Finite Group Databasebeta

← #324473

Subgroups — #324473

Order 512.

588 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 158 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#30220C211yesmaximal, characteristic
2#28725C211yesmaximal, characteristic
3#28512C211yesmaximal, characteristic
4#22238C211yesmaximal, characteristic
5#10486C214yesmaximal
6#9899C211yesmaximal, characteristic
7#9703C211yesmaximal, characteristic
8#9220C214yesmaximal
9#8277C211yesmaximal, characteristic
10C2×#491—21no—
11C2×#475C2214yes—
12C2×C8×Q8C2212yes—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2096—22no—
17#2074—22no—
18#2043—24no—
19#1992—21no—
20#1984C2212yes—
21#1956—21no—
22#1956C2212yes—
23#1956C2212yes—
24#1916—24no—
25#1844—21no—
26#1831C2211yescharacteristic
27#1803C2214yes—
28#1767C2211yescharacteristic
29C4×C4⋊C8C2211yescharacteristic
30#1744C2211yescharacteristic
31C2×#334C2212yes—
32C2×#332C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8D411yescharacteristic
37C2×C4⋊Q8C2312yes—
38C2×#125—21no—
39C2×C4×Q8C2312yes—
40#491—44no—
41#475—28no—
42#475—44no—
43C8×Q8—44no—
44C2×#106—24no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8C2312yes—
47C2×C4⋊C8C2312yes—
48C2×C4⋊C8C2312yes—
49C2×#103—24no—
50C2×#103—24no—
51C2×#103—24no—
52C2×#103—24no—
53C2×#97—21no—
54C2×C4×C8—22no—
55C2×C4×C8—21no—
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#391D412yes—
59#389—22no—
60#389—22no—
61#389—22no—
62#384—21no—
63#384—21no—
64#384—22no—
65#384D411yescharacteristic
66#384C2312yes—
67#382—21no—
68C4×C4⋊C4—22no—
69C43C2311yescharacteristic
70#340—24no—
71#336—24no—
72#336—21no—
73#336—22no—
74#336—21no—
75#336D414yes—
76#334—24no—
77#332—44no—
78C22×Q8—22no—
79C22×Q8C2×D412yes—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8C2×D412yes—
84C22×C8C2×D412yes—
85C4⋊Q8—22no—
86C4⋊Q8—24no—
87C4⋊Q8—24no—
88C4⋊Q8—84no—
89C4⋊Q8SD1614yes—
90#125—42no—
91#125—42no—
92C4×Q8—24no—
93C4×Q8—24no—
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—22no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4C2×D412yes—
106C2×C42—22no—
107C2×C42C2×D412yes—
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42#4112yes—
111C2×C42C2411yescharacteristic
112#106—48no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—24no—
117C4⋊C8—44no—
118C4⋊C8#4114yes—
119#103—48no—
120#103—48no—
121#103—84no—
122#103—84no—
123#97—42no—
124C4×C8—22no—
125C4×C8—22no—
126C4×C8—44no—
127C4×C8—42no—
128#95—24no—
129#95—42no—
130C2×Q8—22no—
131C2×Q8—24no—
132C2×Q8—24no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8C2×SD1614yes—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4#12112yes—
143C22×C4#12112yes—
144C22×C4C22×D411yescharacteristic
145C22×C4C22×D411yescharacteristic
146C22×C4C2×#4111yescharacteristic
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—24no—
150C2×C8—24no—
151C2×C8—24no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—44no—
158C2×C8—42no—
159C4⋊C4—28no—
160C4⋊C4—24no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—44no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—84no—
170C4⋊C4—84no—
171C4⋊C4—84no—
172C42—24no—
173C42—22no—
174C42—42no—
175C42—42no—
176C42—42no—
177C42C2×SD1612yes—
178C42#13512yes—
179C42C2×#4112yes—
180C42H(5,𝔽2)12yes—
181C23C2×#12111yescharacteristic
182Q8—28no—
183Q8—44no—
184Q8—84no—
185Q8—84no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4#46514yes—
204C2×C4#53211yescharacteristic
205C2×C4#53511yescharacteristic
206C2×C4#54612yes—
207C2×C4#55912yes—
208C2×C4C22×SD1611yescharacteristic
209C2×C4C2×#13511yescharacteristic
210C2×C4#57714yes—
211C2×C4#57812yes—
212C8—42no—
213C8—42no—
214C8—44no—
215C8—82no—
216C8—82no—
217C22#254411yescharacteristic
218C22#304312yes—
219C22C2×#46511yescharacteristic
220C22#305911yescharacteristic
221C22#306912yes—
222C4—24no—
223C4—22no—
224C4—42no—
225C4—44no—
226C4—82no—
227C4—162no—
228C4—162no—
229C4#317412yes—
230C4#322112yes—
231C4#329812yes—
232C4#330212yes—
233C2#1939712yes—
234C2#1952312yes—
235C2#2860011yescharacteristic
236C2#2862311yescharacteristic
237C2#2863411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#302202
#287252
#285122
#222382
#104862
#98992
#97032
#92202
#82772