Finite Group Databasebeta

← #7323

Subgroups — #7323

Order 256.

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

normal subgroups only

This group has 128 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#1985C211yesmaximal, characteristic
2#1801C211yesmaximal, characteristic
3#1467C211yesmaximal, characteristic
4C2×C4×Q8C2211yescharacteristic
5#432—21no—
6#423—21no—
7C2×C4⋊C8C411yescharacteristic
8C2×#101C412yes—
9C2×#97C411yescharacteristic
10#343—21no—
11#336—21no—
12#336—21no—
13#324—21no—
14#324C412yes—
15C22×Q8C2×C411yescharacteristic
16C2×M16—22no—
17C22×C8—21no—
18C22×C8—21no—
19C4×Q8—41no—
20C4×Q8—41no—
21C2×C4⋊C4C2×C411yescharacteristic
22C2×C4⋊C4D411yescharacteristic
23C2×C4⋊C4Q811yescharacteristic
24C2×C42C2×C411yescharacteristic
25C2×C42D411yescharacteristic
26C2×C42D411yescharacteristic
27C4⋊C8—41no—
28C4⋊C8—41no—
29C4⋊C8—41no—
30C4⋊C8—41no—
31#101—42no—
32C22⋊C8—22no—
33#97—41no—
34#97—41no—
35C2×Q8—21no—
36C2×Q8—21no—
37C2×Q8—41no—
38C2×Q8—41no—
39C22×C4C4211yescharacteristic
40C22×C4C22⋊C411yescharacteristic
41C22×C4C22⋊C411yescharacteristic
42C22×C4C22⋊C411yescharacteristic
43C22×C4C4⋊C411yescharacteristic
44C22×C4C4⋊C411yescharacteristic
45C22×C4C4⋊C411yescharacteristic
46M16—42no—
47M16—42no—
48C2×C8—22no—
49C2×C8—41no—
50C2×C8—41no—
51C2×C8—42no—
52C2×C8—42no—
53C2×C8—42no—
54C2×C8—41no—
55C4⋊C4—41no—
56C4⋊C4—41no—
57C4⋊C4—41no—
58C42—21no—
59C42—21no—
60C42—21no—
61C42—21no—
62C42—21no—
63C42—21no—
64C23#9511yescharacteristic
65Q8—42no—
66Q8—42no—
67C2×C4—21no—
68C2×C4—21no—
69C2×C4—21no—
70C2×C4—21no—
71C2×C4—21no—
72C2×C4—21no—
73C2×C4—21no—
74C2×C4—21no—
75C2×C4—21no—
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—41no—
81C2×C4—41no—
82C2×C4—41no—
83C2×C4#9911yescharacteristic
84C2×C4#9911yescharacteristic
85C2×C4#10011yescharacteristic
86C2×C4#10111yescharacteristic
87C8—42no—
88C8—82no—
89C8—82no—
90C8—81no—
91C22—21no—
92C22—21no—
93C22#34211yescharacteristic
94C22#34311yescharacteristic
95C22#34411yescharacteristic
96C4—21no—
97C4—21no—
98C4—21no—
99C4—41no—
100C4—41no—
101C4—41no—
102C4—41no—
103C4—41no—
104C4#35511yescharacteristic
105C4#35611yescharacteristic
106C2—21no—
107C2—21no—
108C2#130511yescharacteristic
109C2#139411yescharacteristic
110C2#139511yescharacteristic
subgroupG/H
center, socleC22#342
commutatorC22×C4C42
Frattini, cosocleC2×C4×Q8C22
prime radicalC23#95
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#19852
#18012
#14672