Finite Group Databasebeta

← #8770

Subgroups — #8770

Order 256.

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

normal subgroups only

This group has 213 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#2357C211yesmaximal, characteristic
2#1800C211yesmaximal, characteristic
3#1460C211yesmaximal, characteristic
4#1303C212yesmaximal, complemented
5#1286C212yesmaximal, complemented
6C2×C4×D4C2211yescharacteristic
7#431C2212yes—
8#423C2212yes—
9C2×#100—21no—
10#390—21no—
11#388—22no—
12#382C2211yescharacteristic
13#381C412yes—
14C4×C4⋊C4—21no—
15C4×C4⋊C4C2211yescharacteristic
16C4×C22⋊C4C412yes—
17#343—21no—
18#323—22no—
19C22×D4D411yescharacteristic
20C23×C4C2×C412yes—
21C2×M16—22no—
22C4×D4—22no—
23C4×D4—22no—
24C2×C4⋊C4—21no—
25C2×C4⋊C4—21no—
26C2×C4⋊C4D411yescharacteristic
27C2×C22⋊C4—22no—
28C2×C22⋊C4—22no—
29C2×C22⋊C4D412yes—
30C2×C42—21no—
31C2×C42—22no—
32C2×C42C2311yescharacteristic
33C4⋊C8—22no—
34C4⋊C8—22no—
35C4⋊C8—22no—
36#100—42no—
37C22⋊C8—22no—
38C4×C8—22no—
39#95—22no—
40#95—21no—
41#95C2×C411yescharacteristic
42#95C2×C412yes—
43#95C2×C411yescharacteristic
44C24C22⋊C412yes—
45C2×D4—22no—
46C2×D4—22no—
47C2×D4—21no—
48C2×D4—21no—
49C22×C4—21no—
50C22×C4—22no—
51C22×C4—22no—
52C22×C4—22no—
53C22×C4—21no—
54C22×C4—82no—
55C22×C4C22⋊C412yes—
56C22×C4C22×C411yescharacteristic
57C22×C4C2×D411yescharacteristic
58C22×C4C2×D411yescharacteristic
59M16—42no—
60M16—42no—
61C2×C8—42no—
62C2×C8—42no—
63C4⋊C4—22no—
64C4⋊C4—42no—
65C4⋊C4—42no—
66C22⋊C4—24no—
67C22⋊C4—44no—
68C22⋊C4—82no—
69C42—41no—
70C42—41no—
71C42—82no—
72C42#4112yes—
73C42#4112yes—
74C23—22no—
75C23—22no—
76C23—22no—
77C23—42no—
78C23—42no—
79C23C2×C22⋊C411yescharacteristic
80D4—42no—
81D4—42no—
82C2×C4—22no—
83C2×C4—22no—
84C2×C4—41no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4—42no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—42no—
94C2×C4—41no—
95C2×C4—82no—
96C2×C4—82no—
97C2×C4#11711yescharacteristic
98C2×C4#11711yescharacteristic
99C2×C4#12312yes—
100C2×C4#12312yes—
101C2×C4#13512yes—
102C2×C4AGL(1,ℤ/8)11yescharacteristic
103C2×C4#13711yescharacteristic
104C8—82no—
105C8—82no—
106C22—21no—
107C22—21no—
108C22—42no—
109C22—42no—
110C22—42no—
111C22—82no—
112C22—82no—
113C22#38111yescharacteristic
114C22#41611yescharacteristic
115C22#41711yescharacteristic
116C4—22no—
117C4—21no—
118C4—21no—
119C4—42no—
120C4—42no—
121C4—81no—
122C4—82no—
123C4#48212yes—
124C4#48512yes—
125C2—21no—
126C2—21no—
127C2—82no—
128C2#182511yescharacteristic
129C2#182711yescharacteristic
130C2#182911yescharacteristic
subgroupG/H
center, socleC22#381
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1303C2semidirect
5. #1286C2semidirect

Maximal subgroups

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

subgroupindex
#23572
#18002
#14602
#13032
#12862