Finite Group Databasebeta

← #9030

Subgroups — #9030

Order 256.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#2983C211yesmaximal, characteristic
2#2399C211yesmaximal, characteristic
3C4×#101C211yesmaximal, characteristic
4#1491C212yesmaximal
5#1300C212yesmaximal
6C2×C4×Q8C2211yescharacteristic
7C2×C4×Q8C2211yescharacteristic
8#446—24no—
9#432—21no—
10#432C2212yes—
11#431—21no—
12C2×#101—21no—
13C4×M16C2212yes—
14#398—21no—
15#398C411yescharacteristic
16#398C411yescharacteristic
17#395—21no—
18#395C411yescharacteristic
19#395C411yescharacteristic
20C4×C4⋊C4—21no—
21C4×C4⋊C4—21no—
22#376C2211yescharacteristic
23#343—21no—
24Q8⋊C8—24no—
25C22×Q8C2×C411yescharacteristic
26C22×Q8D411yescharacteristic
27C2×M16—22no—
28C4×Q8—22no—
29C4×Q8—22no—
30C4×Q8—22no—
31C4×Q8—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4D411yescharacteristic
41C2×C42—21no—
42C2×C42—21no—
43C2×C42D411yescharacteristic
44C2×C42D411yescharacteristic
45C2×C42C2311yescharacteristic
46C4⋊C8—22no—
47C4⋊C8—22no—
48C4⋊C8D414yes—
49#101—24no—
50C22⋊C8—22no—
51#97—22no—
52#97—22no—
53#97—22no—
54C4×C8—22no—
55C4×C8—22no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95C2×C411yescharacteristic
62#95C2×C411yescharacteristic
63C2×Q8—22no—
64C2×Q8—22no—
65C2×Q8—22no—
66C2×Q8—41no—
67C2×Q8—42no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22×C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79M16—24no—
80M16—42no—
81C2×C8—24no—
82C2×C8—42no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4—22no—
86C4⋊C4—22no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—42no—
92C42—24no—
93C42—21no—
94C42—21no—
95C42—21no—
96C42—21no—
97C42—41no—
98C42—41no—
99C42C2×D412yes—
100C42#4112yes—
101C23C2×C22⋊C411yescharacteristic
102Q8—24no—
103Q8—44no—
104Q8—42no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4#10412yes—
131C2×C4C4×D412yes—
132C2×C4#12011yescharacteristic
133C2×C4#12112yes—
134C2×C4#12311yescharacteristic
135C8—44no—
136C8—44no—
137C22—21no—
138C22—21no—
139C22#38611yescharacteristic
140C22C2×#10411yescharacteristic
141C22#42111yescharacteristic
142C4—22no—
143C4—22no—
144C4—21no—
145C4—22no—
146C4—42no—
147C4—42no—
148C4—41no—
149C4—41no—
150C4—41no—
151C4—42no—
152C4—81no—
153C4#45612yes—
154C2—21no—
155C2—21no—
156C2#187111yescharacteristic
157C2#188611yescharacteristic
158C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC23C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#29832
#23992
C4×#1012
#14912
#13002