Finite Group Databasebeta

← #324475

Subgroups — #324475

Order 512.

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

normal subgroups only

This group has 572 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#30230C211yesmaximal, characteristic
2#28725C211yesmaximal, characteristic
3#28511C211yesmaximal, characteristic
4#22238C211yesmaximal, characteristic
5#10482C214yesmaximal
6#9899C211yesmaximal, characteristic
7#9702C211yesmaximal, characteristic
8#9216C214yesmaximal
9#8277C211yesmaximal, characteristic
10C2×#491—21no—
11C2×#477C2214yes—
12C2×C8×Q8C2212yes—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2100—24no—
17#2095—22no—
18#2074—22no—
19#1992—21no—
20#1990C2212yes—
21#1990C2212yes—
22#1984C2212yes—
23#1956—21no—
24#1916—24no—
25#1845—21no—
26#1831C2211yescharacteristic
27#1803C2214yes—
28#1767C2211yescharacteristic
29C4×C4⋊C8C2211yescharacteristic
30#1744C2211yescharacteristic
31C2×#334C2212yes—
32C2×#333C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8D411yescharacteristic
37C2×C4⋊Q8C2312yes—
38C2×#125—21no—
39C2×C4×Q8C2312yes—
40#491—44no—
41#477—28no—
42#477—44no—
43C8×Q8—44no—
44C2×#107—24no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8C2312yes—
47C2×C4⋊C8C2312yes—
48C2×C4⋊C8C2312yes—
49C2×#103—24no—
50C2×#103—24no—
51C2×#97—21no—
52C2×C4×C8—22no—
53C2×C4×C8—21no—
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#391D412yes—
57#389—22no—
58#389—22no—
59#389—22no—
60#384—21no—
61#384—22no—
62#384—21no—
63#384D411yescharacteristic
64#384C2312yes—
65#382—21no—
66C4×C4⋊C4—22no—
67C43C2311yescharacteristic
68#340—24no—
69#340—24no—
70#340—24no—
71#336—24no—
72#336—21no—
73#336—22no—
74#336—21no—
75#336D414yes—
76#334—24no—
77#333—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×D411yescharacteristic
108C2×C42C2×D412yes—
109C2×C42C2×D411yescharacteristic
110C2×C42#4112yes—
111C2×C42C2411yescharacteristic
112#107—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#97—42no—
122C4×C8—22no—
123C4×C8—22no—
124C4×C8—44no—
125C4×C8—42no—
126#95—24no—
127#95—42no—
128C2×Q8—22no—
129C2×Q8—24no—
130C2×Q8—24no—
131C2×Q8—44no—
132C2×Q8—84no—
133C2×Q8C2×SD1614yes—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4#12112yes—
139C22×C4#12112yes—
140C22×C4C22×D411yescharacteristic
141C22×C4C22×D411yescharacteristic
142C22×C4C2×#4111yescharacteristic
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—24no—
146C2×C8—24no—
147C2×C8—24no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—44no—
154C2×C8—42no—
155C4⋊C4—28no—
156C4⋊C4—24no—
157C4⋊C4—44no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—44no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—44no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C4⋊C4—84no—
167C4⋊C4—84no—
168C4⋊C4—84no—
169C42—24no—
170C42—22no—
171C42—42no—
172C42—42no—
173C42—42no—
174C42C2×SD1612yes—
175C42#13512yes—
176C42C2×#4112yes—
177C42H(5,𝔽2)12yes—
178C23C2×#12111yescharacteristic
179Q8—28no—
180Q8—44no—
181Q8—164no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—84no—
197C2×C4—82no—
198C2×C4#46514yes—
199C2×C4#53211yescharacteristic
200C2×C4#53511yescharacteristic
201C2×C4#54612yes—
202C2×C4#55912yes—
203C2×C4C22×SD1611yescharacteristic
204C2×C4C2×#13511yescharacteristic
205C2×C4#57814yes—
206C2×C4#57812yes—
207C8—42no—
208C8—42no—
209C8—44no—
210C8—82no—
211C8—82no—
212C22#254411yescharacteristic
213C22#304412yes—
214C22C2×#46511yescharacteristic
215C22#305911yescharacteristic
216C22#306812yes—
217C4—24no—
218C4—22no—
219C4—42no—
220C4—44no—
221C4—82no—
222C4—162no—
223C4—162no—
224C4#317712yes—
225C4#322312yes—
226C4#329812yes—
227C4#330212yes—
228C2#1940312yes—
229C2#1953112yes—
230C2#2860011yescharacteristic
231C2#2862511yescharacteristic
232C2#2863511yescharacteristic
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
#302302
#287252
#285112
#222382
#104822
#98992
#97022
#92162
#82772