Finite Group Databasebeta

← #237784

Subgroups — #237784

Order 512.

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

normal subgroups only

This group has 862 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#34638C211yesmaximal, characteristic
2#21838C212yesmaximal
3#14170C211yesmaximal, characteristic
4#10347C214yesmaximal
5#10346C214yesmaximal
6#8022C212yesmaximal
7#7910C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9C2×#432C2212yes—
10#2647—24no—
11#2646—22no—
12#2646C2214yes—
13#2402—24no—
14#2296—22no—
15#2285C2212yes—
16C2×#376C2211yescharacteristic
17C2×#376C2211yescharacteristic
18#1992—24no—
19#1989—24no—
20#1824C2214yes—
21#1823C2214yes—
22#1445C2212yes—
23#1445C2212yes—
24#1436—22no—
25#1436—22no—
26#1294C2218yes—
27#1280C2214yes—
28C2×#125—22no—
29C2×#125—22no—
30C2×#125—22no—
31C2×#125—24no—
32C2×#125Q814yes—
33C2×#117—22no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4C2311yescharacteristic
37C22×C4⋊C4C2311yescharacteristic
38C22×C42—21no—
39C22×C42D412yes—
40C22×C42C2311yescharacteristic
41#432—24no—
42#432—24no—
43C2×C4⋊C8—22no—
44C2×C22⋊C8C2314yes—
45C2×#97—22no—
46#400—28no—
47#400—28no—
48#398—28no—
49#398—28no—
50#389—24no—
51#389—24no—
52#385—28no—
53#384D414yes—
54#382—22no—
55#380—22no—
56#380C2314yes—
57C4×C4⋊C4—28no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4D414yes—
60#376—22no—
61#376—22no—
62#376—24no—
63C2×#95—24no—
64C2×#95—24no—
65C2×#95—21no—
66C2×#95D412yes—
67C2×#95C2312yes—
68C2×#95C2312yes—
69#340—28no—
70#340—28no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D414yes—
76C23×C4C2×D411yescharacteristic
77C23×C4C2411yescharacteristic
78C22×C8—24no—
79#125—28no—
80#125—28no—
81#125—44no—
82#125—44no—
83#117—48no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C2×D418yes—
102C2×C4⋊C4C2×Q812yes—
103C2×C4⋊C4C2×Q812yes—
104C2×C22⋊C4—24no—
105C2×C42—24no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—44no—
109C2×C42—44no—
110C2×C42—42no—
111C2×C42C2×D412yes—
112C2×C42C2×D412yes—
113C2×C42C2×Q812yes—
114C2×C42#4114yes—
115C2×C42#4112yes—
116C4⋊C8—44no—
117C22⋊C8—24no—
118C22⋊C8—24no—
119#97—44no—
120#95—28no—
121#95—28no—
122#95—24no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—24no—
127#95—48no—
128#95—48no—
129#95—42no—
130C24C22×D411yescharacteristic
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—28no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4#12012yes—
154C22×C4#12011yescharacteristic
155C22×C4#12011yescharacteristic
156C22×C4C22⋊Q812yes—
157C22×C4C22⋊Q812yes—
158C22×C4#12412yes—
159C22×C4#12412yes—
160C22×C4C22×D412yes—
161C22×C4C22×Q811yescharacteristic
162C22×C4C2×#4112yes—
163C22×C4C2×#4111yescharacteristic
164C2×C8—48no—
165C2×C8—44no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—48no—
169C4⋊C4—48no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—48no—
175C4⋊C4—84no—
176C4⋊C4—88no—
177C4⋊C4—88no—
178C22⋊C4—48no—
179C42—44no—
180C42—44no—
181C42—48no—
182C42—48no—
183C42C22⋊Q814yes—
184C42C22⋊Q814yes—
185C42#12414yes—
186C42#12414yes—
187C23—22no—
188C23—22no—
189C23C2×#12011yescharacteristic
190C23C2×C22⋊Q811yescharacteristic
191C23C2×#12411yescharacteristic
192C23#57514yes—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—48no—
206C2×C4—48no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—48no—
213C2×C4—48no—
214C2×C4—44no—
215C2×C4—48no—
216C2×C4—44no—
217C2×C4—84no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—84no—
225C2×C4#39312yes—
226C2×C4#39312yes—
227C2×C4#39314yes—
228C2×C4C2×C22⋊Q812yes—
229C2×C4C2×#12412yes—
230C8—84no—
231C22—24no—
232C22—24no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22C2×#39311yescharacteristic
238C22#300512yes—
239C22#301812yes—
240C22#302812yes—
241C22#308012yes—
242C22#314212yes—
243C4—22no—
244C4—22no—
245C4—24no—
246C4—22no—
247C4—22no—
248C4—84no—
249C4—82no—
250C4—88no—
251C4—84no—
252C4—82no—
253C2—22no—
254C2—22no—
255C2#1834114yes—
256C2#2345311yescharacteristic
257C2#2346111yescharacteristic
258C2#2347011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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
#346382
#218382
#141702
#103472
#103462
#80222
#79102