Finite Group Databasebeta

← #12246

Subgroups — #12246

Order 256.

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

normal subgroups only

This group has 279 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#2537C211yesmaximal, characteristic
2#2060C211yesmaximal, characteristic, complemented
3#1599C212yesmaximal, complemented
4#1345C212yesmaximal, complemented
5#1325C211yesmaximal, characteristic, complemented
6#544C2211yescharacteristic
7C2×#121—22no—
8C2×C4×D4—21no—
9#450—22no—
10#428—21no—
11#394—21no—
12#394C2212yes—
13#384C2211yescharacteristic
14#380—22no—
15#343C2211yescharacteristic
16#339—21no—
17#328—22no—
18#324C2212yes—
19C22×D4—21no—
20C22×D4—21no—
21C22×D4D412yes—
22C23×C4—22no—
23C2×SD16—42no—
24C2×M16—21no—
25C4⋊Q8—41no—
26#125—41no—
27C22⋊Q8—22no—
28#121—48no—
29C4×D4—42no—
30C4×D4—42no—
31#117—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4D411yescharacteristic
35C2×C4⋊C4C2311yescharacteristic
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C42D411yescharacteristic
40#107—41no—
41#106—41no—
42#102—42no—
43C22⋊C8—22no—
44#95—21no—
45#95D412yes—
46C24—22no—
47C2×Q8—22no—
48C2×D4—24no—
49C2×D4—42no—
50C2×D4—42no—
51C2×D4—42no—
52C2×D4—44no—
53C2×D4—44no—
54C2×D4—81no—
55C22×C4—21no—
56C22×C4—22no—
57C22×C4—21no—
58C22×C4—44no—
59C22×C4—44no—
60C22×C4—44no—
61C22×C4C2×D411yescharacteristic
62C22×C4C2×D412yes—
63SD16—84no—
64M16—22no—
65M16—41no—
66C2×C8—41no—
67C2×C8—42no—
68C4⋊C4—22no—
69C4⋊C4—21no—
70C4⋊C4—21no—
71C4⋊C4—41no—
72C4⋊C4—41no—
73C4⋊C4—42no—
74C4⋊C4—44no—
75C4⋊C4—42no—
76C22⋊C4—44no—
77C22⋊C4—44no—
78C22⋊C4—41no—
79C22⋊C4—42no—
80C22⋊C4—42no—
81C42—21no—
82C42—41no—
83C42SD1612yescomplemented
84C23—24no—
85C23—42no—
86C23—42no—
87C23—42no—
88C23—42no—
89C23—42no—
90C23—42no—
91C23#12011yescharacteristic
92Q8—82no—
93D4—48no—
94D4—44no—
95D4—82no—
96D4—84no—
97C2×C4—22no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—88no—
112C2×C4C2×SD1611yescharacteristic
113C2×C4AGL(1,ℤ/8)11yescharacteristic
114C8—42no—
115C8—82no—
116C22—21no—
117C22—21no—
118C22—44no—
119C22—44no—
120C22—44no—
121C22—82no—
122C22—82no—
123C22—82no—
124C22—82no—
125C22#45011yescharacteristic
126C22#45611yescharacteristic
127C22B(4,𝔽2)11yescharacteristic
128C4—24no—
129C4—21no—
130C4—21no—
131C4—42no—
132C4—82no—
133C4—81no—
134C4—81no—
135C4—81no—
136C2—21no—
137C2—21no—
138C2—84no—
139C2#160411yescharacteristic
140C2#219611yescharacteristic
141C2#220411yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#2537C2
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
2. #2060C2semidirect
3. #1599C2semidirect
4. #1345C2semidirect
5. #1325C2semidirect
83. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#25372
#20602
#15992
#13452
#13252