Finite Group Databasebeta

← #10389

Subgroups — #10389

Order 256.

203 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 45 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#2608C211yesmaximal, characteristic
2#1826C212yesmaximal
3#1820C211yesmaximal, characteristic
4#1306C211yesmaximal, characteristic
5#1284C212yesmaximal
6C2×#117C2211yescharacteristic
7#407C2212yes—
8#396—22no—
9#389—22no—
10#387—22no—
11#381C2211yescharacteristic
12#381C2211yescharacteristic
13C4×C22⋊C4C2212yes—
14#342—22no—
15#339—22no—
16#339—22no—
17C23×C4C2311yescharacteristic
18C2×M16—22no—
19#117—22no—
20#117—22no—
21C2×C4⋊C4—22no—
22C2×C4⋊C4—22no—
23C2×C4⋊C4—22no—
24C2×C4⋊C4Q812yes—
25C2×C22⋊C4—21no—
26C2×C22⋊C4—22no—
27C2×C22⋊C4—21no—
28C2×C22⋊C4D412yes—
29C2×C42—21no—
30C2×C42D412yes—
31C2×C42D412yes—
32C22⋊C8—22no—
33C22⋊C8—22no—
34#95—22no—
35#95—22no—
36#95—22no—
37#95—22no—
38C24C2×D411yescharacteristic
39C22×C4—22no—
40C22×C4—22no—
41C22×C4—22no—
42C22×C4—22no—
43C22×C4—42no—
44C22×C4—41no—
45C22×C4C2×D412yes—
46C22×C4C2×Q811yescharacteristic
47C22×C4#4112yes—
48C22×C4#4111yescharacteristic
49M16—42no—
50M16—42no—
51C2×C8—42no—
52C4⋊C4—24no—
53C4⋊C4—44no—
54C4⋊C4—44no—
55C4⋊C4—82no—
56C22⋊C4—44no—
57C22⋊C4—41no—
58C22⋊C4—41no—
59C22⋊C4—42no—
60C22⋊C4—42no—
61C42—42no—
62C42—42no—
63C42—42no—
64C23—22no—
65C23—42no—
66C23#12011yescharacteristic
67C23C22⋊Q811yescharacteristic
68C23#12411yescharacteristic
69C2×C4—22no—
70C2×C4—22no—
71C2×C4—24no—
72C2×C4—41no—
73C2×C4—41no—
74C2×C4—42no—
75C2×C4—42no—
76C2×C4—42no—
77C2×C4—42no—
78C2×C4—42no—
79C2×C4—42no—
80C2×C4—42no—
81C2×C4—42no—
82C2×C4—42no—
83C2×C4—42no—
84C2×C4—82no—
85C2×C4—81no—
86C2×C4—82no—
87C2×C4C22⋊Q812yes—
88C2×C4#12412yes—
89C8—82no—
90C22—22no—
91C22—21no—
92C22—21no—
93C22—22no—
94C22—22no—
95C22—82no—
96C22#39311yescharacteristic
97C22#45812yes—
98C4—22no—
99C4—22no—
100C4—44no—
101C4—82no—
102C4—82no—
103C4—82no—
104C4—161no—
105C2—22no—
106C2—41no—
107C2—41no—
108C2#203011yescharacteristic
109C2#203112yes—
subgroupG/H
center, socleC22#393
commutator, Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#26082
#18262
#18202
#13062
#12842