Finite Group Databasebeta

← #149803

Subgroups — #149803

Order 512.

314 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 62 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#20765C211yesmaximal, characteristic
2#11511C212yesmaximal
3#7366C214yesmaximal, complemented
4C2×C4×M16C411yescharacteristic
5#2285C411yescharacteristic
6C2×#382C2211yescharacteristic
7#1836—22no—
8#1835C412yes—
9#1835C2212yes—
10#1736—22no—
11#1463—22no—
12#1324—22no—
13#1324C2214yes—
14C22×M16—21no—
15C2×#117—21no—
16C22×C4⋊C4D412yes—
17C22×C42C2×C411yescharacteristic
18#407—42no—
19C4×M16—24no—
20C2×#97—21no—
21C2×C4×C8C2×C412yes—
22#384—21no—
23#382—41no—
24#382C2×C412yes—
25#382C2×C411yescharacteristic
26#382D412yes—
27#382C2311yescharacteristic
28#380—41no—
29C4×C4⋊C4—41no—
30C2×#95—41no—
31#340—24no—
32#340—24no—
33#324—24no—
34#324—42no—
35C23×C4—21no—
36C23×C4—22no—
37C23×C4C22⋊C411yescharacteristic
38C2×M16—24no—
39C2×M16—41no—
40C2×M16—41no—
41C2×M16—42no—
42C2×M16—82no—
43C22×C8—22no—
44#117—44no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—42no—
49C2×C4⋊C4—42no—
50C2×C4⋊C4C22⋊C412yes—
51C2×C4⋊C4C2×D412yes—
52C2×C22⋊C4—41no—
53C2×C42—21no—
54C2×C42—22no—
55C2×C42—22no—
56C2×C42—41no—
57C2×C42C22⋊C411yescharacteristic
58C2×C42C22×C411yescharacteristic
59C22⋊C8—44no—
60C22⋊C8—82no—
61#97—24no—
62C4×C8—22no—
63C4×C8M1614yescomplemented
64#95—41no—
65#95—42no—
66#95—41no—
67#95—42no—
68C24#9911yescharacteristic
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—24no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—82no—
83C22×C4—84no—
84C22×C4#9911yescharacteristic
85C22×C4C2×C22⋊C411yescharacteristic
86M16—44no—
87M16—44no—
88M16—164no—
89C2×C8—24no—
90C2×C8—42no—
91C2×C8—42no—
92C2×C8—84no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—44no—
96C4⋊C4—82no—
97C4⋊C4—84no—
98C22⋊C4—82no—
99C42—24no—
100C42—21no—
101C42—21no—
102C42—82no—
103C42#10112yes—
104C42C2×M1612yes—
105C23—21no—
106C23—21no—
107C23—42no—
108C23#35411yescharacteristic
109C23#35411yescharacteristic
110C23C2×#9911yescharacteristic
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—84no—
126C2×C4—84no—
127C2×C4—82no—
128C2×C4—82no—
129C2×C4—84no—
130C2×C4—82no—
131C2×C4—162no—
132C2×C4#35512yes—
133C2×C4#40711yescharacteristic
134C2×C4C2×#10111yescharacteristic
135C8—44no—
136C8—164no—
137C22—22no—
138C22—22no—
139C22—22no—
140C22—21no—
141C22—21no—
142C22—42no—
143C22—42no—
144C22#146311yescharacteristic
145C22C2×#35411yescharacteristic
146C22C2×#35511yescharacteristic
147C4—21no—
148C4—22no—
149C4—21no—
150C4—42no—
151C4—42no—
152C4—82no—
153C4—82no—
154C4—162no—
155C2—21no—
156C2—21no—
157C2—41no—
158C2—41no—
159C2#1151111yescharacteristic
160C2#1360312yes—
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7366C2semidirect
63. C4×C8M16semidirect

Maximal subgroups

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

subgroupindex
#207652
#115112
#73662