Finite Group Databasebeta

← #215673

Subgroups — #215673

Order 512.

760 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 206 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#36640C211yesmaximal, characteristic
2#20778C212yesmaximal
3#9893C218yesmaximal, complemented
4#8722C212yesmaximal
5#8721C212yesmaximal
6#2864C2214yes—
7#2859C412yes—
8#2859C412yes—
9#2851—22no—
10#2851—22no—
11C4×#125—22no—
12C4×#125—22no—
13#2296C412yes—
14#2296C412yes—
15#2285C2211yescharacteristic
16#2285C2211yescharacteristic
17C2×C43C2211yescharacteristic
18#1839—22no—
19#1831—22no—
20#1817—22no—
21#1797—22no—
22#1767C2214yes—
23#1749C2214yes—
24#1745C2214yes—
25#1295C2218yes—
26#1295C2218yes—
27C2×#125—22no—
28C2×#125—22no—
29C2×#117C2×C412yes—
30C2×#117C2×C412yes—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D412yes—
34C22×C42D412yes—
35C22×C42C2311yescharacteristic
36C2×C4⋊C8—24no—
37C2×C22⋊C8—24no—
38C2×#97C2318yes—
39#389—22no—
40#389—22no—
41#389—22no—
42#389—22no—
43#389C2×C414yes—
44#389C2×C414yes—
45#389C2×C414yes—
46#389C2×C414yes—
47#387—24no—
48#387—24no—
49#387—24no—
50#387—24no—
51#385—24no—
52#385—22no—
53#385—22no—
54#384C2×C411yescharacteristic
55#384C2×C411yescharacteristic
56#384C2×C412yes—
57#382—22no—
58#382—22no—
59#382C2312yes—
60#382C2312yes—
61#380—22no—
62#380—22no—
63C4×C4⋊C4—24no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C43C2×C412yes—
69C43D412yes—
70C43D412yes—
71C43C2312yes—
72#336—24no—
73#336—24no—
74#336—24no—
75#336—24no—
76C23×C4C22⋊C412yes—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D412yes—
81C22×C8—28no—
82#125—48no—
83#125—48no—
84#117—44no—
85#117—44no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4C22×C414yes—
95C2×C4⋊C4C22×C412yes—
96C2×C4⋊C4C22×C412yes—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C42—24no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—42no—
103C2×C42—44no—
104C2×C42—44no—
105C2×C42—42no—
106C2×C42C22⋊C412yes—
107C2×C42C22⋊C412yes—
108C2×C42C22⋊C412yes—
109C2×C42C22⋊C412yes—
110C2×C42C22⋊C412yes—
111C2×C42C22⋊C411yescharacteristic
112C2×C42C22⋊C411yescharacteristic
113C2×C42C22×C411yescharacteristic
114C2×C42C22×C412yes—
115C2×C42C22×C412yes—
116C2×C42C2×D412yes—
117C2×C42C2×D412yes—
118C2×C42C2×D412yes—
119C2×C42C2×D412yes—
120C2×C42C2×D412yes—
121C2×C42C2411yescharacteristic
122C4⋊C8—48no—
123C22⋊C8—48no—
124#97—216no—
125#95—24no—
126#95—24no—
127#95—24no—
128#95—24no—
129#95—24no—
130#95—24no—
131C24C2×C22⋊C411yescharacteristic
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4C2×C22⋊C412yes—
149C22×C4C2×C22⋊C412yes—
150C22×C4C2×C22⋊C412yes—
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C2×C22⋊C412yes—
153C22×C4C2×C22⋊C412yes—
154C22×C4C23×C411yescharacteristic
155C22×C4C22×D412yes—
156C2×C8—48no—
157C2×C8—48no—
158C2×C8—48no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—48no—
165C4⋊C4—48no—
166C4⋊C4—48no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—48no—
170C22⋊C4—44no—
171C22⋊C4—44no—
172C42—22no—
173C42—24no—
174C42—24no—
175C42—24no—
176C42—24no—
177C42—24no—
178C42—24no—
179C42—41no—
180C42—41no—
181C42—42no—
182C42—42no—
183C42—42no—
184C42—42no—
185C42—42no—
186C42—42no—
187C42—44no—
188C42—44no—
189C422−1+414yes—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23#41312yes—
195C23C22×C22⋊C411yescharacteristic
196C2×C4—28no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—48no—
217C2×C4—48no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4#41312yes—
228C2×C4#42114yes—
229C2×C4#42114yes—
230C2×C4#51912yes—
231C2×C4#53614yes—
232C8—88no—
233C22—24no—
234C22—22no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22#231011yescharacteristic
239C22C2×#41311yescharacteristic
240C22C2×#41311yescharacteristic
241C22C2×#42112yes—
242C22C2×#42112yes—
243C4—24no—
244C4—22no—
245C4—22no—
246C4—24no—
247C4—24no—
248C4—42no—
249C4—42no—
250C4—42no—
251C4—82no—
252C4—82no—
253C4—84no—
254C2—42no—
255C2#1766511yescharacteristic
256C2#1767311yescharacteristic
257C2#1767712yes—
258C2#2183011yescharacteristic
259C2#2184512yes—
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. #9893C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#207782
#98932
#87222
#87212