Finite Group Databasebeta

← #241263

Subgroups — #241263

Order 512.

764 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 202 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#37642C211yesmaximal, characteristic
2#21988C212yesmaximal
3#20762C211yesmaximal, characteristic
4#10584C218yesmaximal, complemented
5#8034C212yesmaximal
6#8031C211yesmaximal, characteristic
7C2×#432C2212yes—
8C22×#97C2211yescharacteristic
9#2844C2214yes—
10#2628—22no—
11#2628—24no—
12#2571—21no—
13#2571—22no—
14#2551—21no—
15#2551—22no—
16#2285C2211yescharacteristic
17#2285C2211yescharacteristic
18#2285C2212yes—
19#1948—22no—
20#1947—22no—
21#1854—24no—
22#1847C2214yes—
23#1836—21no—
24#1836—21no—
25#1836C2214yes—
26#1836C2214yes—
27#1735—22no—
28#1295C2214yes—
29#1295C2218yes—
30C23×C8—21no—
31C2×C4⋊Q8—24no—
32C2×C4⋊Q8—22no—
33C2×#117—21no—
34C2×#117D412yes—
35C2×#117D414yes—
36C22×C42C2311yescharacteristic
37#432—24no—
38#432Q818yescomplemented
39C2×#107—24no—
40C2×#106—24no—
41C2×C4⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×#97—21no—
44C2×#97—22no—
45C2×#97Q814yescomplemented
46C2×#97C2312yes—
47C2×#97C2314yes—
48#400—24no—
49#400—28no—
50#399—24no—
51#395—24no—
52#395—28no—
53#393—24no—
54#389—24no—
55#389—24no—
56#389—24no—
57#384D411yescharacteristic
58#384D412yes—
59#384D411yescharacteristic
60#384D412yes—
61#384C2312yes—
62#382—24no—
63#382—22no—
64#382—21no—
65#382C2312yes—
66#382C2314yes—
67#380—22no—
68#380—22no—
69#380—22no—
70#380—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75#340—24no—
76#340—24no—
77#340—24no—
78#340—24no—
79C22×Q8—24no—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D412yes—
82C22×C8—24no—
83C22×C8—22no—
84C22×C8—24no—
85C22×C8—42no—
86C22×C8—42no—
87C4⋊Q8—88no—
88C4⋊Q8—84no—
89#117—44no—
90#117—44no—
91#117—84no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4C2×D412yes—
104C2×C4⋊C4C2×D412yes—
105C2×C4⋊C4C2×D414yes—
106C2×C4⋊C4C2×D414yes—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—41no—
110C2×C42—24no—
111C2×C42—22no—
112C2×C42—41no—
113C2×C42C2×D412yes—
114C2×C42C2×D411yescharacteristic
115C2×C42C2×Q812yes—
116C2×C42C2×Q814yes—
117C2×C42#4112yes—
118C2×C42C2411yescharacteristic
119#107—84no—
120#106—84no—
121C4⋊C8—28no—
122C22⋊C8—28no—
123#97—24no—
124#97—22no—
125#97—24no—
126#97C2×Q814yes—
127#97C2×Q818yes—
128#95—24no—
129#95—28no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—24no—
134#95—42no—
135C24#12011yescharacteristic
136C2×Q8—84no—
137C2×Q8—88no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4#12012yes—
148C22×C4#12011yescharacteristic
149C22×C4C22⋊Q812yes—
150C22×C4C22⋊Q814yes—
151C22×C4C4⋊Q812yes—
152C22×C4C4⋊Q814yes—
153C22×C4C22×D411yescharacteristic
154C22×C4C22×D412yes—
155C2×C8—24no—
156C2×C8—28no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—48no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—84no—
177C4⋊C4—88no—
178C22⋊C4—82no—
179C22⋊C4—84no—
180C22⋊C4—82no—
181C42—48no—
182C42—44no—
183C42—82no—
184C42C22⋊Q812yes—
185C42C22⋊Q814yes—
186C42C4⋊Q812yes—
187C42C4⋊Q814yes—
188C42C22×Q811yescharacteristic
189C42C22×Q812yes—
190C42C2×#4111yescharacteristic
191C23—21no—
192C23—21no—
193C23—22no—
194C23—22no—
195C23#39512yes—
196C23C2×#12011yescharacteristic
197Q8—168no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4#39514yes—
222C2×C4#39512yes—
223C2×C4#45412yes—
224C2×C4#45512yes—
225C2×C4C2×C22⋊Q811yescharacteristic
226C2×C4C2×C22⋊Q812yes—
227C2×C4C2×C4⋊Q811yescharacteristic
228C2×C4C2×C4⋊Q812yes—
229C8—48no—
230C8—44no—
231C22—22no—
232C22—24no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22C2×#39511yescharacteristic
237C22C2×#45511yescharacteristic
238C22C2×#45411yescharacteristic
239C22#301912yes—
240C22#302012yes—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—22no—
245C4—24no—
246C4—82no—
247C4—84no—
248C4—82no—
249C4—84no—
250C4—162no—
251C4#206114yes—
252C2—42no—
253C2#1905611yescharacteristic
254C2#1905811yescharacteristic
255C2#1906212yes—
256C2C2×#206111yescharacteristic
257C2#2356112yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10584C2semidirect
38. #432Q8semidirect
45. C2×#97Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#219882
#207622
#105842
#80342
#80312