Finite Group Databasebeta

← #10014

Subgroups — #10014

Order 256.

283 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 61 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#3103C212yesmaximal
2#2399C211yesmaximal, characteristic, complemented
3#1963C211yesmaximal, characteristic
4#1487C212yesmaximal, complemented
5#1274C211yesmaximal, characteristic, complemented
6#532C2211yescharacteristic
7C2×C4×Q8C2212yes—
8#481—22no—
9#475—22no—
10#460—22no—
11#448—22no—
12C4×SD16—24no—
13#431C2212yes—
14#423C2211yescharacteristic
15C2×#104—22no—
16#398—22no—
17#395—22no—
18C4×C4⋊C4—22no—
19#376C2211yescharacteristic
20Q8⋊C8—24no—
21C2×#41—21no—
22C22×Q8D412yes—
23C2×SD16—24no—
24C2×M16—21no—
25C4⋊Q8D411yescharacteristic
26#127D411yescharacteristic
27#124—21no—
28#121—22no—
29C4×Q8—22no—
30C4×Q8—22no—
31C4×Q8—22no—
32C4×Q8—22no—
33C4×D4—22no—
34C2×C4⋊C4—24no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4D412yes—
37C2×C42—22no—
38C2×C42D412yes—
39C2×C42C2311yescharacteristic
40#106—24no—
41C4⋊C8—22no—
42C4⋊C8—22no—
43#104—44no—
44#103—24no—
45#102—24no—
46C22⋊C8—22no—
47C4×C8D414yes—
48#95—22no—
49#95—22no—
50#95—22no—
51#41—42no—
52#41—42no—
53C2×Q8—22no—
54C2×Q8—22no—
55C2×Q8—21no—
56C2×Q8—44no—
57C2×D4—22no—
58C2×D4—21no—
59C2×D4—41no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—41no—
64C22×C4C2×D412yes—
65C22×C4C2×D411yescharacteristic
66SD16—48no—
67M16—42no—
68C2×C8—24no—
69C2×C8—41no—
70C4⋊C4—22no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—48no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C22⋊C4—42no—
77C42—22no—
78C42—42no—
79C42—42no—
80C42SD1614yescomplemented
81C42C2×D411yescharacteristic
82C42C2×D412yes—
83C42#4111yescharacteristic
84C23—41no—
85C23#12011yescharacteristic
86Q8—24no—
87Q8—24no—
88Q8—42no—
89Q8—42no—
90Q8—42no—
91D4—44no—
92D4—42no—
93D4—82no—
94C2×C4—24no—
95C2×C4—24no—
96C2×C4—42no—
97C2×C4—44no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—44no—
101C2×C4—42no—
102C2×C4—41no—
103C2×C4—81no—
104C2×C4—81no—
105C2×C4#12012yes—
106C2×C4#12112yes—
107C2×C4#12111yescharacteristic
108C2×C4#12711yescharacteristic
109C2×C4C2×SD1612yes—
110C2×C4#13712yes—
111C8—44no—
112C8—81no—
113C22—22no—
114C22—82no—
115C22—81no—
116C22#39211yescharacteristic
117C22#44812yes—
118C4—22no—
119C4—22no—
120C4—44no—
121C4—44no—
122C4—82no—
123C4—81no—
124C4#46112yes—
125C4#49211yescharacteristic
126C4#49711yescharacteristic
127C2—22no—
128C2—161no—
129C2#201012yes—
130C2#201311yescharacteristic
subgroupG/H
centerC2×C4#127
commutator, Frattini, cosocleC2×C42C23
socleC22#392
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2399C2semidirect
4. #1487C2semidirect
5. #1274C2semidirect
80. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#31032
#23992
#19632
#14872
#12742