Finite Group Databasebeta

← #18404

Subgroups — #18404

Order 256.

455 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 129 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#3502C211yesmaximal, characteristic
2#3209C212yesmaximal
3#2567C211yesmaximal, characteristic
4#1605C218yesmaximal
5#1503C212yesmaximal
6#1461C211yesmaximal, characteristic
7Q82—22no—
8#557—22no—
9#557—24no—
10#544—21no—
11#544C2214yes—
12#533—22no—
13#533C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15#519—21no—
16C2×C4×Q8C2211yescharacteristic
17#494—22no—
18#487—22no—
19#462—24no—
20#451C2218yes—
21#431C2212yes—
22#395C2214yes—
23#391—21no—
24#389—22no—
25#384—21no—
26C4×C4⋊C4C2211yescharacteristic
27#328C2218yes—
28#324C2214yes—
29C22×Q8D411yescharacteristic
30C22×Q8C2312yes—
31C2×Q16—28no—
32C4⋊Q8—22no—
33C4⋊Q8—22no—
34C4⋊Q8—24no—
35C4⋊Q8—28no—
36C4⋊Q8—22no—
37C4⋊Q8—22no—
38C4⋊Q8D412yes—
39C4⋊Q8D412yes—
40#125—24no—
41#125—24no—
42#125—28no—
43#125—22no—
44C22⋊Q8—22no—
45C22⋊Q8—24no—
46C22⋊Q8C2314yes—
47C4×Q8—24no—
48C4×Q8—24no—
49C4×Q8—24no—
50C4×Q8—24no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×Q8—22no—
54C4×Q8D414yes—
55#117—24no—
56#117—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C42—21no—
64C2×C42D412yes—
65C2×C42C2311yescharacteristic
66C4⋊C8—22no—
67#103—28no—
68C22⋊C8C2314yes—
69C4×C8—22no—
70#95—24no—
71#95C2312yes—
72C2×Q8—24no—
73C2×Q8—22no—
74C2×Q8—22no—
75C2×Q8—22no—
76C2×Q8—24no—
77C2×Q8—82no—
78C2×Q8Q1614yescomplemented
79C2×Q8C2×D414yes—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D412yes—
84C22×C4C2×D412yes—
85C22×C4C2×D411yescharacteristic
86C22×C4C2411yescharacteristic
87Q16—88no—
88C2×C8—24no—
89C4⋊C4—24no—
90C4⋊C4—24no—
91C4⋊C4—24no—
92C4⋊C4—24no—
93C4⋊C4—28no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—24no—
99C4⋊C4—24no—
100C4⋊C4—24no—
101C4⋊C4—41no—
102C4⋊C4—42no—
103C4⋊C4—41no—
104C4⋊C4—84no—
105C4⋊C4C2×D414yes—
106C22⋊C4—24no—
107C42—28no—
108C42—22no—
109C42—22no—
110C42—24no—
111C42—42no—
112C42C2×D412yes—
113C42C2×D412yes—
114C23C22×D411yescharacteristic
115Q8—28no—
116Q8—44no—
117Q8—48no—
118Q8—44no—
119Q8—84no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—24no—
124C2×C4—24no—
125C2×C4—24no—
126C2×C4—22no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—44no—
132C2×C4—42no—
133C2×C4#12011yescharacteristic
134C2×C4#12012yes—
135C2×C4#12011yescharacteristic
136C2×C4C2×Q1614yes—
137C2×C4C2×Q1612yes—
138C2×C4#13712yes—
139C2×C4C22×D412yes—
140C8—44no—
141C22—21no—
142C22—21no—
143C22C2×#12011yescharacteristic
144C22C22×Q1611yescharacteristic
145C22C2×#13711yescharacteristic
146C4—24no—
147C4—21no—
148C4—22no—
149C4—44no—
150C4—44no—
151C4—42no—
152C4—44no—
153C4—82no—
154C4#45114yes—
155C4#45612yes—
156C2—21no—
157C2—21no—
158C2C2×#45111yescharacteristic
159C2C2×#45611yescharacteristic
160C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC23C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
78. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#35022
#32092
#25672
#16052
#15032
#14612