
130C4—41no—
131C4—41no—
132C4—41no—
133C4AΣL(1,𝔽16)11yescharacteristic
134C4#35211yescharacteristic
135C2—21no—
136C2—21no—
137C2—81no—
138C2—81no—
139C2#130511yescharacteristic
140C2#139011yescharacteristic
141C2#139211yescharacteristic
subgroupG/H
center, socleC22#342
commutatorC24C42
Frattini, cosocleC2×C4×D4C22
prime radicalC22×D4C2×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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
#18002
#14652