Finite Group Databasebeta

← #9034

Subgroups — #9034

Order 256.

241 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 67 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#2921C211yesmaximal, characteristic
2#2396C211yesmaximal, characteristic
3#1487C212yesmaximal, complemented
4#1458C211yesmaximal, characteristic
5#1303C212yesmaximal, complemented
6C2×C4×Q8C2211yescharacteristic
7C2×#117C2211yescharacteristic
8#433—24no—
9#431C2212yes—
10#423C2212yes—
11#407—21no—
12C4×M16—21no—
13#400C411yescharacteristic
14#399C411yescharacteristic
15#398—21no—
16#396—21no—
17#395C411yescharacteristic
18#393C411yescharacteristic
19C4×C4⋊C4—21no—
20C4×C4⋊C4C2211yescharacteristic
21C4×C22⋊C4—21no—
22Q8⋊C8—24no—
23#324—21no—
24#323—21no—
25C22×Q8C2×C411yescharacteristic
26C23×C4D411yescharacteristic
27C2×M16—22no—
28C4×Q8—22no—
29C4×Q8—22no—
30#117—22no—
31#117—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4D411yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4C2×C411yescharacteristic
42C2×C22⋊C4D411yescharacteristic
43C2×C42—21no—
44C2×C42—21no—
45C2×C42D411yescharacteristic
46C2×C42C2311yescharacteristic
47C4⋊C8—22no—
48C4⋊C8—22no—
49C4⋊C8D414yes—
50C22⋊C8—24no—
51C22⋊C8—22no—
52#97—22no—
53C4×C8—22no—
54C4×C8—22no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95C2×C411yescharacteristic
60#95C2×C411yescharacteristic
61C24C22⋊C411yescharacteristic
62C2×Q8—22no—
63C2×Q8—42no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22×C411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78M16—42no—
79M16—42no—
80C2×C8—24no—
81C2×C8—42no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C22⋊C4—22no—
90C22⋊C4—22no—
91C22⋊C4—42no—
92C22⋊C4—42no—
93C42—21no—
94C42—21no—
95C42—41no—
96C42—41no—
97C42—42no—
98C42C2×D412yes—
99C42#4112yes—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—41no—
104C23—41no—
105C23C2×C22⋊C411yescharacteristic
106Q8—24no—
107Q8—42no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4C4×D412yes—
132C2×C4#12011yescharacteristic
133C2×C4#12112yes—
134C2×C4#12311yescharacteristic
135C2×C4#13512yes—
136C2×C4#13712yes—
137C8—44no—
138C8—82no—
139C22—21no—
140C22—21no—
141C22—42no—
142C22—41no—
143C22—41no—
144C22—41no—
145C22—81no—
146C22#38611yescharacteristic
147C22#41611yescharacteristic
148C22#41911yescharacteristic
149C4—22no—
150C4—22no—
151C4—42no—
152C4—41no—
153C4—41no—
154C4—41no—
155C4—42no—
156C4—81no—
157C4—81no—
158C4#45212yes—
159C4#48512yes—
160C2—21no—
161C2—21no—
162C2—81no—
163C2#187311yescharacteristic
164C2#188411yescharacteristic
165C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1487C2semidirect
5. #1303C2semidirect

Maximal subgroups

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

subgroupindex
#29212
#23962
#14872
#14582
#13032