Finite Group Databasebeta

← #9015

Subgroups — #9015

Order 256.

400 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#395C211yesmaximal, characteristic
2#2316C211yesmaximal, characteristic
3#1787C212yesmaximal
4C2×#324C211yesmaximal, characteristic
5#1294C212yesmaximal
6C23×Q8C2211yescharacteristic
7C2×#117—21no—
8C22×C4⋊C4—21no—
9C22×C4⋊C4C2211yescharacteristic
10C2×#103—24no—
11C2×C22⋊C8C2212yes—
12#395—22no—
13#395C414yescomplemented
14#389—24no—
15#380—21no—
16#380C2212yes—
17C2×#95—21no—
18C2×#95C2211yescharacteristic
19#324—22no—
20C22×Q8—21no—
21C22×Q8—22no—
22C22×Q8—22no—
23C22×Q8—44no—
24C22×Q8C2×C412yes—
25C23×C4—21no—
26C23×C4—21no—
27C23×C4D411yescharacteristic
28C23×C4D411yescharacteristic
29C23×C4C2311yescharacteristic
30C22×C8—22no—
31#117—42no—
32#117—42no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—24no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—44no—
39C2×C4⋊C4C2×C412yes—
40C2×C4⋊C4D412yes—
41C2×C4⋊C4D414yes—
42C2×C22⋊C4—22no—
43C2×C42—22no—
44#103—48no—
45C22⋊C8—22no—
46C22⋊C8—22no—
47#95—22no—
48#95—24no—
49#95—21no—
50#95—22no—
51#95C2×C412yes—
52C24C2×D411yescharacteristic
53C2×Q8—28no—
54C2×Q8—24no—
55C2×Q8—44no—
56C2×Q8—44no—
57C2×Q8—44no—
58C2×Q8—48no—
59C2×Q8—42no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—24no—
65C22×C4—24no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—24no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—42no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—42no—
79C22×C4C22⋊C412yes—
80C22×C4C22⋊C412yes—
81C22×C4SD1612yes—
82C22×C4Q1612yes—
83C22×C4C22×C411yescharacteristic
84C22×C4C2×D412yes—
85C22×C4C2×D411yescharacteristic
86C22×C4#4112yes—
87C2×C8—44no—
88C2×C8—42no—
89C4⋊C4—24no—
90C4⋊C4—24no—
91C4⋊C4—44no—
92C4⋊C4—48no—
93C4⋊C4—44no—
94C4⋊C4—82no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C42—44no—
98C23—22no—
99C23—22no—
100C23C2×C22⋊C411yescharacteristic
101C23#12011yescharacteristic
102C23#12311yescharacteristic
103C23C2×SD1611yescharacteristic
104C23C2×Q1611yescharacteristic
105C23AGL(1,ℤ/8)11yescharacteristic
106C23#13711yescharacteristic
107Q8—28no—
108Q8—48no—
109Q8—44no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—24no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—24no—
118C2×C4—24no—
119C2×C4—22no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—44no—
125C2×C4—44no—
126C2×C4—42no—
127C2×C4—44no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—44no—
132C2×C4—42no—
133C2×C4—44no—
134C2×C4—44no—
135C2×C4—44no—
136C2×C4—81no—
137C2×C4#10314yescomplemented
138C2×C4C4×D412yes—
139C2×C4#12112yes—
140C8—82no—
141C22—22no—
142C22—22no—
143C22—22no—
144C22—22no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22#38611yescharacteristic
150C22C2×#10311yescharacteristic
151C22#41711yescharacteristic
152C22#45011yescharacteristic
153C22#45111yescharacteristic
154C22#45612yes—
155C22B(4,𝔽2)12yes—
156C22#48311yescharacteristic
157C22#48411yescharacteristic
158C4—24no—
159C4—22no—
160C4—22no—
161C4—44no—
162C4—44no—
163C4—42no—
164C4—84no—
165C2—21no—
166C2—21no—
167C2—21no—
168C2—21no—
169C2#160412yes—
170C2#160512yes—
171C2#187211yescharacteristic
172C2#188611yescharacteristic
173C2#188911yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #395C4semidirect
137. C2×C4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#3952
#23162
#17872
C2×#3242
#12942