Finite Group Databasebeta

← #391736

Subgroups — #391736

Order 512.

1212 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 234 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#20633C212yesmaximal
3#11205C214yesmaximal, complemented
4#11166C212yesmaximal
5#11070C214yesmaximal, complemented
6#10966C212yesmaximal
7C22×#123C2211yescharacteristic
8C2×#518C2211yescharacteristic
9C22×#99C2214yes—
10#2484C414yescomplemented
11#2483C414yescomplemented
12#2419—28no—
13#2315C2212yes—
14#2315C2214yes—
15#2308C2211yescharacteristic
16C2×#381C2212yes—
17C2×#381C2214yes—
18#1799—28no—
19C2×#342C2214yes—
20C2×#342C2218yes—
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⋊C4C2318yes—
33C22×C22⋊C4C2311yescharacteristic
34C22×C22⋊C4C2312yes—
35C22×C22⋊C4C2312yes—
36C2×#99—28no—
37C2×#99—48no—
38#400—24no—
39#398—24no—
40#397—28no—
41#387—28no—
42#386—24no—
43#385—24no—
44#385—24no—
45#385—48no—
46#385C2×C418yes—
47#381—24no—
48#381—24no—
49#381—48no—
50#381C2×C412yes—
51#381C2×C418yes—
52#380—24no—
53#380—28no—
54#379C2×C412yes—
55C2×#95—24no—
56C2×#95—44no—
57#342—28no—
58#342—48no—
59#342—44no—
60C25C22×C411yescharacteristic
61C25C2411yescharacteristic
62C22×D4—24no—
63C22×D4—24no—
64C22×D4—24no—
65C22×D4C2×D414yes—
66C23×C4—22no—
67C23×C4—24no—
68C23×C4—24no—
69C23×C4—24no—
70C23×C4—22no—
71C23×C4—28no—
72C23×C4C22⋊C414yes—
73C23×C4C22×C412yes—
74C23×C4C22×C411yescharacteristic
75C23×C4C22×C412yes—
76C23×C4C2×D414yes—
77#123—416no—
78C4×D4—416no—
79#117—44no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—28no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—48no—
84C2×C4⋊C4C22⋊C414yes—
85C2×C22⋊C4—28no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—28no—
93C2×C22⋊C4—28no—
94C2×C22⋊C4—28no—
95C2×C22⋊C4—28no—
96C2×C22⋊C4—28no—
97C2×C22⋊C4—28no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—44no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4C22⋊C414yes—
102C2×C22⋊C4C22×C414yes—
103C2×C22⋊C4C22×C414yes—
104C2×C22⋊C4C2×D414yes—
105C2×C42C22⋊C414yes—
106#99—416no—
107#95—28no—
108#95—28no—
109#95—28no—
110#95—24no—
111#95—48no—
112#95—416no—
113C24—22no—
114C24—24no—
115C24—82no—
116C24—82no—
117C24C2×C22⋊C412yes—
118C24C23×C411yescharacteristic
119C24C22×D412yes—
120C24H(5,𝔽2)14yes—
121C24H(5,𝔽2)14yes—
122C242−1+414yes—
123C2×D4—28no—
124C2×D4—24no—
125C2×D4—24no—
126C2×D4—28no—
127C2×D4—44no—
128C2×D4—48no—
129C2×D4—44no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—28no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—28no—
138C22×C4—24no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—48no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—48no—
150C22×C4—48no—
151C22×C4—48no—
152C22×C4—82no—
153C22×C4—84no—
154C22×C4—84no—
155C22×C4—84no—
156C22×C4—816no—
157C22×C4C2×C22⋊C412yes—
158C22×C4C2×C22⋊C418yes—
159C4⋊C4—44no—
160C4⋊C4—48no—
161C4⋊C4—88no—
162C4⋊C4—168no—
163C22⋊C4—28no—
164C22⋊C4—28no—
165C22⋊C4—28no—
166C22⋊C4—44no—
167C22⋊C4—48no—
168C22⋊C4—44no—
169C22⋊C4—44no—
170C22⋊C4—48no—
171C22⋊C4—48no—
172C22⋊C4—48no—
173C22⋊C4—48no—
174C22⋊C4—84no—
175C42—44no—
176C23—22no—
177C23—24no—
178C23—44no—
179C23—48no—
180C23—42no—
181C23—44no—
182C23—44no—
183C23—84no—
184C23—84no—
185C23—84no—
186C23—164no—
187C23C22×C22⋊C411yescharacteristic
188C23#51812yes—
189C23#52014yes—
190C23#53414yes—
191C23#53512yes—
192C23#53612yes—
193C23#53714yes—
194C23#56014yes—
195C23#56218yes—
196C23#56314yes—
197D4—48no—
198D4—48no—
199C2×C4—22no—
200C2×C4—28no—
201C2×C4—24no—
202C2×C4—44no—
203C2×C4—48no—
204C2×C4—48no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—48no—
211C2×C4—88no—
212C2×C4—84no—
213C2×C4—84no—
214C2×C4—84no—
215C2×C4—816no—
216C2×C4—816no—
217C2×C4—88no—
218C2×C4—164no—
219C2×C4—162no—
220C2×C4—162no—
221C2×C4—168no—
222C22—22no—
223C22—28no—
224C22—24no—
225C22—44no—
226C22—42no—
227C22—42no—
228C22—48no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—44no—
233C22—42no—
234C22—84no—
235C22—164no—
236C22#230811yescharacteristic
237C22#231112yes—
238C22#237614yes—
239C22#281314yes—
240C22#281414yes—
241C4—48no—
242C4—42no—
243C4—84no—
244C4—82no—
245C4—82no—
246C4—168no—
247C2—22no—
248C2—24no—
249C2—24no—
250C2—24no—
251C2—82no—
252C2#1465811yescharacteristic
253C2#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
3. #11205C2semidirect
5. #11070C2semidirect
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
#206332
#112052
#111662
#110702
#109662