Finite Group Databasebeta

← #92999

Subgroups — #92999

Order 512.

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

normal subgroups only

This group has 237 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#12976C211yesmaximal, characteristic
2#11466C211yesmaximal, characteristic
3#8164C211yesmaximal, characteristic
4C23×C16C411yescharacteristic
5C2×C8×D4C2211yescharacteristic
6#2112—21no—
7C2×C22⋊C16—21no—
8C2×C22⋊C16C411yescharacteristic
9C4×#100C411yescharacteristic
10#1460C411yescharacteristic
11#1451—21no—
12#1374—21no—
13#1322—21no—
14#1314C412yes—
15#1283—21no—
16C23×C8C2×C411yescharacteristic
17C23×C8D411yescharacteristic
18C2×C4×D4C2×C411yescharacteristic
19C2×M32—41no—
20C22×C16—21no—
21C22×C16—21no—
22C22×C16—42no—
23C22×C16—41no—
24C8×D4—41no—
25C8×D4—41no—
26#431—21no—
27C2×C4⋊C8D411yescharacteristic
28C2×#100—21no—
29C2×C22⋊C8C2×C411yescharacteristic
30C2×C22⋊C8Q811yescharacteristic
31C4×M16—21no—
32C2×C4×C8D411yescharacteristic
33C22⋊C16—22no—
34C22⋊C16—42no—
35C22⋊C16—41no—
36C22⋊C16—41no—
37#343C812yes—
38#323—21no—
39#323C812yes—
40#322—41no—
41#322—41no—
42C22×D4M1611yescharacteristic
43C23×C4C4211yescharacteristic
44C23×C4C4⋊C411yescharacteristic
45C2×M16—22no—
46C22×C8—21no—
47C22×C8—21no—
48C22×C8—21no—
49C22×C8—21no—
50C22×C8—41no—
51C22×C8—41no—
52C22×C8—41no—
53C22×C8—41no—
54C22×C8C22⋊C411yescharacteristic
55C22×C8C22⋊C411yescharacteristic
56C22×C8C4⋊C411yescharacteristic
57C22×C8C4⋊C411yescharacteristic
58C4×D4—41no—
59C4×D4—41no—
60C2×C4⋊C4C2×C811yescharacteristic
61C2×C22⋊C4C2×C811yescharacteristic
62C2×C22⋊C4M1611yescharacteristic
63C2×C42C22⋊C411yescharacteristic
64M32—81no—
65M32—81no—
66C2×C16—22no—
67C2×C16—41no—
68C2×C16—42no—
69C2×C16—41no—
70C2×C16—42no—
71C2×C16—42no—
72C2×C16—41no—
73C2×C16—41no—
74C2×C16—41no—
75C4⋊C8—41no—
76C4⋊C8—41no—
77#100—42no—
78C22⋊C8—22no—
79C22⋊C8—41no—
80C22⋊C8—41no—
81#97—41no—
82C4×C8—21no—
83C4×C8—21no—
84C4×C8—41no—
85C4×C8—41no—
86C24#9711yescharacteristic
87C24C4⋊C811yescharacteristic
88C2×D4—22no—
89C2×D4—22no—
90C2×D4—41no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4#9511yescharacteristic
100C22×C4C4×C811yescharacteristic
101C22×C4C22⋊C811yescharacteristic
102C22×C4C22⋊C811yescharacteristic
103C22×C4C4⋊C811yescharacteristic
104M16—42no—
105M16—42no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—42no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—42no—
120C2×C8—42no—
121C2×C8—81no—
122C2×C8#9911yescharacteristic
123C2×C8#9911yescharacteristic
124C4⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C42—21no—
128C42—21no—
129C16—42no—
130C16—81no—
131C16—82no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23#33611yescharacteristic
142C23#34712yes—
143D4—44no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—81no—
160C2×C4#34211yescharacteristic
161C2×C4#34411yescharacteristic
162C2×C4#35012yes—
163C8—21no—
164C8—41no—
165C8—41no—
166C8—41no—
167C8—82no—
168C8—82no—
169C8AΣL(1,𝔽16)11yescharacteristic
170C8#35211yescharacteristic
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22#128211yescharacteristic
184C22#137511yescharacteristic
185C22#137911yescharacteristic
186C4—22no—
187C4—21no—
188C4—41no—
189C4—41no—
190C4—41no—
191C4#139011yescharacteristic
192C4#139211yescharacteristic
193C2—21no—
194C2—21no—
195C2—42no—
196C2—81no—
197C2#720011yescharacteristic
198C2#720311yescharacteristic
199C2#720811yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC22×C4C4×C8
Frattini, cosocleC2×C8×D4C22
socleC22#1282
prime radicalC22×D4M16
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#129762
#114662
#81642