Finite Group Databasebeta

← #94521

Subgroups — #94521

Order 512.

282 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 24 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#8551C211yesmaximal, characteristic
2#7341C212yesmaximal
3C22×C4⋊Q8C411yescharacteristic
4#1784C411yescharacteristic
5#1784C2211yescharacteristic
6#1463—22no—
7#1320—22no—
8C23×Q8C811yescharacteristic
9C23×Q8C2×C411yescharacteristic
10C2×C4⋊Q8—44no—
11C2×C4⋊Q8—41no—
12C2×C4⋊Q8—41no—
13C22×C4⋊C4—21no—
14C22×C4⋊C4—21no—
15C22×C42C811yescharacteristic
16C2×#101—41no—
17C2×#101—41no—
18#407—21no—
19#407D412yes—
20#349—42no—
21#324—44no—
22C22×Q8—21no—
23C22×Q8—21no—
24C22×Q8—42no—
25C22×Q8—42no—
26C22×Q8—41no—
27C22×Q8—82no—
28C22×Q8—82no—
29C23×C4—21no—
30C23×C4—21no—
31C23×C4C22⋊C411yescharacteristic
32C23×C4C2×C811yescharacteristic
33C23×C4M1611yescharacteristic
34C2×M16—22no—
35C2×M16—41no—
36C4⋊Q8—44no—
37C4⋊Q8—44no—
38C4⋊Q8—44no—
39C4⋊Q8—44no—
40C2×C4⋊C4—24no—
41C2×C4⋊C4—42no—
42C2×C4⋊C4—42no—
43C2×C4⋊C4—42no—
44C2×C4⋊C4—42no—
45C2×C4⋊C4—41no—
46C2×C4⋊C4—41no—
47C2×C42—41no—
48C2×C42—42no—
49M32—82no—
50#101—82no—
51#101—82no—
52C22⋊C8—42no—
53C22⋊C8—42no—
54C24C22⋊C811yescharacteristic
55C2×Q8—42no—
56C2×Q8—42no—
57C2×Q8—42no—
58C2×Q8—42no—
59C2×Q8—42no—
60C2×Q8—42no—
61C2×Q8—82no—
62C2×Q8—81no—
63C2×Q8—81no—
64C2×Q8—84no—
65C2×Q8—84no—
66C2×Q8—82no—
67C2×Q8—84no—
68C2×Q8—84no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—44no—
75C22×C4—42no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—81no—
83C22×C4—81no—
84C22×C4—82no—
85C22×C4#9911yescharacteristic
86C22×C4#10011yescharacteristic
87M16—82no—
88M16—82no—
89C2×C8—42no—
90C2×C8—81no—
91C4⋊C4—44no—
92C4⋊C4—48no—
93C4⋊C4—44no—
94C42—42no—
95C42—42no—
96C16—162no—
97C23—21no—
98C23—41no—
99C23—42no—
100C23#32311yescharacteristic
101Q8—84no—
102Q8—84no—
103Q8—84no—
104Q8—84no—
105C2×C4—24no—
106C2×C4—22no—
107C2×C4—21no—
108C2×C4—41no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4—44no—
112C2×C4—44no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—82no—
123C2×C4—82no—
124C2×C4—82no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4AΣL(1,𝔽16)12yes—
128C8—82no—
129C8—161no—
130C22—22no—
131C22—21no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—41no—
137C22#131611yescharacteristic
138C4—22no—
139C4—44no—
140C4—41no—
141C4—42no—
142C4—81no—
143C4—82no—
144C2—21no—
145C2—42no—
146C2—41no—
147C2#733811yescharacteristic
subgroupG/H
center, socleC2#7338
commutatorC23×C4C2×C8
Frattini, cosocle#1784C22
prime radicalC24C22⋊C8
LCSG ▷ C23×C4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1784 ◁ G
LECSG ▷ #1784 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#85512
#73412