Finite Group Databasebeta

← #149787

Subgroups — #149787

Order 512.

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

normal subgroups only

This group has 394 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×#1991C211yesmaximal, characteristic
2#11511C212yesmaximal
3#7359C214yesmaximal, complemented
4C22×#125C411yescharacteristic
5C22×C4×C8C411yescharacteristic
6C2×#382C2211yescharacteristic
7#1991—22no—
8#1991C412yes—
9#1991C2212yes—
10C2×#340—22no—
11#1463—22no—
12#1324—22no—
13#1324C2214yes—
14C23×C8—21no—
15C2×#125—24no—
16C2×#125—21no—
17C2×#125—42no—
18C2×#125—41no—
19C2×#125C2×C412yes—
20C22×C4⋊C4—21no—
21C22×C4⋊C4—41no—
22C22×C4⋊C4D412yes—
23C22×C42C2×C411yescharacteristic
24#407—42no—
25C2×C4×C8—21no—
26C2×C4×C8—42no—
27C2×C4×C8C2×C412yes—
28#382—41no—
29#382C2×C411yescharacteristic
30#382D412yes—
31#382C2311yescharacteristic
32C2×#95—41no—
33#340—24no—
34#340—24no—
35#324—24no—
36#324—42no—
37C23×C4—21no—
38C23×C4—22no—
39C23×C4—41no—
40C23×C4C22⋊C411yescharacteristic
41C2×M16—82no—
42C22×C8—24no—
43C22×C8—22no—
44C22×C8—42no—
45C22×C8—42no—
46#125—24no—
47#125—22no—
48#125—42no—
49#125—48no—
50#125—42no—
51#125M1614yescomplemented
52C2×C4⋊C4—24no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—48no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4C22⋊C412yes—
63C2×C4⋊C4C2×D412yes—
64C2×C42—21no—
65C2×C42—22no—
66C2×C42—22no—
67C2×C42C22⋊C411yescharacteristic
68C2×C42C22×C411yescharacteristic
69C22⋊C8—44no—
70C22⋊C8—82no—
71C4×C8—42no—
72C4×C8—42no—
73#95—41no—
74#95—42no—
75#95—41no—
76C24#9911yescharacteristic
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—24no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—44no—
87C22×C4—44no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—84no—
92C22×C4—84no—
93C22×C4#9911yescharacteristic
94C22×C4C2×C22⋊C411yescharacteristic
95M16—164no—
96C2×C8—24no—
97C2×C8—48no—
98C2×C8—42no—
99C2×C8—42no—
100C2×C8—84no—
101C4⋊C4—24no—
102C4⋊C4—44no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C4⋊C4—44no—
106C4⋊C4—48no—
107C4⋊C4—44no—
108C4⋊C4—42no—
109C4⋊C4—82no—
110C42—24no—
111C42—21no—
112C42—21no—
113C42#10012yes—
114C42C2×M1612yes—
115C23—21no—
116C23—24no—
117C23—21no—
118C23#35412yes—
119C23C2×#9911yescharacteristic
120C2×C4—24no—
121C2×C4—24no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—48no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—44no—
134C2×C4—88no—
135C2×C4—84no—
136C2×C4—84no—
137C2×C4—82no—
138C2×C4—82no—
139C2×C4—162no—
140C2×C4#35612yes—
141C2×C4#40711yescharacteristic
142C2×C4C2×#10011yescharacteristic
143C8—44no—
144C8—164no—
145C22—24no—
146C22—24no—
147C22—22no—
148C22—22no—
149C22—22no—
150C22#141414yes—
151C22#146211yescharacteristic
152C22C2×#35411yescharacteristic
153C22C2×#35611yescharacteristic
154C4—21no—
155C4—22no—
156C4—21no—
157C4—42no—
158C4—42no—
159C4—84no—
160C4—82no—
161C4—82no—
162C2—21no—
163C2—21no—
164C2—24no—
165C2#1150611yescharacteristic
166C2C2×#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. #7359C2semidirect
51. #125M16semidirect

Maximal subgroups

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

subgroupindex
C2×#19912
#115112
#73592