Finite Group Databasebeta

← #12856

Subgroups — #12856

Order 256.

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

normal subgroups only

This group has 457 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
1C2×#390C211yesmaximal, characteristic
2#1781C212yesmaximal
3#1393C214yesmaximal, complemented
4C23×D4C2211yescharacteristic
5C22×C4⋊C4C411yescharacteristic, complemented
6C22×C22⋊C4—22no—
7C22×C42C411yescharacteristic, complemented
8C2×#99—22no—
9C2×#99C2214yes—
10#390—22no—
11#390C412yescomplemented
12#390C2212yes—
13#379—22no—
14C25D412yes—
15C22×D4—24no—
16C22×D4—42no—
17C22×D4—41no—
18C22×D4—41no—
19C22×D4C2×C411yescharacteristic
20C22×D4D412yes—
21C22×D4C2311yescharacteristic
22C23×C4—21no—
23C23×C4—21no—
24C23×C4C2×C411yescharacteristic
25C2×C4⋊C4—21no—
26C2×C4⋊C4—41no—
27C2×C4⋊C4—41no—
28C2×C4⋊C4C2×C412yes—
29C2×C22⋊C4—24no—
30C2×C22⋊C4—24no—
31C2×C22⋊C4—24no—
32C2×C22⋊C4—48no—
33C2×C22⋊C4—42no—
34C2×C22⋊C4—42no—
35C2×C42—21no—
36C2×C42—42no—
37C2×C42C2×C412yes—
38#99—28no—
39#99—44no—
40C24—24no—
41C24—22no—
42C24—22no—
43C24—42no—
44C24—44no—
45C24—44no—
46C24C22⋊C412yes—
47C24C22⋊C411yescharacteristic
48C24C2×D412yes—
49C2×D4—24no—
50C2×D4—24no—
51C2×D4—22no—
52C2×D4—41no—
53C2×D4—41no—
54C2×D4—42no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4—48no—
58C2×D4—48no—
59C2×D4#4114yes—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—24no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—44no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—44no—
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22×C411yescharacteristic
75C4⋊C4—42no—
76C4⋊C4—42no—
77C22⋊C4—48no—
78C22⋊C4—48no—
79C22⋊C4—44no—
80C22⋊C4—44no—
81C22⋊C4—84no—
82C22⋊C4—84no—
83C42—42no—
84C42—42no—
85C23—22no—
86C23—22no—
87C23—21no—
88C23—21no—
89C23—22no—
90C23—22no—
91C23—22no—
92C23—22no—
93C23—22no—
94C23—44no—
95C23—44no—
96C23—42no—
97C23—42no—
98C23—42no—
99C23—42no—
100C23—416no—
101C23—44no—
102C23—44no—
103C23—82no—
104C23—82no—
105C23—88no—
106C23#9911yescharacteristic
107C23#9911yescharacteristic
108C23C2×C22⋊C411yescharacteristic
109C23#12314yes—
110D4—44no—
111D4—48no—
112D4—44no—
113C2×C4—24no—
114C2×C4—24no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—24no—
120C2×C4—48no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—44no—
124C2×C4—42no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4—84no—
128C2×C4—84no—
129C2×C4—84no—
130C2×C4#9912yes—
131C2×C4#11712yes—
132C22—24no—
133C22—24no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—24no—
141C22—44no—
142C22—44no—
143C22—42no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—42no—
148C22—82no—
149C22—88no—
150C22—88no—
151C22—82no—
152C22#35312yes—
153C22#35412yes—
154C22#38111yescharacteristic
155C22C2×#9911yescharacteristic
156C22C2×#9911yescharacteristic
157C4—22no—
158C4—22no—
159C4—44no—
160C4—82no—
161C4—164no—
162C2—21no—
163C2—21no—
164C2—22no—
165C2—22no—
166C2—44no—
167C2—82no—
168C2#181911yescharacteristic
169C2C2×#35311yescharacteristic
170C2C2×#35411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC22×D4C23
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1393C2semidirect
5. C22×C4⋊C4C4semidirect
7. C22×C42C4semidirect
11. #390C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3902
#17812
#13932