Finite Group Databasebeta

← #330780

Subgroups — #330780

Order 512.

538 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 162 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#33310C211yesmaximal, characteristic
2#29983C212yesmaximal
3#21341C211yesmaximal, characteristic
4#10654C214yesmaximal
5#10620C214yesmaximal
6#9905C212yesmaximal
7#8785C211yesmaximal, characteristic
8C2×#498—21no—
9C2×#498—21no—
10#2868C2212yes—
11#2295C2211yescharacteristic
12#2099—22no—
13#2096—22no—
14#2058—22no—
15#2024—22no—
16#1984C2212yes—
17#1948C2214yes—
18#1939C2212yes—
19#1916—22no—
20#1844—22no—
21#1844C2214yes—
22#1831C2212yes—
23#1803C2214yes—
24#1768—21no—
25#1767C2211yescharacteristic
26#1746—21no—
27C42×C8C2211yescharacteristic
28#1296C2218yes—
29#1277C2214yes—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4×Q8C2311yescharacteristic
34#498—44no—
35#498—44no—
36C2×#106—24no—
37C2×#106—24no—
38C2×#106—24no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8C2314yes—
41C2×C4⋊C8C2312yes—
42C2×#103—24no—
43C2×#97—21no—
44C2×C4×C8D412yes—
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8C2312yes—
48#391D412yes—
49#389D412yes—
50#387—22no—
51#384—22no—
52#384—22no—
53#384—22no—
54#384C2314yes—
55C4×C4⋊C4—21no—
56C4×C4⋊C4C2311yescharacteristic
57#376—21no—
58C43C2311yescharacteristic
59#340—24no—
60#340—24no—
61#336—22no—
62#336—22no—
63#336—22no—
64#336—22no—
65#336—22no—
66#336—21no—
67C22×Q8—42no—
68C22×Q8C2×D411yescharacteristic
69C22×C8—22no—
70C22×C8—24no—
71C22×C8C2×D412yes—
72C22×C8C2×D412yes—
73C4⋊Q8—44no—
74C4⋊Q8—44no—
75C4⋊Q8—84no—
76C4⋊Q8—84no—
77C4×Q8—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4C2×D411yescharacteristic
87C2×C4⋊C4C2×D412yes—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42C2×D412yes—
91C2×C42C2×D411yescharacteristic
92C2×C42C2×D411yescharacteristic
93C2×C42#4112yes—
94C2×C42C2411yescharacteristic
95#106—48no—
96#106—48no—
97#106—84no—
98C4⋊C8—28no—
99C4⋊C8—24no—
100C4⋊C8—44no—
101#103—84no—
102#97—42no—
103C4×C8—24no—
104C4×C8—22no—
105C4×C8—24no—
106C4×C8SD1614yes—
107#95—22no—
108#95—22no—
109#95—22no—
110#95—21no—
111C2×Q8—22no—
112C2×Q8—24no—
113C2×Q8—84no—
114C2×Q8—84no—
115C2×Q8—84no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—44no—
120C22×C4#12112yes—
121C22×C4#12112yes—
122C22×C4C22×D411yescharacteristic
123C22×C4C22×D411yescharacteristic
124C22×C4C2×#4111yescharacteristic
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—42no—
130C2×C8—44no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8—44no—
134C2×C8—44no—
135C2×C8C2×SD1614yes—
136C2×C8#13714yes—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—42no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C4⋊C4—44no—
145C4⋊C4—84no—
146C4⋊C4—84no—
147C4⋊C4—84no—
148C4⋊C4—84no—
149C42—22no—
150C42—24no—
151C42—24no—
152C42—42no—
153C42—42no—
154C42C2×SD1612yes—
155C42AGL(1,ℤ/8)12yes—
156C422−1+414yes—
157C23C2×#12111yescharacteristic
158Q8—44no—
159Q8—168no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—84no—
173C2×C4—84no—
174C2×C4—84no—
175C2×C4#46014yes—
176C2×C4#47114yes—
177C2×C4#53612yes—
178C2×C4#53612yes—
179C2×C4#54112yes—
180C2×C4C22×SD1611yescharacteristic
181C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
182C2×C4C2×#13712yes—
183C2×C4#57714yes—
184C8—24no—
185C8—24no—
186C8—82no—
187C8—84no—
188C22#262111yescharacteristic
189C22C2×#46011yescharacteristic
190C22#303911yescharacteristic
191C22#305712yes—
192C22C2×#47112yes—
193C4—24no—
194C4—24no—
195C4—22no—
196C4—82no—
197C4—81no—
198C4—81no—
199C4—164no—
200C4#322912yes—
201C4#323912yes—
202C4#324414yes—
203C2#1920212yes—
204C2#1935712yes—
205C2#2986611yescharacteristic
206C2#2988111yescharacteristic
207C2#2988711yescharacteristic
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
#333102
#299832
#213412
#106542
#106202
#99052
#87852