Finite Group Databasebeta

← #213910

Subgroups — #213910

Order 512.

954 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 190 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#20729C212yesmaximal
3#9514C218yesmaximal, complemented
4#8751C212yesmaximal
5#8548C212yesmaximal
6C22×#123C2211yescharacteristic
7C2×#518C2211yescharacteristic
8#2921C2214yes—
9#2484C2214yes—
10#2483C412yes—
11#2483C412yes—
12#2419—24no—
13#2419—24no—
14#2315C412yes—
15#2308C2211yescharacteristic
16C2×#381C412yes—
17#1822C2214yes—
18#1800—24no—
19#1783C2214yes—
20C4×#100—24no—
21#1463C2218yes—
22#1306C2218yes—
23C23×D4C2311yescharacteristic
24C24×C4C2×C411yescharacteristic
25C2×#123—24no—
26C2×#123—44no—
27#518—44no—
28C2×C4×D4D414yes—
29C2×C4×D4D414yes—
30C2×#117C2312yes—
31C22×C4⋊C4C2312yes—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4C2×C412yes—
34C22×C22⋊C4C2×C412yes—
35C22×C22⋊C4C2×C412yes—
36C22×C22⋊C4C2×C412yes—
37#433—48no—
38#423—24no—
39C2×#100—28no—
40#407C2318yes—
41C4×M16—24no—
42#400—24no—
43#398—24no—
44#397—24no—
45#397—24no—
46#387—24no—
47#387—24no—
48#386—24no—
49#385—24no—
50#385—22no—
51#385—22no—
52#385C2×C414yes—
53#385C2×C414yes—
54#381—22no—
55#381—22no—
56#381—22no—
57#381—22no—
58#381C2×C414yes—
59#381C2×C414yes—
60#381C2312yes—
61#380—22no—
62#380—22no—
63#379C2×C411yescharacteristic
64#379C2×C411yescharacteristic
65C2×#95—24no—
66#343—28no—
67#324—48no—
68C25C22×C411yescharacteristic
69C25C22×C411yescharacteristic
70C22×D4—24no—
71C22×D4—44no—
72C22×D4C2×D414yes—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—42no—
76C23×C4—44no—
77C23×C4C22⋊C414yes—
78C23×C4C22⋊C412yes—
79C23×C4C22⋊C412yes—
80C23×C4C22×C412yes—
81C23×C4C22×C412yes—
82C23×C4C2411yescharacteristic
83C2×M16—28no—
84#123—416no—
85C4×D4—48no—
86C4×D4—48no—
87#117—28no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—28no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4C2×D414yes—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—44no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4—44no—
104C2×C22⋊C4C22⋊C412yes—
105C2×C22⋊C4C22⋊C412yes—
106C2×C22⋊C4C22⋊C414yes—
107C2×C22⋊C4C22×C414yes—
108C2×C22⋊C4C22×C412yes—
109C2×C22⋊C4C22×C412yes—
110C2×C42C2×D412yes—
111C2×C42C2×D412yes—
112C4⋊C8—48no—
113#100—416no—
114C22⋊C8—216no—
115#97—44no—
116C4×C8—44no—
117#95—24no—
118#95—24no—
119#95—24no—
120#95—24no—
121#95—24no—
122#95—24no—
123#95—24no—
124C24—22no—
125C24—24no—
126C24—22no—
127C24—42no—
128C24—82no—
129C24—82no—
130C24C2×C22⋊C412yes—
131C24C2×C22⋊C412yes—
132C24C23×C411yescharacteristic
133C2×D4—28no—
134C2×D4—28no—
135C2×D4—44no—
136C2×D4—416no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—82no—
153C22×C4—84no—
154C22×C4—84no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4—88no—
158C22×C4C2×C22⋊C414yes—
159C22×C4C2×C22⋊C414yes—
160C22×C4C22×D412yes—
161C22×C4H(5,𝔽2)14yes—
162M16—416no—
163C2×C8—48no—
164C4⋊C4—28no—
165C4⋊C4—48no—
166C4⋊C4—84no—
167C4⋊C4—84no—
168C22⋊C4—28no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—44no—
172C22⋊C4—44no—
173C22⋊C4—44no—
174C22⋊C4—44no—
175C22⋊C4—44no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—82no—
179C22⋊C4—82no—
180C42—24no—
181C42—24no—
182C23—28no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—24no—
188C23—42no—
189C23—44no—
190C23—44no—
191C23—42no—
192C23—42no—
193C23—84no—
194C23—84no—
195C23—88no—
196C23—164no—
197C23#41014yes—
198C23C22×C22⋊C411yescharacteristic
199C23#51812yes—
200D4—416no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—48no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—88no—
217C2×C4—84no—
218C2×C4—164no—
219C2×C4—162no—
220C2×C4—162no—
221C2×C4—164no—
222C2×C4#53414yes—
223C8—88no—
224C22—22no—
225C22—24no—
226C22—22no—
227C22—24no—
228C22—44no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—84no—
236C22—88no—
237C22—164no—
238C22#230811yescharacteristic
239C22C2×#41012yes—
240C4—24no—
241C4—44no—
242C4—44no—
243C4—82no—
244C4—82no—
245C4—82no—
246C4—82no—
247C2—22no—
248C2—42no—
249C2—44no—
250C2—82no—
251C2#2174711yescharacteristic
252C2#2177912yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9514C2semidirect

Maximal subgroups

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

subgroupindex
#354482
#207292
#95142
#87512
#85482