Finite Group Databasebeta

← #213933

Subgroups — #213933

Order 512.

753 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 753 conjugacy classes of subgroups — too many to draw. Only its 186 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#36959C211yesmaximal, characteristic
2#21832C212yesmaximal
3#9514C214yesmaximal
4#9514C214yesmaximal
5#8755C212yesmaximal
6#8554C212yesmaximal
7C2×#544C2211yescharacteristic
8C22×#125C2211yescharacteristic
9#2921C2214yes—
10#2855C412yes—
11#2719—24no—
12#2713—24no—
13#2627C412yes—
14#2583C412yes—
15#2503C412yes—
16#2484C2212yes—
17#2484C2212yes—
18#2313C2211yescharacteristic
19#1998—24no—
20#1997—24no—
21#1822C2214yes—
22#1821C2214yes—
23#1463C2214yes—
24#1463C2214yes—
25#1306C2218yes—
26#544—44no—
27C2×C4⋊Q8D414yes—
28C2×#125—24no—
29C2×#125—44no—
30C2×#125D414yes—
31C2×C22⋊Q8C2×C412yes—
32C2×C22⋊Q8C2×C412yes—
33C2×#117C2312yes—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2311yescharacteristic
38C22×C4⋊C4C2311yescharacteristic
39C22×C4⋊C4C2311yescharacteristic
40C22×C42C2×C411yescharacteristic
41#433—48no—
42#423—24no—
43#407C2318yes—
44C4×M16—24no—
45#400—22no—
46#400—24no—
47#400—22no—
48#400—22no—
49#398—22no—
50#398—24no—
51#398—22no—
52#398—22no—
53#395—22no—
54#395—22no—
55#391—22no—
56#391—22no—
57#389—24no—
58#389—22no—
59#387—24no—
60#387—24no—
61#387—24no—
62#387—24no—
63#385C2×C414yes—
64#385C2×C414yes—
65#382—22no—
66#381C2×C414yes—
67#381C2×C414yes—
68#381C2312yes—
69#380C2×C412yes—
70C4×C22⋊C4C2×C412yes—
71#343—28no—
72#343—24no—
73#343—24no—
74#324—44no—
75#324—44no—
76C22×Q8C22⋊C412yes—
77C22×Q8C22⋊C412yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2411yescharacteristic
85C2×M16—28no—
86C4⋊Q8—44no—
87C4⋊Q8—44no—
88#125—44no—
89#125—44no—
90#125—48no—
91#125—48no—
92C22⋊Q8—44no—
93C22⋊Q8—44no—
94#117—28no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4C22⋊C412yes—
110C2×C4⋊C4C22⋊C412yes—
111C2×C4⋊C4C22⋊C414yes—
112C2×C4⋊C4C22⋊C414yes—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C22⋊C4C22×C414yes—
117C2×C22⋊C4C22×C414yes—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—42no—
121C2×C42C2×D414yes—
122C4⋊C8—48no—
123C22⋊C8—28no—
124C22⋊C8—28no—
125#97—44no—
126C4×C8—44no—
127#95—22no—
128#95—22no—
129#95—24no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—24no—
134#95—24no—
135#95—22no—
136#95—22no—
137C24C23×C411yescharacteristic
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—82no—
141C2×Q8—82no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—81no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4C2×C22⋊C414yes—
161C22×C4C2×C22⋊C414yes—
162C22×C4C2×C22⋊C412yes—
163C22×C4C2×C22⋊C412yes—
164C22×C4C22×D412yes—
165C22×C4H(5,𝔽2)14yes—
166M16—416no—
167C2×C8—48no—
168C4⋊C4—24no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—48no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—82no—
183C4⋊C4—84no—
184C22⋊C4—28no—
185C22⋊C4—44no—
186C42—24no—
187C42—24no—
188C42—44no—
189C42—82no—
190C23—22no—
191C23—41no—
192C23—41no—
193C23C22×C22⋊C411yescharacteristic
194C23#51812yes—
195Q8—162no—
196Q8—162no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—44no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4#53414yes—
227C8—88no—
228C22—21no—
229C22—21no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—44no—
234C22#230811yescharacteristic
235C22#288412yes—
236C4—24no—
237C4—42no—
238C4—42no—
239C4—44no—
240C4—42no—
241C4—44no—
242C4—84no—
243C4—81no—
244C4—81no—
245C4—81no—
246C2—22no—
247C2—42no—
248C2#2175111yescharacteristic
249C2#2177911yescharacteristic
250C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#369592
#218322
#95142
#87552
#85542