Finite Group Databasebeta

← #324463

Subgroups — #324463

Order 512.

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

normal subgroups only

This group has 578 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#29150C211yesmaximal, characteristic
3#28512C211yesmaximal, characteristic
4#22238C211yesmaximal, characteristic
5#10487C214yesmaximal
6#9897C211yesmaximal, characteristic
7#9703C211yesmaximal, characteristic
8#9220C214yesmaximal
9#8278C211yesmaximal, characteristic
10C2×#475C2214yes—
11C2×C8×Q8C2212yes—
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2096—22no—
16#2043—24no—
17#2026—22no—
18#1984C2212yes—
19#1960—21no—
20#1956—21no—
21#1956C2212yes—
22#1956C2212yes—
23#1916—24no—
24#1844—21no—
25#1836—21no—
26#1831C2211yescharacteristic
27#1803C2214yes—
28#1767C2211yescharacteristic
29C4×C4⋊C8C2211yescharacteristic
30#1744C2211yescharacteristic
31C2×#335C2212yes—
32C2×#332C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8D411yescharacteristic
37C2×C4⋊Q8C2312yes—
38C2×#125—21no—
39C2×C4×Q8C2312yes—
40#475—28no—
41#475—44no—
42C8×Q8—44no—
43C2×#106—24no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8C2312yes—
46C2×C4⋊C8C2312yes—
47C2×C4⋊C8C2312yes—
48C2×#103—24no—
49C2×#103—24no—
50C2×#103—24no—
51C2×#103—24no—
52C2×#97—21no—
53C2×C4×C8—22no—
54C2×C4×C8—21no—
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#391D412yes—
58#389—22no—
59#389—22no—
60#389—22no—
61#384—21no—
62#384—21no—
63#384—22no—
64#384D411yescharacteristic
65#384C2312yes—
66#382—21no—
67C4×C4⋊C4—22no—
68C43C2311yescharacteristic
69#340—24no—
70#336—24no—
71#336—22no—
72#336—21no—
73#336—21no—
74#336D414yes—
75#335—24no—
76#332—44no—
77C22×Q8—22no—
78C22×Q8C2×D412yes—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8C2×D412yes—
83C22×C8C2×D412yes—
84C4⋊Q8—22no—
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—84no—
88C4⋊Q8Q1614yes—
89#125—82no—
90C4×Q8—24no—
91C4×Q8—24no—
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—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C2×D412yes—
104C2×C42—22no—
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C42#4112yes—
109C2×C42C2411yescharacteristic
110#106—48no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—24no—
115C4⋊C8—44no—
116C4⋊C8#4114yes—
117#103—48no—
118#103—84no—
119#103—84no—
120#103—84no—
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—42no—
132C2×Q8—42no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×Q8C2×Q1614yes—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4#12112yes—
141C22×C4#12112yes—
142C22×C4C22×D411yescharacteristic
143C22×C4C22×D411yescharacteristic
144C22×C4C2×#4111yescharacteristic
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—24no—
148C2×C8—24no—
149C2×C8—24no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—44no—
157C4⋊C4—28no—
158C4⋊C4—24no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C4⋊C4—84no—
167C4⋊C4—84no—
168C4⋊C4—84no—
169C42—22no—
170C42—24no—
171C42—42no—
172C42—42no—
173C42—42no—
174C42C2×Q1612yes—
175C42#13512yes—
176C42C2×#4112yes—
177C42H(5,𝔽2)12yes—
178C23C2×#12111yescharacteristic
179Q8—28no—
180Q8—44no—
181Q8—84no—
182Q8—84no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4#46714yes—
201C2×C4#53211yescharacteristic
202C2×C4#53511yescharacteristic
203C2×C4#54612yes—
204C2×C4#55912yes—
205C2×C4C22×Q1611yescharacteristic
206C2×C4C2×#13511yescharacteristic
207C2×C4#57714yes—
208C2×C4#57712yes—
209C8—42no—
210C8—42no—
211C8—44no—
212C8—82no—
213C8—82no—
214C22#254411yescharacteristic
215C22#304312yes—
216C22C2×#46711yescharacteristic
217C22#305711yescharacteristic
218C22#306912yes—
219C4—24no—
220C4—22no—
221C4—42no—
222C4—44no—
223C4—82no—
224C4—162no—
225C4—162no—
226C4#317412yes—
227C4#322112yes—
228C4#329912yes—
229C4#330312yes—
230C2#1939912yes—
231C2#1952112yes—
232C2#2859811yescharacteristic
233C2#2862311yescharacteristic
234C2#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
#291502
#285122
#222382
#104872
#98972
#97032
#92202
#82782