Finite Group Databasebeta

← #17398

Subgroups — #17398

Order 256.

517 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 179 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#2920C212yesmaximal, complemented
3#1473C218yesmaximal, complemented
4#1463C212yesmaximal, complemented
5#1462C212yesmaximal
6C2×#126—22no—
7C2×#126—22no—
8C2×#126—22no—
9C2×#126—22no—
10C2×#126C414yes—
11C2×#126C2214yes—
12C22×C4⋊C4C411yescharacteristic
13C22×C4⋊C4C411yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C22⋊C4C411yescharacteristic
16C22×C22⋊C4C411yescharacteristic
17C22×C22⋊C4C2211yescharacteristic
18C22×C42C2211yescharacteristic
19#432—22no—
20#432C2214yescomplemented
21#431—22no—
22#431C2214yescomplemented
23#423—22no—
24#407C2214yes—
25C4×M16—22no—
26#324C2218yescomplemented
27#323C2218yes—
28C25C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4C2×C411yescharacteristic
31C23×C4C2×C411yescharacteristic
32C23×C4C2×C411yescharacteristic
33C23×C4C2×C411yescharacteristic
34C23×C4C2311yescharacteristic
35C2×M16—24no—
36#126—24no—
37#126—24no—
38#126—24no—
39#126—24no—
40#126—24no—
41#126—24no—
42#126—24no—
43#126—24no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4C2×C414yes—
53C2×C4⋊C4C2×C414yes—
54C2×C4⋊C4C2×C412yes—
55C2×C4⋊C4C2312yes—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4C2×C414yes—
65C2×C22⋊C4C2×C414yes—
66C2×C22⋊C4C2×C412yes—
67C2×C22⋊C4C2312yes—
68C2×C42C2×C412yes—
69C2×C42D412yes—
70C2×C42D412yes—
71C2×C42D412yes—
72C2×C42D412yes—
73C2×C42C2312yes—
74C4⋊C8—24no—
75C4⋊C8—24no—
76C22⋊C8—24no—
77C22⋊C8C2318yes—
78#97—24no—
79C4×C8—24no—
80C24—21no—
81C24—21no—
82C24—42no—
83C24—42no—
84C24—42no—
85C24C22×C411yescharacteristic
86C24C22×C412yes—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4C22⋊C412yes—
100C22×C4C22⋊C411yescharacteristic
101C22×C4C22⋊C412yes—
102C22×C4C22⋊C411yescharacteristic
103C22×C4C22⋊C411yescharacteristic
104C22×C4C22⋊C411yescharacteristic
105C22×C4C22×C412yes—
106C22×C4C22×C412yes—
107C22×C4C22×C412yes—
108C22×C4C22×C412yes—
109C22×C4C22×C411yescharacteristic
110C22×C4C22×C412yes—
111C22×C4C2×D412yes—
112C22×C4C2×D412yes—
113C22×C4C2411yescharacteristic
114M16—48no—
115C2×C8—28no—
116C4⋊C4—24no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C22⋊C4—24no—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—24no—
127C22⋊C4—24no—
128C42C22⋊C412yes—
129C42C22⋊C412yes—
130C42C22⋊C412yes—
131C42C22⋊C412yes—
132C42C2×D412yes—
133C42C2×D412yes—
134C42C2×D412yes—
135C42C2×D412yes—
136C23—22no—
137C23—22no—
138C23—22no—
139C23—22no—
140C23—24no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23—88no—
150C23C2×C22⋊C411yescharacteristic
151C23C2×C22⋊C411yescharacteristic
152C23C23×C411yescharacteristic
153C2×C4—24no—
154C2×C4—24no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4C2×C22⋊C412yes—
178C2×C4C2×C22⋊C412yes—
179C2×C4C2×C22⋊C412yes—
180C2×C4C2×C22⋊C412yes—
181C2×C4C2×C22⋊C412yes—
182C2×C4C22×D412yes—
183C8—48no—
184C22—28no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—84no—
195C22—84no—
196C22—84no—
197C22#41012yes—
198C22#41312yes—
199C22C22×C22⋊C411yescharacteristic
200C22#56812yes—
201C4—22no—
202C4—22no—
203C4—22no—
204C4—22no—
205C4—22no—
206C4—22no—
207C4—42no—
208C4—42no—
209C4—42no—
210C2—22no—
211C2—22no—
212C2—22no—
213C2—82no—
214C2#287911yescharacteristic
215C2C2×#41011yescharacteristic
216C2C2×#41311yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC22×C4C24
prime radicalC25C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2920C2semidirect
3. #1473C2semidirect
4. #1463C2semidirect
20. #432C22semidirect
22. #431C22semidirect
26. #324C22semidirect

Maximal subgroups

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

subgroupindex
C22×#1262
#29202
#14732
#14632
#14622