Finite Group Databasebeta

← #8100

Subgroups — #8100

Order 256.

497 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 123 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#2280C212yesmaximal
2#2280C211yesmaximal, characteristic
3#1304C214yesmaximal
4C24×C4C2211yescharacteristic
5C2×#117—22no—
6C2×#117C412yescomplemented
7#381—22no—
8#381C412yescomplemented
9#380—22no—
10#380—22no—
11#380C414yescomplemented
12#379C2214yes—
13#379C2212yes—
14C4×C22⋊C4—22no—
15C4×C22⋊C4—22no—
16C4×C22⋊C4C414yescomplemented
17C25C2311yescharacteristic
18C23×C4—44no—
19C23×C4C2×C411yescharacteristic
20C23×C4C2×C412yes—
21C23×C4C2×C412yes—
22C23×C4C2×C411yescharacteristic
23C23×C4D412yes—
24C23×C4D411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4D412yes—
27C23×C4Q811yescharacteristic
28C23×C4Q811yescharacteristic
29#117—48no—
30#117—44no—
31C2×C4⋊C4—24no—
32C2×C4⋊C4—22no—
33C2×C22⋊C4—24no—
34C2×C22⋊C4—24no—
35C2×C22⋊C4—24no—
36C2×C22⋊C4C2×C414yes—
37C2×C22⋊C4C2×C418yes—
38C2×C42—24no—
39C2×C42—22no—
40#95—24no—
41#95—24no—
42#95—24no—
43C24—42no—
44C24—44no—
45C24C22×C411yescharacteristic
46C24C22×C412yes—
47C24C2×D412yes—
48C24C2×D411yescharacteristic
49C24C2×Q811yescharacteristic
50C22×C4—24no—
51C22×C4—28no—
52C22×C4—28no—
53C22×C4—24no—
54C22×C4—44no—
55C22×C4—44no—
56C22×C4—42no—
57C22×C4—42no—
58C22×C4—42no—
59C22×C4—42no—
60C22×C4—42no—
61C22×C4—44no—
62C22×C4C4211yescharacteristic
63C22×C4C4211yescharacteristic
64C22×C4C4212yes—
65C22×C4C22⋊C412yes—
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22⋊C412yes—
68C22×C4C22⋊C412yes—
69C22×C4C22⋊C412yes—
70C22×C4C22⋊C412yes—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C4⋊C411yescharacteristic
73C22×C4C4⋊C412yes—
74C22×C4C4⋊C411yescharacteristic
75C22×C4C4⋊C412yes—
76C22×C4C4⋊C412yes—
77C22×C4C4⋊C412yes—
78C22×C4C4⋊C411yescharacteristic
79C22×C4C4⋊C411yescharacteristic
80C4⋊C4—44no—
81C4⋊C4—48no—
82C22⋊C4—48no—
83C22⋊C4—48no—
84C22⋊C4—48no—
85C22⋊C4—48no—
86C22⋊C4—44no—
87C42—48no—
88C42—44no—
89C23—22no—
90C23—24no—
91C23—24no—
92C23—24no—
93C23—24no—
94C23—42no—
95C23—41no—
96C23—41no—
97C23—42no—
98C23—42no—
99C23—42no—
100C23—42no—
101C23—416no—
102C23C2×C4211yescharacteristic
103C23C2×C22⋊C411yescharacteristic
104C23C2×C22⋊C412yes—
105C23C2×C4⋊C411yescharacteristic
106C23C2×C4⋊C412yes—
107C2×C4—22no—
108C2×C4—24no—
109C2×C4—24no—
110C2×C4—24no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—24no—
115C2×C4—22no—
116C2×C4—24no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—48no—
125C2×C4—48no—
126C2×C4—48no—
127C2×C4—44no—
128C2×C4—48no—
129C2×C4#9511yescharacteristic
130C2×C4#9511yescharacteristic
131C2×C4#9512yes—
132C2×C4#9512yes—
133C2×C4#9511yescharacteristic
134C2×C4#9511yescharacteristic
135C2×C4#9914yes—
136C22—22no—
137C22—22no—
138C22—22no—
139C22—21no—
140C22—21no—
141C22—22no—
142C22—22no—
143C22—22no—
144C22—22no—
145C22—22no—
146C22—48no—
147C22—44no—
148C22—44no—
149C22—48no—
150C22—44no—
151C22C2×#9511yescharacteristic
152C22C2×#9912yes—
153C22#41014yes—
154C4—21no—
155C4—21no—
156C4—22no—
157C4—22no—
158C4—84no—
159C4—88no—
160C4#34212yes—
161C4#34212yes—
162C2—22no—
163C2—24no—
164C2—41no—
165C2—41no—
166C2—42no—
167C2C2×#34211yescharacteristic
168C2#174012yes—
subgroupG/H
centerC2×C4#95
commutatorC23C2×C42
Frattini, prime radical, cosocleC25C23
socleC22C2×#95
LCSG ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#117C4semidirect
8. #381C4semidirect
11. #380C4semidirect
16. C4×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#22802
#13042