Finite Group Databasebeta

← #9508

Subgroups — #9508

Order 256.

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

normal subgroups only

This group has 271 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#2975C211yesmaximal, characteristic
2#2486C211yesmaximal, characteristic
3#1821C212yesmaximal, complemented
4#1465C211yesmaximal, characteristic
5#1305C212yesmaximal, complemented
6C2×#123C412yes—
7C2×C4×D4C2211yescharacteristic
8C22×C4⋊C4C2211yescharacteristic
9#436—24no—
10#431—21no—
11#407—21no—
12#407C2212yes—
13#407C2212yes—
14#398—21no—
15#397—21no—
16#389—21no—
17#381C412yes—
18#380—21no—
19#380C2211yescharacteristic
20#343—24no—
21#324—21no—
22#323—21no—
23C22×D4C2×C411yescharacteristic
24C23×C4C2×C412yes—
25C23×C4D411yescharacteristic
26C23×C4C2311yescharacteristic
27C2×M16—22no—
28C2×M16—22no—
29C2×M16D414yes—
30#123—44no—
31C4×D4—22no—
32C4×D4—22no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—22no—
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4—22no—
41C2×C22⋊C4C2×C411yescharacteristic
42C2×C22⋊C4C2×C412yes—
43C2×C22⋊C4D411yescharacteristic
44C2×C42D411yescharacteristic
45C4⋊C8—22no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50C22⋊C8—22no—
51C4×C8—22no—
52#95—22no—
53#95—22no—
54#95—21no—
55C24C22⋊C411yescharacteristic
56C24C22×C411yescharacteristic
57C2×D4—22no—
58C2×D4—22no—
59C2×D4—41no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—24no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C2×D412yes—
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4#4112yes—
76M16—44no—
77M16—44no—
78M16—82no—
79C2×C8—24no—
80C2×C8—42no—
81C4⋊C4—22no—
82C4⋊C4—42no—
83C4⋊C4—44no—
84C4⋊C4—41no—
85C4⋊C4—42no—
86C4⋊C4—41no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C22⋊C4—44no—
90C22⋊C4—42no—
91C42—21no—
92C42—21no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—41no—
99C23—41no—
100C23—41no—
101C23—41no—
102C23C2×C22⋊C411yescharacteristic
103C23C4×D412yes—
104D4—44no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—22no—
108C2×C4—24no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—81no—
123C2×C4—84no—
124C2×C4#10012yes—
125C2×C4#12011yescharacteristic
126C2×C4#12112yes—
127C2×C4#12311yescharacteristic
128C8—44no—
129C8—82no—
130C22—21no—
131C22—21no—
132C22—22no—
133C22—22no—
134C22—22no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—41no—
139C22—41no—
140C22—41no—
141C22—81no—
142C22#38611yescharacteristic
143C22C2×#10011yescharacteristic
144C22#41311yescharacteristic
145C4—22no—
146C4—21no—
147C4—22no—
148C4—41no—
149C4—41no—
150C4—82no—
151C4—82no—
152C4#45812yes—
153C2—21no—
154C2—21no—
155C2—42no—
156C2—81no—
157C2#189011yescharacteristic
158C2#190011yescharacteristic
159C2#191511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1821C2semidirect
5. #1305C2semidirect

Maximal subgroups

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

subgroupindex
#29752
#24862
#18212
#14652
#13052