Finite Group Databasebeta

← #10217

Subgroups — #10217

Order 256.

215 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 49 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#3255C211yesmaximal, characteristic
2#3237C212yesmaximal
3#1995C211yesmaximal, characteristic
4#1506C212yesmaximal
5#1307C211yesmaximal, characteristic
6#533C2211yescharacteristic
7#533C2212yes—
8#501—22no—
9#491—22no—
10#485—24no—
11#477—22no—
12#475—22no—
13#475—22no—
14#470—22no—
15#462—22no—
16#432C2212yes—
17#423C2211yescharacteristic
18#376C2211yescharacteristic
19#339—22no—
20#328—24no—
21C2×Q16—22no—
22C2×M16D412yes—
23C4⋊Q8D412yes—
24C4⋊Q8D412yes—
25C4⋊Q8D412yes—
26#125—22no—
27#125—21no—
28C22⋊Q8—22no—
29C22⋊Q8—22no—
30C22⋊Q8—22no—
31C4×Q8—22no—
32C4×Q8—22no—
33C4×Q8—22no—
34#117—21no—
35#117—22no—
36C2×C42—21no—
37C2×C42C2311yescharacteristic
38#107—24no—
39#106—22no—
40#106—24no—
41C4⋊C8—22no—
42C4⋊C8—22no—
43#103—24no—
44#103—24no—
45#103—24no—
46C22⋊C8D414yes—
47#97—22no—
48#95—22no—
49C2×Q8—22no—
50C2×Q8—22no—
51C2×Q8—22no—
52C22×C4—41no—
53C22×C4C2×D411yescharacteristic
54C22×C4C2×D412yes—
55Q16—82no—
56M16—42no—
57C2×C8—24no—
58C2×C8—22no—
59C4⋊C4—22no—
60C4⋊C4—22no—
61C4⋊C4—22no—
62C4⋊C4—22no—
63C4⋊C4—22no—
64C4⋊C4—22no—
65C4⋊C4—42no—
66C4⋊C4—42no—
67C4⋊C4—42no—
68C22⋊C4—41no—
69C22⋊C4—42no—
70C42—22no—
71C42—41no—
72C42—41no—
73C42—42no—
74C42C2×D411yescharacteristic
75C42C2×D412yes—
76C42#4111yescharacteristic
77C23#12711yescharacteristic
78Q8—44no—
79Q8—42no—
80Q8—42no—
81Q8—82no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—42no—
85C2×C4—42no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—82no—
90C2×C4#12012yes—
91C2×C4#12011yescharacteristic
92C2×C4#12112yes—
93C2×C4#12111yescharacteristic
94C8—44no—
95C8—42no—
96C22—21no—
97C22—22no—
98C22#39211yescharacteristic
99C4—21no—
100C4—22no—
101C4—22no—
102C4—82no—
103C4—82no—
104C4—82no—
105C4—82no—
106C4#45511yescharacteristic
107C4#45611yescharacteristic
108C2—41no—
109C2#200711yescharacteristic
110C2#200912yes—
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC23#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#32552
#32372
#19952
#15062
#13072