Finite Group Databasebeta

← #18189

Subgroups — #18189

Order 256.

451 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 117 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#3113C212yesmaximal
3#2399C211yesmaximal, characteristic
4#1611C214yesmaximal
5#1605C214yesmaximal
6#1506C212yesmaximal
7#1467C211yesmaximal, characteristic
8Q82—22no—
9#557—22no—
10#557—24no—
11#544—21no—
12#544C2214yes—
13#533—22no—
14#533C2212yes—
15#519—21no—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18#483C2214yes—
19#475—22no—
20#462—22no—
21#451C2214yes—
22#441—24no—
23#432C2212yes—
24#398—21no—
25#398C2212yes—
26#395—21no—
27#395C2212yes—
28C4×C4⋊C4—22no—
29#376C2211yescharacteristic
30#328C2218yes—
31#324C2214yes—
32C22×Q8D411yescharacteristic
33C22×Q8C2311yescharacteristic
34C2×Q16—24no—
35C4⋊Q8—22no—
36C4⋊Q8—22no—
37C4⋊Q8—24no—
38C4⋊Q8—28no—
39C4⋊Q8D412yes—
40C4⋊Q8D412yes—
41#125—24no—
42#125—24no—
43#125—28no—
44#125—22no—
45C22⋊Q8—22no—
46C22⋊Q8—24no—
47C22⋊Q8C2314yes—
48C4×Q8—24no—
49C4×Q8—24no—
50C4×Q8—24no—
51C4×Q8—24no—
52C4×Q8—22no—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8D414yes—
58#117—24no—
59#117—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42D412yes—
69C2×C42C2311yescharacteristic
70#106—24no—
71C4⋊C8—22no—
72#103—24no—
73#103—24no—
74C22⋊C8C2314yes—
75#97—22no—
76#95—21no—
77#95—22no—
78#95—22no—
79#95C2312yes—
80C2×Q8—24no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—24no—
84C2×Q8—22no—
85C2×Q8—42no—
86C2×Q8C2×D414yes—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D412yes—
93C22×C4C2×D412yes—
94C22×C4C2411yescharacteristic
95Q16—48no—
96C2×C8—24no—
97C4⋊C4—24no—
98C4⋊C4—24no—
99C4⋊C4—24no—
100C4⋊C4—28no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—24no—
106C4⋊C4—24no—
107C4⋊C4—24no—
108C4⋊C4—22no—
109C4⋊C4—44no—
110C4⋊C4—42no—
111C4⋊C4—44no—
112C4⋊C4C2×D414yes—
113C22⋊C4—24no—
114C42—24no—
115C42—28no—
116C42—22no—
117C42—22no—
118C42—24no—
119C42—41no—
120C42—41no—
121C42C2×D412yes—
122C42C2×D412yes—
123C23C22×D411yescharacteristic
124Q8—28no—
125Q8—24no—
126Q8—44no—
127Q8—42no—
128Q8—44no—
129Q8—84no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—24no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4#12011yescharacteristic
148C2×C4#12012yes—
149C2×C4#12011yescharacteristic
150C2×C4#13714yes—
151C2×C4C22×D412yes—
152C8—44no—
153C22—21no—
154C22—21no—
155C22C2×#12011yescharacteristic
156C22C2×#13712yes—
157C4—22no—
158C4—24no—
159C4—21no—
160C4—22no—
161C4—44no—
162C4—44no—
163C4—42no—
164C4—42no—
165C4—44no—
166C4—81no—
167C4#45612yes—
168C2—22no—
169C2#300411yescharacteristic
170C2C2×#45611yescharacteristic
171C2#302511yescharacteristic
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

Maximal subgroups

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

subgroupindex
#35022
#31132
#23992
#16112
#16052
#15062
#14672