Finite Group Databasebeta

← #241319

Subgroups — #241319

Order 512.

792 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 218 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#37643C211yesmaximal, characteristic
2#21964C212yesmaximal
3#20760C211yesmaximal, characteristic
4#10582C214yesmaximal, complemented
5#10581C214yesmaximal, complemented
6#8041C211yesmaximal, characteristic
7#7897C212yesmaximal
8C2×#423C2212yes—
9C22×C4×C8C2211yescharacteristic
10#2845—22no—
11#2629—24no—
12#2629—22no—
13#2552—22no—
14#2552—22no—
15#2552C2214yes—
16#2285C2211yescharacteristic
17#2285C2212yes—
18#2285C2211yescharacteristic
19#1948—22no—
20#1948C2214yes—
21#1947—22no—
22#1947C2214yes—
23#1835—22no—
24#1834C2212yes—
25#1834C2212yes—
26#1809—24no—
27#1735—22no—
28#1296C2214yes—
29#1279C2218yes—
30C22×M16—22no—
31C23×C8—21no—
32C2×#125—22no—
33C2×#125—24no—
34C2×#117—22no—
35C2×#117—21no—
36C2×#117D412yes—
37C22×C42C2311yescharacteristic
38C2×#107—24no—
39C2×#106—24no—
40#423—24no—
41#423Q818yes—
42C2×C4⋊C8—22no—
43C2×C4⋊C8C2314yes—
44C2×C4×C8—21no—
45C2×C4×C8—22no—
46C2×C4×C8Q814yes—
47C2×C4×C8C2312yes—
48#398—28no—
49#398—28no—
50#398—28no—
51#396—28no—
52#389—24no—
53#389—24no—
54#389—24no—
55#384—22no—
56#384—24no—
57#384D412yes—
58#384D412yes—
59#384C2311yescharacteristic
60#384C2311yescharacteristic
61#384C2312yes—
62#382—22no—
63#382—21no—
64#382D412yes—
65#380—22no—
66#380—22no—
67#380—22no—
68#380D414yes—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2314yes—
73#340—24no—
74#340—24no—
75#340—24no—
76#340—24no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C2×M16—28no—
80C2×M16—82no—
81C2×M16—82no—
82C22×C8—24no—
83C22×C8—22no—
84C22×C8—24no—
85C22×C8—42no—
86C22×C8—42no—
87#125—84no—
88#125—84no—
89#125—84no—
90#117—44no—
91#117—44no—
92#117—82no—
93#117—82no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4C2×D414yes—
108C2×C4⋊C4C2×D414yes—
109C2×C4⋊C4C2×D414yes—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—41no—
113C2×C42—22no—
114C2×C42—24no—
115C2×C42—41no—
116C2×C42C2×D412yes—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×Q812yes—
119C2×C42C2×Q814yes—
120C2×C42#4112yes—
121C2×C42C2411yescharacteristic
122#107—84no—
123#106—84no—
124C4⋊C8—28no—
125C4⋊C8—24no—
126C4⋊C8C2×Q818yes—
127C4×C8—24no—
128C4×C8—22no—
129C4×C8C2×Q814yes—
130#95—28no—
131#95—24no—
132#95—24no—
133#95—24no—
134#95—42no—
135C24#12011yescharacteristic
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4#12012yes—
148C22×C4#12011yescharacteristic
149C22×C4C22⋊Q812yes—
150C22×C4C22⋊Q814yes—
151C22×C4C4⋊Q812yes—
152C22×C4C4⋊Q814yes—
153C22×C4C22×D411yescharacteristic
154C22×C4C22×D412yes—
155M16—88no—
156C2×C8—24no—
157C2×C8—28no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—48no—
161C2×C8—44no—
162C2×C8—44no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—44no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—88no—
183C4⋊C4—88no—
184C4⋊C4—88no—
185C22⋊C4—82no—
186C22⋊C4—84no—
187C22⋊C4—82no—
188C42—44no—
189C42—44no—
190C42—44no—
191C42—82no—
192C42C22⋊Q812yes—
193C42C22⋊Q814yes—
194C42C4⋊Q812yes—
195C42C4⋊Q814yes—
196C42C22×Q811yescharacteristic
197C42C22×Q812yes—
198C42C2×#4111yescharacteristic
199C23—22no—
200C23—22no—
201C23—21no—
202C23—21no—
203C23#39512yes—
204C23C2×#12011yescharacteristic
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4#39514yes—
229C2×C4#39512yes—
230C2×C4#45412yes—
231C2×C4#45512yes—
232C2×C4C2×C22⋊Q811yescharacteristic
233C2×C4C2×C22⋊Q812yes—
234C2×C4C2×C4⋊Q811yescharacteristic
235C2×C4C2×C4⋊Q812yes—
236C2×C4#57612yes—
237C2×C4#57712yes—
238C2×C4#57712yes—
239C2×C4#57812yes—
240C8—44no—
241C8—48no—
242C22—24no—
243C22—22no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22C2×#39511yescharacteristic
248C22#301212yes—
249C22#301312yes—
250C22C2×#45511yescharacteristic
251C22C2×#45411yescharacteristic
252C4—24no—
253C4—21no—
254C4—21no—
255C4—84no—
256C4—84no—
257C4—84no—
258C4—161no—
259C4—161no—
260C4#206114yes—
261C4#308112yes—
262C4#308212yes—
263C4#316312yes—
264C4#316412yes—
265C2—42no—
266C2#1869212yes—
267C2#1869611yescharacteristic
268C2#1869811yescharacteristic
269C2#2355712yes—
270C2C2×#206111yescharacteristic
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. #10582C2semidirect
5. #10581C2semidirect

Maximal subgroups

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

subgroupindex
#376432
#219642
#207602
#105822
#105812
#80412
#78972