Finite Group Databasebeta

← #375667

Subgroups — #375667

Order 512.

1300 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 250 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#35448C211yesmaximal, characteristic
2#20392C212yesmaximal
3#11166C212yesmaximal
4#10966C212yesmaximal
5#10936C214yesmaximal, complemented
6#10915C214yesmaximal, complemented
7C22×#123C2211yescharacteristic
8C2×#518C2211yescharacteristic
9C22×#99C2214yes—
10#2484C414yescomplemented
11#2483C414yescomplemented
12#2419—28no—
13#2315C2212yes—
14#2308C2211yescharacteristic
15C2×#381C2212yes—
16C2×#381C2214yes—
17#2273C2214yes—
18C4×#99—28no—
19C2×#342C2218yes—
20C2×#342C2214yes—
21C2×#342C2214yes—
22C23×D4C2311yescharacteristic
23C24×C4C2311yescharacteristic
24C2×#123—24no—
25C2×#123—44no—
26#518—44no—
27C2×C4×D4D418yes—
28C2×#117C2×C412yes—
29C22×C4⋊C4C2×C412yes—
30C22×C22⋊C4C2×C412yes—
31C22×C22⋊C4C2×C412yes—
32C22×C22⋊C4C2311yescharacteristic
33C22×C22⋊C4C2312yes—
34C22×C22⋊C4C2318yes—
35C22×C22⋊C4C2312yes—
36C2×#99—28no—
37C2×#99—216no—
38#400—24no—
39#398—24no—
40#397—28no—
41#387—28no—
42#386—24no—
43#385—24no—
44#385—24no—
45#385C2×C418yes—
46#381—24no—
47#381—24no—
48#381—216no—
49#381C2×C412yes—
50#381C2×C418yes—
51#380—24no—
52#379C2×C412yes—
53C4×C22⋊C4—28no—
54C4×C22⋊C4—48no—
55C2×#95—24no—
56C2×#95—44no—
57#342—28no—
58#342—216no—
59#342—44no—
60C25C22×C411yescharacteristic
61C25C2411yescharacteristic
62C22×D4—28no—
63C22×D4—24no—
64C22×D4C2×D414yes—
65C23×C4—22no—
66C23×C4—28no—
67C23×C4—24no—
68C23×C4—22no—
69C23×C4—28no—
70C23×C4C22⋊C414yes—
71C23×C4C22×C412yes—
72C23×C4C22×C411yescharacteristic
73C23×C4C22×C412yes—
74C23×C4C2×D414yes—
75#123—416no—
76C4×D4—416no—
77#117—44no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—28no—
80C2×C4⋊C4—44no—
81C2×C4⋊C4C22⋊C414yes—
82C2×C22⋊C4—28no—
83C2×C22⋊C4—28no—
84C2×C22⋊C4—28no—
85C2×C22⋊C4—28no—
86C2×C22⋊C4—28no—
87C2×C22⋊C4—28no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—28no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—44no—
95C2×C22⋊C4—44no—
96C2×C22⋊C4C22⋊C414yes—
97C2×C22⋊C4C22×C414yes—
98C2×C22⋊C4C22×C414yes—
99C2×C22⋊C4C2×D414yes—
100C2×C22⋊C4#41116yes—
101C2×C42—48no—
102C2×C42C22⋊C414yes—
103#99—232no—
104#95—28no—
105#95—28no—
106#95—28no—
107#95—24no—
108#95—416no—
109#95—48no—
110C24—22no—
111C24—24no—
112C24—82no—
113C24—82no—
114C24C2×C22⋊C412yes—
115C24C23×C411yescharacteristic
116C24C22×D412yes—
117C24C2×#4118yes—
118C24H(5,𝔽2)14yes—
119C2×D4—28no—
120C2×D4—28no—
121C2×D4—28no—
122C2×D4—44no—
123C2×D4—48no—
124C2×D4—44no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—28no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—28no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—216no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—48no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—48no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—48no—
145C22×C4—82no—
146C22×C4—84no—
147C22×C4—84no—
148C22×C4—84no—
149C22×C4—816no—
150C22×C4C2×C22⋊C412yes—
151C22×C4C2×C22⋊C418yes—
152C4⋊C4—44no—
153C4⋊C4—48no—
154C4⋊C4—88no—
155C22⋊C4—28no—
156C22⋊C4—28no—
157C22⋊C4—28no—
158C22⋊C4—216no—
159C22⋊C4—216no—
160C22⋊C4—44no—
161C22⋊C4—48no—
162C22⋊C4—48no—
163C22⋊C4—44no—
164C22⋊C4—44no—
165C22⋊C4—48no—
166C22⋊C4—84no—
167C42—44no—
168C42—168no—
169C23—22no—
170C23—24no—
171C23—44no—
172C23—48no—
173C23—42no—
174C23—44no—
175C23—44no—
176C23—84no—
177C23—88no—
178C23—164no—
179C23C22×C22⋊C411yescharacteristic
180C23C4×#4114yes—
181C23#51812yes—
182C23#52514yes—
183C23#53214yes—
184C23#53414yes—
185C23#55114yes—
186C23#55318yes—
187C23#55914yes—
188D4—416no—
189C2×C4—22no—
190C2×C4—28no—
191C2×C4—24no—
192C2×C4—44no—
193C2×C4—48no—
194C2×C4—48no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—416no—
199C2×C4—48no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—88no—
203C2×C4—816no—
204C2×C4—88no—
205C2×C4—84no—
206C2×C4—816no—
207C2×C4—164no—
208C2×C4—164no—
209C2×C4—168no—
210C22—22no—
211C22—28no—
212C22—24no—
213C22—44no—
214C22—44no—
215C22—48no—
216C22—44no—
217C22—44no—
218C22—44no—
219C22—44no—
220C22—42no—
221C22—84no—
222C22—164no—
223C22#229712yes—
224C22#230811yescharacteristic
225C22#237414yes—
226C22#260218yes—
227C4—48no—
228C4—42no—
229C4—84no—
230C4—816no—
231C4—82no—
232C4—82no—
233C2—22no—
234C2—28no—
235C2—24no—
236C2—82no—
237C2#1416711yescharacteristic
238C2#2177712yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10936C2semidirect
6. #10915C2semidirect
10. #2484C4semidirect
11. #2483C4semidirect

Maximal subgroups

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

subgroupindex
#354482
#203922
#111662
#109662
#109362
#109152