Finite Group Databasebeta

← #215660

Subgroups — #215660

Order 512.

864 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 222 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#20463C212yesmaximal
3#9892C218yesmaximal, complemented
4#8722C212yesmaximal
5#8721C212yesmaximal
6C2×#431—22no—
7#2864C2214yes—
8#2859C412yes—
9#2859C412yes—
10#2851—22no—
11#2851—22no—
12C4×#125—22no—
13C4×#125—22no—
14#2296C412yes—
15#2296C412yes—
16#2285C2211yescharacteristic
17#2285C2211yescharacteristic
18C2×C43C2211yescharacteristic
19#1767—22no—
20#1767C2214yes—
21#1749—22no—
22#1749C2214yes—
23#1744—22no—
24C4×#97C2214yes—
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
36#431—216no—
37C2×C4⋊C8—24no—
38C2×C22⋊C8—24no—
39C2×#97—24no—
40C2×#97C2318yes—
41C2×C4×C8—24no—
42#389—22no—
43#389—22no—
44#389—22no—
45#389—22no—
46#389C2×C414yes—
47#389C2×C414yes—
48#389C2×C414yes—
49#389C2×C414yes—
50#387—24no—
51#387—24no—
52#387—24no—
53#387—24no—
54#385—24no—
55#385—22no—
56#385—22no—
57#384C2×C411yescharacteristic
58#384C2×C411yescharacteristic
59#384C2×C412yes—
60#382—22no—
61#382—22no—
62#382C2312yes—
63#382C2312yes—
64#380—22no—
65#380—22no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C43C2×C412yes—
72C43D412yes—
73C43D412yes—
74C43C2312yes—
75#336—24no—
76#336—24no—
77C23×C4C22⋊C412yes—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D412yes—
82C22×C8—28no—
83#125—48no—
84#125—48no—
85#117—44no—
86#117—44no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4C22×C414yes—
96C2×C4⋊C4C22×C412yes—
97C2×C4⋊C4C22×C412yes—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C42—28no—
101C2×C42—24no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—42no—
105C2×C42—44no—
106C2×C42—42no—
107C2×C42C22⋊C412yes—
108C2×C42C22⋊C412yes—
109C2×C42C22⋊C412yes—
110C2×C42C22⋊C412yes—
111C2×C42C22⋊C411yescharacteristic
112C2×C42C22⋊C411yescharacteristic
113C2×C42C22⋊C412yes—
114C2×C42C22×C411yescharacteristic
115C2×C42C22×C412yes—
116C2×C42C22×C412yes—
117C2×C42C2×D412yes—
118C2×C42C2×D412yes—
119C2×C42C2×D412yes—
120C2×C42C2×D412yes—
121C2×C42C2×D412yes—
122C2×C42C2411yescharacteristic
123C4⋊C8—216no—
124C22⋊C8—216no—
125#97—216no—
126#97—48no—
127C4×C8—216no—
128#95—24no—
129#95—24no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—24no—
134C24C2×C22⋊C411yescharacteristic
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4C2×C22⋊C412yes—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×C22⋊C412yes—
154C22×C4C2×C22⋊C412yes—
155C22×C4C2×C22⋊C412yes—
156C22×C4C2×C22⋊C412yes—
157C22×C4C23×C411yescharacteristic
158C22×C4C22×D412yes—
159C2×C8—216no—
160C2×C8—48no—
161C2×C8—48no—
162C4⋊C4—44no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—48no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C4⋊C4—48no—
172C4⋊C4—48no—
173C22⋊C4—44no—
174C22⋊C4—44no—
175C42—22no—
176C42—24no—
177C42—24no—
178C42—24no—
179C42—24no—
180C42—24no—
181C42—24no—
182C42—24no—
183C42—41no—
184C42—41no—
185C42—42no—
186C42—42no—
187C42—42no—
188C42—42no—
189C42—44no—
190C42—44no—
191C42#10418yes—
192C422−1+414yes—
193C23—21no—
194C23—21no—
195C23—22no—
196C23—22no—
197C23#41312yes—
198C23C22×C22⋊C411yescharacteristic
199C2×C4—28no—
200C2×C4—216no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
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—22no—
217C2×C4—22no—
218C2×C4—48no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4#41312yes—
229C2×C4C2×#10414yes—
230C2×C4C2×#10414yes—
231C2×C4C2×#10414yes—
232C2×C4#42114yes—
233C2×C4#51912yes—
234C2×C4#53614yes—
235C8—416no—
236C22—22no—
237C22—24no—
238C22—41no—
239C22—42no—
240C22—41no—
241C22#231011yescharacteristic
242C22C2×#41311yescharacteristic
243C22C2×#41311yescharacteristic
244C22C22×#10412yes—
245C22C2×#42112yes—
246C4—22no—
247C4—22no—
248C4—24no—
249C4—24no—
250C4—24no—
251C4—24no—
252C4—24no—
253C4—42no—
254C4—82no—
255C4—82no—
256C4—84no—
257C2—42no—
258C2#1763811yescharacteristic
259C2#1764612yes—
260C2#1767311yescharacteristic
261C2#2183011yescharacteristic
262C2#2184412yes—
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. #9892C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#204632
#98922
#87222
#87212