Finite Group Databasebeta

← #375736

Subgroups — #375736

Order 512.

1256 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 258 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
1C2×#2646C211yesmaximal, characteristic
2#13796C212yesmaximal
3#11353C214yesmaximal
4#10912C218yesmaximal
5#2646—28no—
6C2×#400C414yescomplemented
7C2×#398C414yescomplemented
8C2×#385C2214yes—
9C2×#380C2212yes—
10C2×C4×C22⋊C4C2214yes—
11C22×#95C2211yescharacteristic
12C2×#342C22116yes—
13#1434—28no—
14C22⋊#95C2214yes—
15C22⋊#95C2214yes—
16C24×C4C2311yescharacteristic
17C24×C4C2312yes—
18C22×C4⋊C4C2×C412yes—
19C22×C4⋊C4C2×C414yes—
20C22×C4⋊C4C2×C414yes—
21C22×C4⋊C4C2×C412yes—
22C22×C22⋊C4C2314yes—
23C22×C22⋊C4C2318yes—
24C22×C42—24no—
25#400—216no—
26#398—216no—
27#385—28no—
28#385—28no—
29#380—24no—
30#380—24no—
31C4×C22⋊C4—216no—
32C2×#95—24no—
33C2×#95—24no—
34C2×#95—24no—
35C2×#95—24no—
36C2×#95—24no—
37C2×#95C2×C414yes—
38C2×#95C2×C414yes—
39C2×#95C2×C414yes—
40C2×#95C2×C414yes—
41C2×#95D418yes—
42#342—232no—
43C25C2411yescharacteristic
44C23×C4—28no—
45C23×C4—24no—
46C23×C4—28no—
47C23×C4—82no—
48C23×C4—82no—
49C23×C4—82no—
50C23×C4C22×C414yes—
51C23×C4C22×C414yes—
52C23×C4C22×C412yes—
53C23×C4C22×C414yes—
54C23×C4C2×D414yes—
55C23×C4C2×D414yes—
56C23×C4C2×D414yes—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—28no—
60C2×C4⋊C4—28no—
61C2×C4⋊C4—82no—
62C2×C4⋊C4—82no—
63C2×C4⋊C4—84no—
64C2×C4⋊C4—84no—
65C2×C22⋊C4—216no—
66C2×C22⋊C4—216no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—216no—
70C2×C22⋊C4—28no—
71C2×C22⋊C4#41116yes—
72C2×C42—216no—
73C2×C42—84no—
74C2×C42—84no—
75#95—216no—
76#95—28no—
77#95—28no—
78#95—28no—
79#95—84no—
80#95—84no—
81#95—84no—
82#95—84no—
83#95C22⋊C4116yes—
84C24—41no—
85C24—42no—
86C24—41no—
87C24C23×C411yescharacteristic
88C24C22×D412yes—
89C24C2×#4118yes—
90C242−1+414yes—
91C22×C4—24no—
92C22×C4—24no—
93C22×C4—24no—
94C22×C4—28no—
95C22×C4—28no—
96C22×C4—28no—
97C22×C4—24no—
98C22×C4—24no—
99C22×C4—216no—
100C22×C4—28no—
101C22×C4—28no—
102C22×C4—44no—
103C22×C4—44no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—44no—
107C22×C4—44no—
108C22×C4—44no—
109C22×C4—44no—
110C22×C4—44no—
111C22×C4—44no—
112C22×C4—48no—
113C22×C4—48no—
114C22×C4—48no—
115C22×C4—48no—
116C22×C4—48no—
117C22×C4—48no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—816no—
123C22×C4—88no—
124C22×C4C2×C22⋊C414yes—
125C22×C4C2×C22⋊C418yes—
126C4⋊C4—84no—
127C4⋊C4—84no—
128C4⋊C4—88no—
129C4⋊C4—88no—
130C22⋊C4—232no—
131C22⋊C4—48no—
132C22⋊C4—48no—
133C22⋊C4—416no—
134C42—816no—
135C23—22no—
136C23—21no—
137C23—22no—
138C23—22no—
139C23—22no—
140C23—21no—
141C23—22no—
142C23—48no—
143C23—44no—
144C23—48no—
145C23—44no—
146C23C22×C22⋊C411yescharacteristic
147C23C4×#4114yes—
148C23#51912yes—
149C23#52514yes—
150C23#53214yes—
151C23#53614yes—
152C23#552116yes—
153C2×C4—24no—
154C2×C4—24no—
155C2×C4—24no—
156C2×C4—28no—
157C2×C4—28no—
158C2×C4—28no—
159C2×C4—48no—
160C2×C4—48no—
161C2×C4—48no—
162C2×C4—416no—
163C2×C4—416no—
164C2×C4—416no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—88no—
172C2×C4—88no—
173C2×C4—88no—
174C2×C4—88no—
175C2×C4—88no—
176C2×C4—88no—
177C2×C4—88no—
178C2×C4—88no—
179C2×C4—88no—
180C22—22no—
181C22—24no—
182C22—24no—
183C22—22no—
184C22—24no—
185C22—24no—
186C22—24no—
187C22—24no—
188C22—24no—
189C22—24no—
190C22—416no—
191C22#229712yes—
192C22#231011yescharacteristic
193C22#237114yes—
194C22#261118yes—
195C22#288312yes—
196C22#288311yescharacteristic
197C22#288411yescharacteristic
198C4—44no—
199C4—48no—
200C4—816no—
201C4—84no—
202C4—84no—
203C2—28no—
204C2—24no—
205C2#1417011yescharacteristic
206C2#2142212yes—
207C2#2142312yes—
208C2#2182711yescharacteristic
209C2#2182811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C2×#400C4semidirect
7. C2×#398C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
#137962
#113532
#109122