Finite Group Databasebeta

← #330789

Subgroups — #330789

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#29984C212yesmaximal
3#21341C211yesmaximal, characteristic
4#10653C214yesmaximal
5#10620C214yesmaximal
6#9903C212yesmaximal
7#8785C211yesmaximal, characteristic
8C2×#498—22no—
9#2868C2212yes—
10#2295C2211yescharacteristic
11#2100—22no—
12#2095—22no—
13#2057—22no—
14#2024—22no—
15#1984C2212yes—
16#1947C2214yes—
17#1939C2212yes—
18#1916—22no—
19#1844—21no—
20#1844—21no—
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—48no—
35C2×#107—24no—
36C2×#106—24no—
37C2×#106—24no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8C2314yes—
40C2×C4⋊C8C2312yes—
41C2×#103—24no—
42C2×#97—21no—
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D412yes—
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2312yes—
47#391D412yes—
48#389D412yes—
49#387—22no—
50#384—22no—
51#384—22no—
52#384—22no—
53#384C2314yes—
54C4×C4⋊C4—21no—
55C4×C4⋊C4C2311yescharacteristic
56#376—21no—
57C43C2311yescharacteristic
58#340—24no—
59#340—24no—
60#336—22no—
61#336—22no—
62#336—22no—
63#336—22no—
64#336—22no—
65#336—21no—
66C22×Q8—42no—
67C22×Q8C2×D411yescharacteristic
68C22×C8—22no—
69C22×C8—24no—
70C22×C8C2×D412yes—
71C22×C8C2×D412yes—
72C4⋊Q8—48no—
73C4⋊Q8—84no—
74C4⋊Q8—84no—
75C4×Q8—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4C2×D411yescharacteristic
85C2×C4⋊C4C2×D412yes—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42C2×D411yescharacteristic
89C2×C42C2×D412yes—
90C2×C42C2×D411yescharacteristic
91C2×C42#4112yes—
92C2×C42C2411yescharacteristic
93#107—84no—
94#106—48no—
95#106—48no—
96C4⋊C8—28no—
97C4⋊C8—24no—
98C4⋊C8—44no—
99#103—84no—
100#97—42no—
101C4×C8—24no—
102C4×C8—22no—
103C4×C8—24no—
104C4×C8Q1614yes—
105#95—22no—
106#95—22no—
107#95—22no—
108#95—21no—
109C2×Q8—22no—
110C2×Q8—24no—
111C2×Q8—84no—
112C2×Q8—84no—
113C2×Q8—84no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—44no—
118C22×C4#12112yes—
119C22×C4#12112yes—
120C22×C4C22×D411yescharacteristic
121C22×C4C22×D411yescharacteristic
122C22×C4C2×#4111yescharacteristic
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—42no—
128C2×C8—44no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—44no—
132C2×C8—44no—
133C2×C8C2×Q1614yes—
134C2×C8AGL(1,ℤ/8)14yes—
135C4⋊C4—22no—
136C4⋊C4—24no—
137C4⋊C4—42no—
138C4⋊C4—44no—
139C4⋊C4—48no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C4⋊C4—84no—
143C4⋊C4—84no—
144C4⋊C4—84no—
145C4⋊C4—84no—
146C42—22no—
147C42—24no—
148C42—24no—
149C42—42no—
150C42—42no—
151C42C2×Q1612yes—
152C42#13712yes—
153C422−1+414yes—
154C23C2×#12111yescharacteristic
155Q8—44no—
156Q8—168no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—84no—
170C2×C4—84no—
171C2×C4—84no—
172C2×C4#46214yes—
173C2×C4#47114yes—
174C2×C4#53612yes—
175C2×C4#53612yes—
176C2×C4#54112yes—
177C2×C4C22×Q1611yescharacteristic
178C2×C4C2×AGL(1,ℤ/8)12yes—
179C2×C4C2×#13711yescharacteristic
180C2×C4#57714yes—
181C8—24no—
182C8—24no—
183C8—82no—
184C8—84no—
185C22#262111yescharacteristic
186C22C2×#46211yescharacteristic
187C22#303811yescharacteristic
188C22#305712yes—
189C22C2×#47112yes—
190C4—24no—
191C4—24no—
192C4—22no—
193C4—82no—
194C4—81no—
195C4—81no—
196C4—164no—
197C4#323012yes—
198C4#323812yes—
199C4#324414yes—
200C2#1920712yes—
201C2#1935212yes—
202C2#2986911yescharacteristic
203C2#2988111yescharacteristic
204C2#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
#299842
#213412
#106532
#106202
#99032
#87852