Finite Group Databasebeta

← #20700

Subgroups — #20700

Order 256.

887 conjugacy classes of subgroups; only the 247 normal subgroups are shown.

normal subgroups only

This group has 887 conjugacy classes of subgroups — too many to draw. Only its 247 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
1C23×C4⋊C4C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic
3#2280C211yesmaximal, characteristic
4#1824C214yesmaximal, complemented
5#1823C214yesmaximal, complemented
6#1736C214yesmaximal, complemented
7C24×C4—21no—
8C24×C4C2211yescharacteristic
9C22×M16C2212yes—
10C2×#117C2212yes—
11C22×C4⋊C4—21no—
12C22×C4⋊C4—21no—
13C22×C4⋊C4—21no—
14C22×C4⋊C4—24no—
15C22×C4⋊C4C418yescomplemented
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2212yes—
19#407—24no—
20C2×C22⋊C8C2212yes—
21C2×C22⋊C8C2212yes—
22#381—21no—
23#380C2212yes—
24#380C2212yes—
25#379—21no—
26C4×C22⋊C4—21no—
27C4×C22⋊C4—21no—
28#340C22116yescomplemented
29C25D411yescharacteristic
30C23×C4—28no—
31C23×C4—28no—
32C23×C4—22no—
33C23×C4—22no—
34C23×C4—22no—
35C23×C4—48no—
36C23×C4C2×C414yes—
37C23×C4D411yescharacteristic
38C23×C4D411yescharacteristic
39C23×C4D411yescharacteristic
40C23×C4D414yes—
41C23×C4C2311yescharacteristic
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C2×M16—24no—
45C2×M16—28no—
46C22×C8C2314yes—
47#117—28no—
48C2×C4⋊C4—28no—
49C2×C4⋊C4—28no—
50C2×C4⋊C4—28no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4C2×C418yes—
58C2×C4⋊C4C2×C418yes—
59C2×C4⋊C4C2×C418yes—
60C2×C4⋊C4C2314yes—
61C2×C4⋊C4C2314yes—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C42—22no—
66C22⋊C8—24no—
67C22⋊C8—24no—
68#95—22no—
69#95—22no—
70C24—24no—
71C24—24no—
72C24—24no—
73C24C22⋊C414yes—
74C24C2×D411yescharacteristic
75C24C2×D411yescharacteristic
76C24C2×D411yescharacteristic
77C22×C4—24no—
78C22×C4—216no—
79C22×C4—24no—
80C22×C4—24no—
81C22×C4—216no—
82C22×C4—24no—
83C22×C4—24no—
84C22×C4—28no—
85C22×C4—28no—
86C22×C4—28no—
87C22×C4—24no—
88C22×C4—24no—
89C22×C4—48no—
90C22×C4—48no—
91C22×C4—432no—
92C22×C4—48no—
93C22×C4C22⋊C414yes—
94C22×C4C22⋊C414yes—
95C22×C4C22⋊C414yes—
96C22×C4C22×C412yes—
97C22×C4C22×C412yes—
98C22×C4C22×C412yes—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D412yes—
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D412yes—
105C22×C4#4118yes—
106C22×C4C2411yescharacteristic
107M16—48no—
108C2×C8—24no—
109C2×C8—28no—
110C4⋊C4—28no—
111C4⋊C4—28no—
112C4⋊C4—28no—
113C4⋊C4—24no—
114C4⋊C4—28no—
115C4⋊C4C22×C418yes—
116C22⋊C4—44no—
117C22⋊C4—44no—
118C22⋊C4—44no—
119C42—44no—
120C23—232no—
121C23—22no—
122C23—22no—
123C23—22no—
124C23—22no—
125C23—22no—
126C23—22no—
127C23—28no—
128C23—28no—
129C23—28no—
130C23C2×C22⋊C412yes—
131C23C2×C22⋊C412yes—
132C23C2×C22⋊C412yes—
133C23#12314yes—
134C23AGL(1,ℤ/8)14yes—
135C23#13714yes—
136C23C22×D411yescharacteristic
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—28no—
140C2×C4—24no—
141C2×C4—24no—
142C2×C4—24no—
143C2×C4—24no—
144C2×C4—28no—
145C2×C4—28no—
146C2×C4—28no—
147C2×C4—416no—
148C2×C4—416no—
149C2×C4—48no—
150C2×C4—416no—
151C2×C4—44no—
152C2×C4—44no—
153C2×C4—44no—
154C2×C4C2×C22⋊C412yes—
155C2×C4C2×C22⋊C412yes—
156C2×C4C2×C22⋊C412yes—
157C2×C4#11718yes—
158C2×C4#12318yes—
159C2×C4#12314yes—
160C2×C4C23×C411yescharacteristic
161C2×C4C22×D411yescharacteristic
162C2×C4C2×#4114yes—
163C8—44no—
164C22—232no—
165C22—28no—
166C22—28no—
167C22—28no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22#38114yes—
175C22#41814yes—
176C22#41914yes—
177C22C22×C22⋊C411yescharacteristic
178C22C2×#12312yes—
179C22C2×AGL(1,ℤ/8)12yes—
180C22C2×#13712yes—
181C4—22no—
182C4—22no—
183C4—48no—
184C4—84no—
185C4#38114yes—
186C4C2×#11712yes—
187C4C2×#12312yes—
188C2—24no—
189C2—24no—
190C2—24no—
191C2C2×#38111yescharacteristic
192C2C2×#41811yescharacteristic
193C2C2×#41911yescharacteristic
194C2#309214yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1824C2semidirect
5. #1823C2semidirect
6. #1736C2semidirect
15. C22×C4⋊C4C4semidirect
28. #340C22semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C42
C2×#4072
#22802
#18242
#18232
#17362