Finite Group Databasebeta

← #97036

Subgroups — #97036

Order 512.

594 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2647C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic
3#8699C211yesmaximal, characteristic
4#7593C214yesmaximal
5#2647—28no—
6C2×#400—22no—
7C2×#398—22no—
8C2×#389C412yes—
9C2×#389C2212yes—
10C2×#384C411yescharacteristic
11C2×#382C411yescharacteristic
12C2×#376C2211yescharacteristic
13#1817C2212yes—
14#1817C2212yes—
15#1280—22no—
16#1280—22no—
17#1271—28no—
18C22×C4⋊C4—22no—
19C22×C4⋊C4—22no—
20C22×C4⋊C4—22no—
21C22×C4⋊C4C2×C412yes—
22C22×C4⋊C4C2×C412yes—
23C22×C42C2×C411yescharacteristic
24C22×C42C2×C411yescharacteristic
25C22×C42C2311yescharacteristic
26C2×C22⋊C8—22no—
27C2×C22⋊C8—22no—
28#400—216no—
29#398—216no—
30#389—24no—
31#389—24no—
32#389—24no—
33#389D418yes—
34#384—24no—
35#382—24no—
36#376—24no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95D412yes—
41C2×#95D412yes—
42#336—24no—
43#336—24no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4C22⋊C412yes—
47C23×C4C22⋊C412yes—
48C23×C4C22×C411yescharacteristic
49C23×C4C2×D411yescharacteristic
50C23×C4C2×D411yescharacteristic
51C22×C8—42no—
52C22×C8—42no—
53C2×C4⋊C4—28no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—28no—
57C2×C4⋊C4—28no—
58C2×C4⋊C4—28no—
59C2×C4⋊C4—44no—
60C2×C4⋊C4—84no—
61C2×C4⋊C4—84no—
62C2×C4⋊C4—84no—
63C2×C4⋊C4C22⋊C414yes—
64C2×C4⋊C4C22⋊C414yes—
65C2×C42—24no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—24no—
69C2×C42—42no—
70C2×C42C2×D414yes—
71C22⋊C8—44no—
72C22⋊C8—44no—
73#95—28no—
74#95—28no—
75#95—28no—
76#95—24no—
77#95—24no—
78C24C2×C22⋊C411yescharacteristic
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—24no—
87C22×C4—24no—
88C22×C4—24no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—44no—
94C22×C4—44no—
95C22×C4—44no—
96C22×C4—44no—
97C22×C4—44no—
98C22×C4—44no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4#10114yes—
102C22×C4C2×C22⋊C412yes—
103C22×C4#12012yes—
104C22×C4#12012yes—
105C22×C4#13114yes—
106C2×C8—82no—
107C2×C8—82no—
108C2×C8—84no—
109C2×C8—84no—
110C4⋊C4—28no—
111C4⋊C4—44no—
112C4⋊C4—44no—
113C4⋊C4—44no—
114C4⋊C4—88no—
115C4⋊C4—88no—
116C4⋊C4—88no—
117C42—28no—
118C42—44no—
119C42—44no—
120C23—22no—
121C23—22no—
122C23#37911yescharacteristic
123C23#40812yes—
124C23#41012yes—
125C23C2×#10112yes—
126C2×C4—28no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—24no—
130C2×C4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—24no—
134C2×C4—44no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—44no—
138C2×C4—42no—
139C2×C4—44no—
140C2×C4—44no—
141C2×C4—44no—
142C2×C4—42no—
143C2×C4—44no—
144C2×C4—42no—
145C2×C4—44no—
146C2×C4—44no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—88no—
150C2×C4—88no—
151C2×C4—84no—
152C2×C4#40814yes—
153C8—162no—
154C8—162no—
155C22—22no—
156C22—22no—
157C22—22no—
158C22—22no—
159C22—24no—
160C22#147014yes—
161C22#178011yescharacteristic
162C22#178211yescharacteristic
163C22#178411yescharacteristic
164C22#179512yes—
165C22#179612yes—
166C4—24no—
167C4—42no—
168C4—42no—
169C4—44no—
170C4—84no—
171C4—82no—
172C4—84no—
173C2—24no—
174C2#830611yescharacteristic
175C2#832512yes—
176C2#851811yescharacteristic
177C2#852111yescharacteristic
178C2#852512yes—
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#26472
#86992
#75932