Finite Group Databasebeta

← #12209

Subgroups — #12209

Order 256.

202 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 37 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#2724C211yesmaximal, characteristic
2#2027C211yesmaximal, characteristic
3#1611C212yesmaximal, complemented
4#1346C212yesmaximal, complemented
5#1318C211yesmaximal, characteristic
6#544C2211yescharacteristic
7C2×#123—21no—
8#483—22no—
9#418—21no—
10#398C2212yes—
11#397—22no—
12#390C2211yescharacteristic
13#389—21no—
14#385—22no—
15#343C2211yescharacteristic
16#339—21no—
17#328—22no—
18#324C2212yes—
19C22×D4D411yescharacteristic
20C23×C4—21no—
21C2×M16—21no—
22C4⋊Q8—41no—
23#125—41no—
24#123—42no—
25#123—42no—
26C22⋊Q8—22no—
27#117—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4D412yes—
32C2×C4⋊C4C2311yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C42D411yescharacteristic
37#106—42no—
38#103—42no—
39#102—41no—
40#102—41no—
41C22⋊C8—22no—
42#95—22no—
43#95—22no—
44#95D412yes—
45C24—21no—
46C2×Q8—22no—
47C2×D4—21no—
48C2×D4—41no—
49C2×D4—41no—
50C2×D4SD1612yes—
51C22×C4—21no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—42no—
55C22×C4—81no—
56C22×C4—81no—
57C22×C4C2×D411yescharacteristic
58C22×C4C2×D412yes—
59M16—41no—
60M16—41no—
61C2×C8—42no—
62C2×C8—41no—
63C4⋊C4—21no—
64C4⋊C4—21no—
65C4⋊C4—22no—
66C4⋊C4—41no—
67C4⋊C4—42no—
68C4⋊C4—42no—
69C4⋊C4—42no—
70C4⋊C4—42no—
71C4⋊C4—42no—
72C4⋊C4—81no—
73C22⋊C4—44no—
74C22⋊C4—42no—
75C22⋊C4—44no—
76C22⋊C4—41no—
77C42—41no—
78C42—41no—
79C23—22no—
80C23—42no—
81C23—42no—
82C23—41no—
83C23—41no—
84C23#12011yescharacteristic
85Q8—82no—
86D4—42no—
87D4—81no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—84no—
104C2×C4C2×SD1611yescharacteristic
105C2×C4AGL(1,ℤ/8)11yescharacteristic
106C8—81no—
107C8—82no—
108C22—21no—
109C22—21no—
110C22—44no—
111C22—42no—
112C22—82no—
113C22—81no—
114C22—81no—
115C22#45011yescharacteristic
116C22#45511yescharacteristic
117C22B(4,𝔽2)11yescharacteristic
118C4—21no—
119C4—21no—
120C4—42no—
121C4—42no—
122C4—82no—
123C4—82no—
124C4—81no—
125C4—81no—
126C2—21no—
127C2—21no—
128C2—82no—
129C2#160111yescharacteristic
130C2#219811yescharacteristic
131C2#220311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1611C2semidirect
4. #1346C2semidirect

Maximal subgroups

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

subgroupindex
#27242
#20272
#16112
#13462
#13182