Finite Group Databasebeta

← #303021

Subgroups — #303021

Order 512.

799 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 799 conjugacy classes of subgroups — too many to draw. Only its 172 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#59864C211yesmaximal, characteristic
2#29319C212yesmaximal, complemented
3#29143C211yesmaximal, characteristic
4#10578C214yesmaximal, complemented
5#9869C212yesmaximal, complemented
6#8921C214yesmaximal
7#8231C211yesmaximal, characteristic
8#2859C2211yescharacteristic
9#2804—22no—
10#2580—24no—
11#2571C2212yes—
12#2563—22no—
13#2487—22no—
14#2366C2212yes—
15D4×C4⋊C4—22no—
16C4×C4⋊Q8C2212yes—
17C4×#127C2211yescharacteristic
18#2084—22no—
19#2057—22no—
20#2025—22no—
21#1957C2212yes—
22#1948C2214yes—
23#1947C2214yes—
24#1943—24no—
25#1834C2212yes—
26#1804—22no—
27#1767C2212yes—
28C4×C4⋊C8C2211yescharacteristic
29C2×#331C2212yes—
30#1296C2212yes—
31#1279C2218yes—
32C2×C4⋊Q8D412yes—
33C2×#127C2311yescharacteristic
34C2×#125—21no—
35C2×C4×Q8—22no—
36C2×C4×D4—24no—
37C2×C4×D4—22no—
38C2×C4×D4D412yes—
39C22×C4⋊C4—22no—
40C2×#107—24no—
41C2×#106—24no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8C2314yes—
44C2×#102—24no—
45C2×#97—22no—
46C2×C4×C8C2312yes—
47#400—24no—
48#399—24no—
49#393—28no—
50#391—22no—
51#390—24no—
52#390—22no—
53#390D412yes—
54#389—24no—
55#389—22no—
56#389—22no—
57#389—22no—
58#389D414yes—
59#386—24no—
60#385—24no—
61#385—28no—
62#384D412yes—
63#384C2312yes—
64#384C2311yescharacteristic
65#382—24no—
66#382—21no—
67#380—22no—
68#380—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2314yes—
71C43C2311yescharacteristic
72#340—24no—
73#340—24no—
74#340—24no—
75#336—22no—
76#336—21no—
77#331Q818yescomplemented
78C22×Q8—22no—
79C22×D4—22no—
80C22×D4C2×D412yes—
81C23×C4—24no—
82C23×C4—22no—
83C22×C8—24no—
84C22×C8—22no—
85C4⋊Q8—44no—
86#127—22no—
87#127C2×Q814yes—
88#125—82no—
89C4×Q8—82no—
90C4×Q8—82no—
91C4×D4—48no—
92C4×D4—44no—
93C4×D4—84no—
94C4×D4—84no—
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—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—84no—
106C2×C4⋊C4C2×D414yes—
107C2×C4⋊C4C2×D414yes—
108C2×C4⋊C4C2×D412yes—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C42—24no—
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D412yes—
117C2×C42C2×D411yescharacteristic
118C2×C42C2411yescharacteristic
119#107—48no—
120#106—84no—
121C4⋊C8—24no—
122C4⋊C8—42no—
123C4⋊C8C2×Q818yes—
124#102—84no—
125#97—82no—
126C4×C8—24no—
127#95—24no—
128#95—28no—
129#95—24no—
130#95—24no—
131#95—41no—
132#95—41no—
133C24—22no—
134C2×Q8—44no—
135C2×Q8—82no—
136C2×Q8—82no—
137C2×D4—22no—
138C2×D4—28no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—44no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4C4⋊Q814yes—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—48no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4—84no—
152C22×C4—88no—
153C22×C4#12012yes—
154C22×C4#12011yescharacteristic
155C22×C4#12011yescharacteristic
156C22×C4C22×D411yescharacteristic
157C22×C4C22×D412yes—
158C2×C8—28no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—42no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—84no—
177C4⋊C4—82no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C22⋊C4—48no—
181C22⋊C4—82no—
182C22⋊C4—84no—
183C22⋊C4—88no—
184C42—24no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42—41no—
189C42—44no—
190C42—44no—
191C42C22×Q812yes—
192C422−1+412yes—
193C23—24no—
194C23—44no—
195C23—44no—
196C23—42no—
197C23—42no—
198C23C2×#12011yescharacteristic
199Q8—88no—
200D4—44no—
201D4—48no—
202D4—48no—
203D4—82no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—88no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—168no—
225C2×C4#45512yes—
226C2×C4C2×C4⋊Q811yescharacteristic
227C2×C4#53611yescharacteristic
228C2×C4D4×Q814yes—
229C2×C4#57514yes—
230C2×C4#57612yes—
231C2×C4#57712yes—
232C8—48no—
233C8—82no—
234C22—44no—
235C22—48no—
236C22—82no—
237C22—82no—
238C22—84no—
239C22#243411yescharacteristic
240C22#300512yes—
241C22#301212yes—
242C22C2×#45511yescharacteristic
243C22#302111yescharacteristic
244C4—22no—
245C4—21no—
246C4—21no—
247C4—41no—
248C4—42no—
249C4—84no—
250C4—82no—
251C4—84no—
252C4—162no—
253C4#338814yes—
254C4#338914yes—
255C2—84no—
256C2#1853111yescharacteristic
257C2#1869111yescharacteristic
258C2#1885612yes—
259C2#2582311yescharacteristic
260C2#2584512yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29319C2semidirect
4. #10578C2semidirect
5. #9869C2semidirect
77. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#598642
#293192
#291432
#105782
#98692
#89212
#82312