Finite Group Databasebeta

← #9654

Subgroups — #9654

Order 256.

320 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 85 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×#389C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic
3#1749C211yesmaximal, characteristic
4C2×#324C214yesmaximal
5C22×C4⋊C4C411yescharacteristic
6C22×C4⋊C4C411yescharacteristic
7C22×C4⋊C4C2212yes—
8C22×C42C2211yescharacteristic
9C2×C4⋊C8—21no—
10C2×C22⋊C8C2212yes—
11C2×C22⋊C8C2212yes—
12C2×#97—21no—
13#389—21no—
14#389—21no—
15#389—21no—
16#389—21no—
17C2×#95C412yes—
18#336—21no—
19#336—21no—
20#324—28no—
21C23×C4C2×C411yescharacteristic
22C23×C4C2×C412yes—
23C23×C4C2×C411yescharacteristic
24C23×C4C2×C412yes—
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27C22×C8—22no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—24no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—24no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—24no—
37C2×C4⋊C4—42no—
38C2×C42—42no—
39C2×C42D411yescharacteristic
40C2×C42D411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42D411yescharacteristic
43C4⋊C8—42no—
44C22⋊C8—22no—
45C22⋊C8—24no—
46C22⋊C8D414yes—
47#97—42no—
48#95—22no—
49#95—22no—
50C24C22×C411yescharacteristic
51C22×C4—22no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—24no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—24no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—24no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4M1614yes—
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D412yes—
75C22×C4#4112yes—
76C2×C8—24no—
77C2×C8—42no—
78C2×C8—42no—
79C2×C8—42no—
80C2×C8—42no—
81C2×C8—42no—
82C4⋊C4—24no—
83C4⋊C4—24no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C4⋊C4—44no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C42—22no—
90C42—22no—
91C42—41no—
92C42—41no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23—21no—
97C23C2×C22⋊C411yescharacteristic
98C23C4×D412yes—
99C23C2×M1612yes—
100C23#13112yes—
101C2×C4—28no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—24no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—28no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—44no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—84no—
127C2×C4#9912yes—
128C2×C4#10112yes—
129C2×C4#10112yes—
130C2×C4#12112yes—
131C2×C4#12411yescharacteristic
132C2×C4#12711yescharacteristic
133C8—44no—
134C8—82no—
135C22—22no—
136C22—22no—
137C22—22no—
138C22—22no—
139C22—22no—
140C22—22no—
141C22#39011yescharacteristic
142C22#40711yescharacteristic
143C22#40811yescharacteristic
144C22C2×#9911yescharacteristic
145C22#41011yescharacteristic
146C22C2×#10111yescharacteristic
147C22C2×#10111yescharacteristic
148C22#43514yes—
149C4—22no—
150C4—21no—
151C4—21no—
152C4—22no—
153C4—22no—
154C4—22no—
155C4—44no—
156C4—41no—
157C4—82no—
158C2—24no—
159C2#146311yescharacteristic
160C2#146711yescharacteristic
161C2#147011yescharacteristic
162C2#147211yescharacteristic
163C2#195511yescharacteristic
164C2#196411yescharacteristic
165C2#196611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutator, prime radicalC24C22×C4
Frattini, cosocleC23×C4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#3892
#17972
#17492
C2×#3242