Finite Group Databasebeta

← #97469

Subgroups — #97469

Order 512.

772 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 98 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#35674C211yesmaximal, characteristic
2#11382C212yesmaximal
3#7632C214yesmaximal
4C22×C4⋊Q8C2211yescharacteristic
5C2×#519C411yescharacteristic
6#2484—24no—
7#2446—28no—
8#2310C411yescharacteristic
9#2310C412yes—
10#2296C2212yes—
11C2×#343—24no—
12#1728C2214yes—
13#1271—28no—
14C23×Q8C2×C411yescharacteristic
15C23×Q8C2×C411yescharacteristic
16C22×M16—24no—
17C2×C4⋊Q8—24no—
18C2×C4⋊Q8—44no—
19#519—44no—
20C2×C4×Q8—24no—
21C2×#117—22no—
22C2×#117C2×C412yes—
23C2×#117C2×C412yes—
24C22×C4⋊C4D414yes—
25C22×C42C2311yescharacteristic
26#400—28no—
27#398—28no—
28#391—24no—
29#391—24no—
30#391—24no—
31#389D418yes—
32#387—216no—
33#385—24no—
34#381—24no—
35#381—22no—
36C4×C4⋊C4—24no—
37#343—48no—
38#336—28no—
39C22×Q8—24no—
40C22×Q8—22no—
41C22×Q8—22no—
42C22×Q8—42no—
43C22×Q8—44no—
44C22×Q8—44no—
45C23×C4C22⋊C412yes—
46C23×C4C22⋊C412yes—
47C23×C4C22×C411yescharacteristic
48C23×C4C2×D412yes—
49C2×M16—28no—
50C2×M16—44no—
51C2×M16—48no—
52C2×M16—48no—
53C22×C8—44no—
54C4⋊Q8—416no—
55C4×Q8—416no—
56#117—44no—
57#117—44no—
58#117—44no—
59#117—44no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—28no—
62C2×C4⋊C4—28no—
63C2×C4⋊C4—28no—
64C2×C4⋊C4—44no—
65C2×C4⋊C4C22⋊C414yes—
66C2×C4⋊C4C22⋊C414yes—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C42—24no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—42no—
73C2×C42C2×D414yes—
74#95—28no—
75#95—216no—
76#95—28no—
77C24C2×C22⋊C411yescharacteristic
78C2×Q8—28no—
79C2×Q8—416no—
80C2×Q8—44no—
81C2×Q8—44no—
82C2×Q8—44no—
83C2×Q8—44no—
84C2×Q8—44no—
85C2×Q8—88no—
86C22×C4—24no—
87C22×C4—22no—
88C22×C4—24no—
89C22×C4—24no—
90C22×C4—24no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—24no—
96C22×C4—24no—
97C22×C4—44no—
98C22×C4—44no—
99C22×C4#9914yes—
100C22×C4C2×C22⋊C412yes—
101C22×C4#12014yes—
102M16—416no—
103M16—88no—
104C2×C8—48no—
105C2×C8—84no—
106C2×C8—84no—
107C4⋊C4—28no—
108C4⋊C4—416no—
109C4⋊C4—44no—
110C4⋊C4—48no—
111C4⋊C4—48no—
112C4⋊C4—48no—
113C22⋊C4—44no—
114C22⋊C4—44no—
115C42—44no—
116C42—44no—
117C42—44no—
118C42—48no—
119C23—24no—
120C23—22no—
121C23#37911yescharacteristic
122C23C2×#9912yes—
123Q8—416no—
124Q8—88no—
125C2×C4—28no—
126C2×C4—24no—
127C2×C4—24no—
128C2×C4—28no—
129C2×C4—24no—
130C2×C4—22no—
131C2×C4—24no—
132C2×C4—22no—
133C2×C4—24no—
134C2×C4—44no—
135C2×C4—44no—
136C2×C4—44no—
137C2×C4—48no—
138C2×C4—48no—
139C2×C4—44no—
140C2×C4—42no—
141C2×C4—44no—
142C2×C4—44no—
143C2×C4—84no—
144C2×C4#41314yes—
145C2×C4#45414yes—
146C2×C4#45614yes—
147C8—88no—
148C22—24no—
149C22—22no—
150C22—44no—
151C22#178111yescharacteristic
152C22#178512yes—
153C22#179314yes—
154C4—24no—
155C4—24no—
156C4—24no—
157C4—44no—
158C4—84no—
159C4—84no—
160C4—82no—
161C2—42no—
162C2#845512yes—
163C2#847414yes—
164C2#854811yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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
#356742
#113822
#76322