Finite Group Databasebeta

← #8764

Subgroups — #8764

Order 256.

249 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 71 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#2349C211yesmaximal, characteristic
2#1800C211yesmaximal, characteristic
3#1460C211yesmaximal, characteristic
4#1303C212yesmaximal, complemented
5#1286C212yesmaximal, complemented
6C2×C4×D4—21no—
7C2×C4×D4C2211yescharacteristic
8#431C2212yes—
9#423C2212yes—
10C2×#100—21no—
11#389—21no—
12#385—22no—
13#382C2211yescharacteristic
14#380C412yes—
15C4×C4⋊C4C2211yescharacteristic
16C2×#95C412yes—
17#343—21no—
18#323—22no—
19C22×D4D411yescharacteristic
20C23×C4—22no—
21C23×C4C2×C412yes—
22C2×M16—22no—
23C4×D4—22no—
24C4×D4—22no—
25C4×D4—42no—
26C4×D4—42no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4D411yescharacteristic
31C2×C22⋊C4—22no—
32C2×C22⋊C4D412yes—
33C2×C42—21no—
34C2×C42C2311yescharacteristic
35C4⋊C8—22no—
36C4⋊C8—22no—
37C4⋊C8—22no—
38#100—42no—
39C22⋊C8—22no—
40C4×C8—22no—
41#95—22no—
42#95—21no—
43#95C2×C411yescharacteristic
44#95C2×C412yes—
45#95C2×C411yescharacteristic
46C24C22⋊C412yes—
47C2×D4—22no—
48C2×D4—22no—
49C2×D4D812yes—
50C2×D4SD1612yes—
51C22×C4—21no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—21no—
56C22×C4—44no—
57C22×C4—44no—
58C22×C4—82no—
59C22×C4—82no—
60C22×C4C22⋊C412yes—
61C22×C4C22×C411yescharacteristic
62C22×C4C2×D411yescharacteristic
63C22×C4C2×D411yescharacteristic
64M16—42no—
65M16—42no—
66C2×C8—42no—
67C2×C8—42no—
68C4⋊C4—22no—
69C4⋊C4—42no—
70C4⋊C4—41no—
71C4⋊C4—41no—
72C4⋊C4—82no—
73C22⋊C4—24no—
74C22⋊C4—44no—
75C42—41no—
76C42—41no—
77C42#4112yes—
78C42#4112yes—
79C23—22no—
80C23—22no—
81C23—42no—
82C23—42no—
83C23#10214yes—
84C23C2×C22⋊C411yescharacteristic
85D4—24no—
86D4—24no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4—41no—
90C2×C4—41no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—42no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—41no—
99C2×C4—42no—
100C2×C4—82no—
101C2×C4—82no—
102C2×C4—88no—
103C2×C4#11711yescharacteristic
104C2×C4#11711yescharacteristic
105C2×C4#12312yes—
106C2×C4#12312yes—
107C2×C4C2×D811yescharacteristic
108C2×C4C2×SD1611yescharacteristic
109C2×C4#13712yes—
110C8—82no—
111C8—82no—
112C22—21no—
113C22—21no—
114C22—24no—
115C22—44no—
116C22—44no—
117C22—42no—
118C22—42no—
119C22#38111yescharacteristic
120C22C2×#10211yescharacteristic
121C22#41911yescharacteristic
122C4—22no—
123C4—21no—
124C4—21no—
125C4—42no—
126C4—42no—
127C4—81no—
128C4—82no—
129C4#48012yes—
130C4#48312yes—
131C2—21no—
132C2—21no—
133C2—44no—
134C2#182311yescharacteristic
135C2#182711yescharacteristic
136C2#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
#23492
#18002
#14602
#13032
#12862