Finite Group Databasebeta

← #149801

Subgroups — #149801

Order 512.

334 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 66 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×#1835C211yesmaximal, characteristic
2#11511C212yesmaximal
3#7366C214yesmaximal, complemented
4C22×C4×C8C411yescharacteristic
5C2×#382C411yescharacteristic
6C2×#382C2211yescharacteristic
7#1835—22no—
8#1835C412yes—
9#1835C2212yes—
10C2×#340—22no—
11#1463—22no—
12#1324—22no—
13#1324C2214yes—
14C23×C8—21no—
15C22×C4⋊C4—21no—
16C22×C4⋊C4D412yes—
17C22×C42C2×C411yescharacteristic
18#407—42no—
19C2×C4×C8—21no—
20C2×C4×C8—24no—
21C2×C4×C8C2×C412yes—
22#382—21no—
23#382—41no—
24#382—41no—
25#382C2×C412yes—
26#382C2×C411yescharacteristic
27#382D412yes—
28#382C2311yescharacteristic
29C2×#95—41no—
30C2×#95—41no—
31#340—24no—
32#340—24no—
33#324—24no—
34#324—42no—
35C23×C4—21no—
36C23×C4—22no—
37C23×C4—41no—
38C23×C4C22⋊C411yescharacteristic
39C2×M16—82no—
40C22×C8—24no—
41C22×C8—22no—
42C22×C8—42no—
43C22×C8—42no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—44no—
49C2×C4⋊C4—41no—
50C2×C4⋊C4C22⋊C412yes—
51C2×C4⋊C4C2×D412yes—
52C2×C42—21no—
53C2×C42—22no—
54C2×C42—22no—
55C2×C42C22⋊C411yescharacteristic
56C2×C42C22×C411yescharacteristic
57C22⋊C8—44no—
58C22⋊C8—82no—
59C4×C8—24no—
60C4×C8—22no—
61C4×C8M1614yescomplemented
62#95—41no—
63#95—42no—
64#95—41no—
65#95—42no—
66#95—42no—
67C24#9911yescharacteristic
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—24no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—44no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—84no—
82C22×C4—82no—
83C22×C4—84no—
84C22×C4#9911yescharacteristic
85C22×C4C2×C22⋊C411yescharacteristic
86M16—164no—
87C2×C8—24no—
88C2×C8—48no—
89C2×C8—42no—
90C2×C8—42no—
91C2×C8—84no—
92C4⋊C4—42no—
93C4⋊C4—42no—
94C4⋊C4—44no—
95C4⋊C4—82no—
96C4⋊C4—82no—
97C42—24no—
98C42—21no—
99C42—21no—
100C42#10112yes—
101C42C2×M1612yes—
102C23—21no—
103C23—24no—
104C23—21no—
105C23#35412yes—
106C23C2×#9911yescharacteristic
107C2×C4—24no—
108C2×C4—24no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—48no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—84no—
121C2×C4—82no—
122C2×C4—82no—
123C2×C4—84no—
124C2×C4—82no—
125C2×C4—162no—
126C2×C4—164no—
127C2×C4#35512yes—
128C2×C4#40711yescharacteristic
129C2×C4C2×#10111yescharacteristic
130C8—44no—
131C8—164no—
132C22—24no—
133C22—24no—
134C22—22no—
135C22—22no—
136C22—22no—
137C22#141414yes—
138C22#146311yescharacteristic
139C22C2×#35411yescharacteristic
140C22C2×#35511yescharacteristic
141C4—21no—
142C4—22no—
143C4—21no—
144C4—42no—
145C4—42no—
146C4—82no—
147C4—82no—
148C4—162no—
149C2—21no—
150C2—21no—
151C2—24no—
152C2#1151111yescharacteristic
153C2C2×#141412yes—
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
61. C4×C8M16semidirect

Maximal subgroups

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

subgroupindex
C2×#18352
#115112
#73662