Finite Group Databasebeta

← #149789

Subgroups — #149789

Order 512.

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

normal subgroups only

This group has 378 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#21323C211yesmaximal, characteristic
2#11511C212yesmaximal
3#7359C214yesmaximal, complemented
4C2×#533C411yescharacteristic
5C2×C4×M16C411yescharacteristic
6C2×#382C2211yescharacteristic
7#1993—22no—
8#1991C412yes—
9#1991C2212yes—
10#1736—22no—
11#1463—22no—
12#1324—22no—
13#1324C2214yes—
14C22×M16—21no—
15#533—24no—
16#533—42no—
17C2×C4⋊Q8—21no—
18C2×#125C2×C412yes—
19C2×C22⋊Q8—41no—
20C2×C4×Q8—41no—
21C2×#117—21no—
22C22×C4⋊C4D412yes—
23C22×C42C2×C411yescharacteristic
24#407—42no—
25C4×M16—42no—
26C2×#97—21no—
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—
37C22×Q8—41no—
38C23×C4—21no—
39C23×C4—22no—
40C23×C4C22⋊C411yescharacteristic
41C2×M16—24no—
42C2×M16—41no—
43C2×M16—41no—
44C2×M16—42no—
45C2×M16—82no—
46C22×C8—22no—
47C4⋊Q8—24no—
48C4⋊Q8—42no—
49#125—22no—
50#125—42no—
51#125M1614yescomplemented
52C22⋊Q8—48no—
53C4×Q8—48no—
54#117—24no—
55#117—42no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
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×C22⋊C4—41no—
65C2×C42—21no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—41no—
69C2×C42C22⋊C411yescharacteristic
70C2×C42C22×C411yescharacteristic
71C22⋊C8—44no—
72C22⋊C8—82no—
73#97—42no—
74C4×C8—42no—
75#95—41no—
76#95—42no—
77#95—41no—
78C24#9911yescharacteristic
79C2×Q8—44no—
80C2×Q8—84no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—24no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—82no—
95C22×C4—84no—
96C22×C4#9911yescharacteristic
97C22×C4C2×C22⋊C411yescharacteristic
98M16—44no—
99M16—44no—
100M16—164no—
101C2×C8—24no—
102C2×C8—42no—
103C2×C8—42no—
104C2×C8—84no—
105C4⋊C4—24no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—44no—
109C4⋊C4—44no—
110C4⋊C4—44no—
111C4⋊C4—42no—
112C4⋊C4—82no—
113C22⋊C4—44no—
114C42—24no—
115C42—21no—
116C42—21no—
117C42—44no—
118C42#10012yes—
119C42C2×M1612yes—
120C23—21no—
121C23—21no—
122C23—42no—
123C23#35411yescharacteristic
124C23#35411yescharacteristic
125C23C2×#9911yescharacteristic
126Q8—88no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—44no—
142C2×C4—84no—
143C2×C4—84no—
144C2×C4—84no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4—162no—
148C2×C4#35612yes—
149C2×C4#40711yescharacteristic
150C2×C4C2×#10011yescharacteristic
151C8—44no—
152C8—164no—
153C22—22no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—21no—
158C22—42no—
159C22—42no—
160C22#146211yescharacteristic
161C22C2×#35411yescharacteristic
162C22C2×#35611yescharacteristic
163C4—21no—
164C4—22no—
165C4—21no—
166C4—42no—
167C4—42no—
168C4—84no—
169C4—82no—
170C4—82no—
171C2—21no—
172C2—21no—
173C2—41no—
174C2—41no—
175C2#1150611yescharacteristic
176C2#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. #7359C2semidirect
51. #125M16semidirect

Maximal subgroups

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

subgroupindex
#213232
#115112
#73592