Finite Group Databasebeta

← #8556

Subgroups — #8556

Order 256.

421 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C22×#126C211yesmaximal, characteristic
2#1821C212yesmaximal, complemented
3#1463C212yesmaximal, complemented
4#1462C212yesmaximal
5C2×#126—24no—
6C2×#126—24no—
7C2×#126—24no—
8C22×C4⋊C4C411yescharacteristic
9C22×C4⋊C4C2211yescharacteristic
10C22×C4⋊C4C2211yescharacteristic
11C22×C22⋊C4C411yescharacteristic
12C22×C22⋊C4C411yescharacteristic
13C22×C22⋊C4C2211yescharacteristic
14C22×C42C411yescharacteristic
15#407C2214yes—
16#343—24no—
17#324—24no—
18#323—24no—
19C25C2×C411yescharacteristic
20C23×C4C2×C411yescharacteristic
21C23×C4C2×C411yescharacteristic
22C23×C4C2×C411yescharacteristic
23C23×C4C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4C2311yescharacteristic
26C2×M16—24no—
27#126—232no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4D414yes—
36C2×C4⋊C4D414yes—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4D414yes—
46C2×C42—22no—
47C2×C42—22no—
48C2×C42—22no—
49C22⋊C8—24no—
50C22⋊C8—24no—
51C24—21no—
52C24—21no—
53C24—42no—
54C24—42no—
55C24—42no—
56C24C22⋊C412yes—
57C24C22×C411yescharacteristic
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—42no—
74C22×C4C22⋊C412yes—
75C22×C4C22⋊C412yes—
76C22×C4C22⋊C412yes—
77C22×C4C22⋊C412yes—
78C22×C4C22⋊C412yes—
79C22×C4C2×D412yes—
80C22×C4C2×D412yes—
81C22×C4C2×D412yes—
82M16—48no—
83C2×C8—44no—
84C4⋊C4—28no—
85C4⋊C4—28no—
86C4⋊C4—28no—
87C22⋊C4—28no—
88C22⋊C4—28no—
89C22⋊C4—28no—
90C42—28no—
91C23—22no—
92C23—22no—
93C23—22no—
94C23—22no—
95C23—24no—
96C23—42no—
97C23—42no—
98C23—42no—
99C23—42no—
100C23—42no—
101C23—42no—
102C23—42no—
103C23—42no—
104C23—88no—
105C23C2×C22⋊C411yescharacteristic
106C23C2×C22⋊C411yescharacteristic
107C23C2×C22⋊C411yescharacteristic
108C2×C4—24no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—24no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4#12014yes—
132C8—84no—
133C22—28no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—42no—
141C22—42no—
142C22—42no—
143C22—84no—
144C22—84no—
145C22—84no—
146C22#37911yescharacteristic
147C22#41012yes—
148C22#41312yes—
149C22#41312yes—
150C4—24no—
151C4—42no—
152C4—42no—
153C4—42no—
154C4—42no—
155C4—42no—
156C2—22no—
157C2—22no—
158C2—22no—
159C2—82no—
160C2#178211yescharacteristic
161C2#178511yescharacteristic
162C2#178511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25C2×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
2. #1821C2semidirect
3. #1463C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1262
#18212
#14632
#14622