Finite Group Databasebeta

← #149754

Subgroups — #149754

Order 512.

391 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 60 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#20691C211yesmaximal, characteristic
2#11504C212yesmaximal
3#7348C214yesmaximal
4C2×#519C411yescharacteristic
5#2310C411yescharacteristic
6#2310C2211yescharacteristic
7#1820—22no—
8#1820C412yes—
9#1820C2212yes—
10#1740—22no—
11#1464—22no—
12#1317—22no—
13#1317C2214yes—
14C23×Q8C2×C411yescharacteristic
15#519—42no—
16#519—42no—
17C2×C4×Q8—24no—
18C2×#117—21no—
19C2×#117—21no—
20C2×#117—21no—
21C2×#117C2×C412yes—
22C2×#117D412yes—
23#432—42no—
24#391—42no—
25#391—42no—
26#381—21no—
27#381C2×C412yes—
28#381C2×C411yescharacteristic
29#381D412yes—
30#381C2311yescharacteristic
31#342—24no—
32#342—24no—
33#324—42no—
34#323—24no—
35C22×Q8—22no—
36C22×Q8—22no—
37C22×Q8—22no—
38C22×Q8—42no—
39C22×Q8—42no—
40C23×C4C22⋊C411yescharacteristic
41C23×C4C22⋊C411yescharacteristic
42C23×C4C22×C411yescharacteristic
43C4×Q8—416no—
44#117—42no—
45#117—42no—
46#117—42no—
47#117—42no—
48#117—42no—
49#117—42no—
50#117—42no—
51#117—42no—
52#117—42no—
53#117—42no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—41no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4C22⋊C412yes—
62C2×C22⋊C4C2×D412yes—
63C2×C42—22no—
64C2×C42—22no—
65C2×C42—22no—
66C2×C42—41no—
67C2×C42M1614yes—
68C4⋊C8—82no—
69C22⋊C8—44no—
70#97—82no—
71#95—41no—
72#95—42no—
73#95—41no—
74#95—42no—
75#95—42no—
76C24C2×C22⋊C411yescharacteristic
77C2×Q8—28no—
78C2×Q8—42no—
79C2×Q8—42no—
80C2×Q8—44no—
81C2×Q8—42no—
82C2×Q8—44no—
83C2×Q8—42no—
84C2×Q8—44no—
85C2×Q8—42no—
86C2×Q8—42no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—24no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4#9912yes—
102C22×C4#10012yes—
103C22×C4C2×M1612yes—
104C2×C8—84no—
105C4⋊C4—44no—
106C4⋊C4—44no—
107C4⋊C4—44no—
108C4⋊C4—82no—
109C4⋊C4—82no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C22⋊C4—44no—
115C42—44no—
116C42—44no—
117C42—44no—
118C42—42no—
119C42—42no—
120C42—81no—
121C42—81no—
122C23—22no—
123C23—81no—
124C23#40711yescharacteristic
125C23C2×#9911yescharacteristic
126C23C2×#10011yescharacteristic
127Q8—48no—
128Q8—48no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—84no—
151C2×C4—82no—
152C2×C4—84no—
153C2×C4#35212yes—
154C2×C4#35212yes—
155C8—164no—
156C22—21no—
157C22—21no—
158C22—41no—
159C22—41no—
160C22—41no—
161C22—82no—
162C22#146211yescharacteristic
163C22C2×#35212yes—
164C4—22no—
165C4—22no—
166C4—42no—
167C4—42no—
168C4—82no—
169C4—84no—
170C4—82no—
171C4—162no—
172C2—21no—
173C2—22no—
174C2—41no—
175C2—41no—
176C2#1150011yescharacteristic
subgroupG/H
center, socleC2#11500
commutatorC23×C4C22×C4
Frattini, cosocle#381C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2310 ◁ G
LECSG ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#206912
#115042
#73482