Finite Group Databasebeta

← #13438

Subgroups — #13438

Order 256.

203 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 43 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#3202C211yesmaximal, characteristic
2#2123C212yesmaximal
3#1909C212yesmaximal
4#1858C211yesmaximal, characteristic
5#1391C211yesmaximal, characteristic
6#568C2211yescharacteristic
7#536C2212yes—
8#485—22no—
9#470—21no—
10#467—21no—
11#451—22no—
12#430—21no—
13#419—22no—
14#413C2211yescharacteristic
15#410C2212yes—
16#408C2211yescharacteristic
17#352—24no—
18#341—21no—
19H(3,ℤ/4)—22no—
20C2×#41C2311yescharacteristic
21C22×Q8D412yes—
22C2×Q16—21no—
23C2×Q16—21no—
24#131—42no—
25C2×M16—22no—
26C2×M16—21no—
27C2×M16—21no—
28C2×M16D411yescharacteristic
29C22×C8D411yescharacteristic
30C4⋊Q8—22no—
31#124—22no—
32#123—22no—
33#123—22no—
34C22⋊Q8—22no—
35C22⋊Q8—22no—
36#117—22no—
37#117D412yes—
38#108—42no—
39#107—21no—
40#106—21no—
41#103—22no—
42#103—22no—
43#103—44no—
44#101—21no—
45#100D412yes—
46#99—22no—
47#99—22no—
48C22⋊C8—22no—
49#41—41no—
50#41—41no—
51C2×Q8—22no—
52C2×Q8—22no—
53C2×Q8—42no—
54C2×Q8—42no—
55C2×Q8#4111yescharacteristic
56C2×D4C2×D411yescharacteristic
57C2×D4#4112yes—
58C22×C4—22no—
59C22×C4C2×D411yescharacteristic
60C22×C4C2×D412yes—
61Q16—81no—
62Q16—81no—
63M16—22no—
64M16—22no—
65M16—41no—
66M16—42no—
67M16—41no—
68M16—41no—
69C2×C8—24no—
70C2×C8—21no—
71C2×C8—21no—
72C2×C8—41no—
73C2×C8—41no—
74C2×C8—41no—
75C4⋊C4—22no—
76C4⋊C4—42no—
77C4⋊C4—42no—
78C4⋊C4—42no—
79C22⋊C4—22no—
80C22⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—42no—
83C42—22no—
84C42—42no—
85C23#12112yes—
86C23#12411yescharacteristic
87Q8—41no—
88Q8—44no—
89Q8—42no—
90Q8—42no—
91D4—41no—
92D4—42no—
93C2×C4—22no—
94C2×C4—21no—
95C2×C4—22no—
96C2×C4—21no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—82no—
100C2×C4—82no—
101C2×C4#12011yescharacteristic
102C2×C4#12111yescharacteristic
103C2×C4#12312yes—
104C8—42no—
105C8—42no—
106C8—41no—
107C8—41no—
108C22—22no—
109C22—21no—
110C22—42no—
111C22—41no—
112C22#39411yescharacteristic
113C4—21no—
114C4—22no—
115C4—82no—
116C4—82no—
117C4—162no—
118C4B(4,𝔽2)11yescharacteristic
119C4#45811yescharacteristic
120C2—21no—
121C2—42no—
122C2—41no—
123C2#204411yescharacteristic
subgroupG/H
center, socleC2#2044
commutator, Frattini, prime radical, cosocleC2×#41C23
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#32022
#21232
#19092
#18582
#13912