Finite Group Databasebeta

← #215685

Subgroups — #215685

Order 512.

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

normal subgroups only

This group has 926 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
1C2×#2859C211yesmaximal, characteristic
2#20777C212yesmaximal
3#9901C218yesmaximal, complemented
4#8725C212yesmaximal
5#8723C212yesmaximal
6C22×#125C412yes—
7#2859—22no—
8#2859—22no—
9#2859C414yes—
10#2859C2214yes—
11C2×#389—22no—
12C2×#389—22no—
13C2×#384C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×#382C412yes—
16C2×C43C2211yescharacteristic
17#1839—22no—
18#1831C2214yes—
19#1817—22no—
20#1797C2214yes—
21C4×C4⋊C8C2214yes—
22C4×C22⋊C8—22no—
23#1744—22no—
24#1296C2218yes—
25#1296C2218yes—
26C2×#125—24no—
27C2×#125—48no—
28C2×#125C2×C418yes—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4C2×C412yes—
35C22×C42C2×C411yescharacteristic
36C22×C42C2×C411yescharacteristic
37C22×C42D412yes—
38C22×C42D412yes—
39C22×C42C2311yescharacteristic
40C2×C4⋊C8C2318yes—
41C2×C22⋊C8—24no—
42C2×C4×C8—24no—
43#389—24no—
44#389—24no—
45#389—24no—
46#389—24no—
47#389—24no—
48#389—24no—
49#389—24no—
50#389—24no—
51#384—22no—
52#384—22no—
53#384C2×C412yes—
54#384C2×C412yes—
55#384C2312yes—
56#384C2312yes—
57#382—22no—
58#382—22no—
59#382C2×C418yes—
60C2×#95—24no—
61C43C2×C412yes—
62C43D412yes—
63C43D412yes—
64C43C2312yes—
65#336—24no—
66#336—24no—
67#336—24no—
68#336—24no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C412yes—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D412yes—
76C22×C8—28no—
77#125—416no—
78#125—416no—
79C2×C4⋊C4—28no—
80C2×C4⋊C4—28no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—48no—
90C2×C4⋊C4—48no—
91C2×C4⋊C4—48no—
92C2×C4⋊C4—48no—
93C2×C4⋊C4C22×C414yes—
94C2×C4⋊C4C22×C414yes—
95C2×C42—24no—
96C2×C42—28no—
97C2×C42—24no—
98C2×C42—24no—
99C2×C42—44no—
100C2×C42C22⋊C412yes—
101C2×C42C22⋊C412yes—
102C2×C42C22⋊C412yes—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C22⋊C412yes—
105C2×C42C22⋊C412yes—
106C2×C42C22⋊C411yescharacteristic
107C2×C42C22×C412yes—
108C2×C42C22×C411yescharacteristic
109C2×C42C22×C412yes—
110C2×C42C2×D412yes—
111C2×C42C2×D412yes—
112C2×C42C2×D412yes—
113C2×C42C2×D412yes—
114C2×C42C2×D412yes—
115C2×C42C2411yescharacteristic
116C4⋊C8—216no—
117C22⋊C8—48no—
118C4×C8—48no—
119#95—28no—
120#95—28no—
121C24C2×C22⋊C411yescharacteristic
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—48no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—48no—
139C22×C4#10014yes—
140C22×C4C2×C22⋊C412yes—
141C22×C4C2×C22⋊C412yes—
142C22×C4C2×C22⋊C412yes—
143C22×C4C2×C22⋊C412yes—
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C2×C22⋊C412yes—
146C22×C4#13518yes—
147C22×C4C23×C411yescharacteristic
148C22×C4C22×D412yes—
149C2×C8—48no—
150C2×C8—48no—
151C2×C8—48no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—48no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—416no—
158C4⋊C4—48no—
159C4⋊C4—48no—
160C4⋊C4—416no—
161C4⋊C4—48no—
162C42—22no—
163C42—24no—
164C42—24no—
165C42—24no—
166C42—24no—
167C42—24no—
168C42—24no—
169C42—24no—
170C42—22no—
171C42—24no—
172C42—24no—
173C42—24no—
174C42—42no—
175C42—42no—
176C42#10114yes—
177C422−1+414yes—
178C23—21no—
179C23—21no—
180C23—22no—
181C23C2×#10012yes—
182C23#41614yes—
183C23C22×C22⋊C411yescharacteristic
184C2×C4—28no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—216no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—44no—
203C2×C4—48no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—48no—
208C2×C4—48no—
209C2×C4—88no—
210C2×C4—88no—
211C2×C4C2×#10014yes—
212C2×C4C2×#10114yes—
213C2×C4C2×#10112yes—
214C2×C4#41614yes—
215C2×C4#51912yes—
216C2×C4#53614yes—
217C2×C4C2×#13514yes—
218C2×C4#57514yes—
219C8—88no—
220C22—24no—
221C22—22no—
222C22—24no—
223C22—22no—
224C22#154314yes—
225C22#231011yescharacteristic
226C22C22×#10011yescharacteristic
227C22C22×#10111yescharacteristic
228C22C2×#41612yes—
229C22#289612yes—
230C4—24no—
231C4—22no—
232C4—22no—
233C4—24no—
234C4—24no—
235C4—42no—
236C4—42no—
237C4—84no—
238C4—84no—
239C4#323218yes—
240C2—24no—
241C2C2×#154311yescharacteristic
242C2#1771512yes—
243C2#1775411yescharacteristic
244C2#2182911yescharacteristic
245C2#2183612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9901C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
#207772
#99012
#87252
#87232