Finite Group Databasebeta

← #93702

Subgroups — #93702

Order 512.

323 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 30 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#8308C211yesmaximal, characteristic
2#7054C212yesmaximal
3C22×C22⋊Q8C411yescharacteristic
4#1779C411yescharacteristic
5#1779C2211yescharacteristic
6C2×#324—22no—
7#1314—22no—
8C23×Q8C811yescharacteristic
9C24×C4C2×C411yescharacteristic
10C2×C22⋊Q8—44no—
11C2×C22⋊Q8—41no—
12C2×C22⋊Q8—41no—
13C22×C4⋊C4—21no—
14C22×C4⋊C4C811yescharacteristic
15C22×C22⋊C4—21no—
16C2×C22⋊C8—21no—
17C2×C22⋊C8—41no—
18C2×C22⋊C8—41no—
19C2×C22⋊C8D412yes—
20C22⋊C16—42no—
21#324—44no—
22C25M1611yescharacteristic
23C22×Q8—42no—
24C22×Q8—41no—
25C22×Q8—82no—
26C23×C4—21no—
27C23×C4—21no—
28C23×C4—21no—
29C23×C4—21no—
30C23×C4—42no—
31C23×C4—82no—
32C23×C4C22⋊C411yescharacteristic
33C23×C4C2×C811yescharacteristic
34C22×C8—22no—
35C22×C8—41no—
36C22⋊Q8—44no—
37C22⋊Q8—44no—
38C22⋊Q8—44no—
39C22⋊Q8—44no—
40C2×C4⋊C4—24no—
41C2×C4⋊C4—41no—
42C2×C4⋊C4—42no—
43C2×C4⋊C4—41no—
44C2×C4⋊C4—41no—
45C2×C4⋊C4—42no—
46C2×C22⋊C4—42no—
47C2×C22⋊C4—42no—
48C2×C22⋊C4—42no—
49C2×C16—82no—
50C22⋊C8—42no—
51C22⋊C8—42no—
52C22⋊C8—82no—
53C22⋊C8—82no—
54C24—21no—
55C24—22no—
56C24—41no—
57C24—42no—
58C24—41no—
59C24—81no—
60C24C22⋊C811yescharacteristic
61C2×Q8—42no—
62C2×Q8—42no—
63C2×Q8—82no—
64C2×Q8—84no—
65C2×Q8—82no—
66C2×Q8—84no—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—42no—
74C22×C4—42no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—42no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—81no—
84C22×C4—81no—
85C22×C4—81no—
86C22×C4—84no—
87C22×C4—84no—
88C22×C4—82no—
89C22×C4#9911yescharacteristic
90C22×C4#10011yescharacteristic
91C2×C8—42no—
92C2×C8—82no—
93C2×C8—81no—
94C2×C8—82no—
95C4⋊C4—44no—
96C4⋊C4—42no—
97C4⋊C4—42no—
98C4⋊C4—44no—
99C22⋊C4—48no—
100C16—162no—
101C23—21no—
102C23—21no—
103C23—22no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—42no—
110C23—42no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—82no—
115C23—84no—
116C23—84no—
117C23—82no—
118C23#32311yescharacteristic
119C23#35012yes—
120Q8—84no—
121Q8—84no—
122C2×C4—22no—
123C2×C4—21no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—44no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—82no—
134C2×C4—84no—
135C2×C4—84no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4—82no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4—82no—
142C2×C4AΣL(1,𝔽16)12yes—
143C8—82no—
144C8—161no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—42no—
151C22—42no—
152C22—41no—
153C22—41no—
154C22—42no—
155C22—81no—
156C22—81no—
157C22—84no—
158C22—84no—
159C22—81no—
160C22—81no—
161C22#131511yescharacteristic
162C22#131611yescharacteristic
163C22#132111yescharacteristic
164C4—22no—
165C4—41no—
166C4—42no—
167C4—82no—
168C4—82no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2—82no—
173C2#705811yescharacteristic
174C2#733811yescharacteristic
175C2#734311yescharacteristic
subgroupG/H
center, socleC22#1316
commutatorC23×C4C2×C8
Frattini, cosocle#1779C22
prime radicalC25M16
LCSG ▷ C23×C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1779 ◁ G
LECSG ▷ #1779 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#83082
#70542