Finite Group Databasebeta

← #93053

Subgroups — #93053

Order 512.

282 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 64 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#11440C211yesmaximal, characteristic
2#11233C211yesmaximal, characteristic
3C2×#1290C211yesmaximal, characteristic
4C2×#432—21no—
5C22×#108C411yescharacteristic
6C22×C4⋊C8C2211yescharacteristic
7#2112—21no—
8C2×#341—21no—
9C2×#341C411yescharacteristic
10#1728—21no—
11#1322—21no—
12#1322C412yes—
13#1290—21no—
14#1290C412yes—
15C22×M16—21no—
16C22×M16—21no—
17C23×C8C2×C411yescharacteristic
18C23×C8D411yescharacteristic
19C22×C42D411yescharacteristic
20C2×M32—41no—
21#432—22no—
22#432—22no—
23#432—41no—
24#432—41no—
25C2×#108—41no—
26C2×#108—42no—
27C2×C4⋊C8—41no—
28C2×C4⋊C8—41no—
29C2×C4⋊C8—41no—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C4⋊C8D411yescharacteristic
33C2×C4⋊C8Q811yescharacteristic
34C2×C22⋊C8—41no—
35C2×#97—22no—
36C22⋊C16—22no—
37C22⋊C16—41no—
38#341—41no—
39#341—42no—
40#336—41no—
41#336—41no—
42C23×C4C22⋊C411yescharacteristic
43C23×C4SD1611yescharacteristic
44C23×C4Q1611yescharacteristic
45C2×M16—22no—
46C2×M16—22no—
47C2×M16—41no—
48C2×M16—41no—
49C2×M16—41no—
50C2×M16—41no—
51C2×M16—41no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C22×C8—21no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—42no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8C4211yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C4⋊C411yescharacteristic
71C22×C8C4⋊C411yescharacteristic
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C22⋊C411yescharacteristic
77C2×C42C4⋊C411yescharacteristic
78C2×C42D811yescharacteristic
79C2×C42SD1611yescharacteristic
80M32—81no—
81M32—81no—
82C2×C16—42no—
83#108—42no—
84#108—42no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89C4⋊C8—42no—
90C4⋊C8—42no—
91C22⋊C8—42no—
92C22⋊C8—42no—
93#97—24no—
94#97—42no—
95C24#10311yescharacteristic
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—81no—
107C22×C4#9511yescharacteristic
108C22×C4#9911yescharacteristic
109C22×C4#9911yescharacteristic
110C22×C4#10211yescharacteristic
111C22×C4#10211yescharacteristic
112C22×C4#10311yescharacteristic
113C22×C4#10411yescharacteristic
114C22×C4#10411yescharacteristic
115C22×C4#10611yescharacteristic
116C22×C4#10711yescharacteristic
117M16—42no—
118M16—42no—
119M16—42no—
120M16—42no—
121M16—81no—
122M16—81no—
123M16—81no—
124M16—81no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—44no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—81no—
142C2×C8—81no—
143C2×C8—81no—
144C2×C8—84no—
145C2×C8—81no—
146C2×C8—81no—
147C42—21no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—21no—
155C16—82no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—41no—
161C23#32811yescharacteristic
162C23#32811yescharacteristic
163C23#34011yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—81no—
188C2×C4#32711yescharacteristic
189C2×C4#32711yescharacteristic
190C2×C4#33911yescharacteristic
191C2×C4#34211yescharacteristic
192C2×C4AΣL(1,𝔽16)11yescharacteristic
193C2×C4#35211yescharacteristic
194C2×C4#35511yescharacteristic
195C2×C4#35611yescharacteristic
196C8—41no—
197C8—41no—
198C8—41no—
199C8—41no—
200C8—82no—
201C8—84no—
202C8—82no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22#129411yescharacteristic
215C22#135311yescharacteristic
216C22#135411yescharacteristic
217C22#138511yescharacteristic
218C22#138811yescharacteristic
219C4—21no—
220C4—21no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4#134511yescharacteristic
226C4#134611yescharacteristic
227C4#139011yescharacteristic
228C4#139411yescharacteristic
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—41no—
234C2#725111yescharacteristic
235C2#725611yescharacteristic
236C2#726511yescharacteristic
subgroupG/H
center, socleC22#1294
commutatorC22×C8C42
Frattini, cosocleC22×C4⋊C8C22
prime radicalC24#103
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C22×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#114402
#112332
C2×#12902