Finite Group Databasebeta

← #93054

Subgroups — #93054

Order 512.

304 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 68 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×#1292C211yesmaximal, characteristic
4C22×#108C411yescharacteristic
5C2×#423—21no—
6C22×C4⋊C8C2211yescharacteristic
7#2112—21no—
8C2×#341—21no—
9C2×#341C411yescharacteristic
10#1728—21no—
11#1322—21no—
12#1322C412yes—
13#1292—21no—
14#1292C412yes—
15C22×M16—21no—
16C22×M16—21no—
17C22×M16—41no—
18C23×C8C2×C411yescharacteristic
19C23×C8D411yescharacteristic
20C22×C42D411yescharacteristic
21C2×M32—41no—
22C2×#108—41no—
23C2×#108—42no—
24#423—22no—
25#423—22no—
26#423—41no—
27#423—41no—
28C2×C4⋊C8—22no—
29C2×C4⋊C8—41no—
30C2×C4⋊C8—41no—
31C2×C4⋊C8—41no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8D411yescharacteristic
35C2×C4⋊C8Q811yescharacteristic
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—42no—
50C2×M16—42no—
51C2×M16—41no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—82no—
60C2×M16—82no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8C4211yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C4⋊C411yescharacteristic
75C22×C8C4⋊C411yescharacteristic
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42C22⋊C411yescharacteristic
81C2×C42C4⋊C411yescharacteristic
82C2×C42D811yescharacteristic
83C2×C42SD1611yescharacteristic
84M32—81no—
85M32—81no—
86C2×C16—42no—
87#108—42no—
88#108—42no—
89C4⋊C8—24no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97C24#10311yescharacteristic
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—81no—
110C22×C4#9511yescharacteristic
111C22×C4#9911yescharacteristic
112C22×C4#9911yescharacteristic
113C22×C4#10211yescharacteristic
114C22×C4#10211yescharacteristic
115C22×C4#10311yescharacteristic
116C22×C4#10411yescharacteristic
117C22×C4#10411yescharacteristic
118C22×C4#10611yescharacteristic
119C22×C4#10711yescharacteristic
120M16—42no—
121M16—42no—
122M16—42no—
123M16—42no—
124M16—81no—
125M16—81no—
126M16—81no—
127M16—84no—
128M16—82no—
129M16—81no—
130M16—82no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—42no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—44no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—84no—
151C2×C8—81no—
152C2×C8—81no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C16—82no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—41no—
167C23#32811yescharacteristic
168C23#32811yescharacteristic
169C23#34011yescharacteristic
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
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—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4#32711yescharacteristic
195C2×C4#32711yescharacteristic
196C2×C4#33911yescharacteristic
197C2×C4#34211yescharacteristic
198C2×C4AΣL(1,𝔽16)11yescharacteristic
199C2×C4#35211yescharacteristic
200C2×C4#35511yescharacteristic
201C2×C4#35611yescharacteristic
202C2×C4#36111yescharacteristic
203C2×C4#36211yescharacteristic
204C2×C4#36512yes—
205C8—41no—
206C8—41no—
207C8—41no—
208C8—41no—
209C8—82no—
210C8—84no—
211C8—82no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22#129411yescharacteristic
224C22#135311yescharacteristic
225C22#135411yescharacteristic
226C22#138311yescharacteristic
227C22#138711yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—21no—
232C4—22no—
233C4—41no—
234C4—41no—
235C4#134511yescharacteristic
236C4#134611yescharacteristic
237C4#139011yescharacteristic
238C4#139411yescharacteristic
239C2—21no—
240C2—21no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2#725111yescharacteristic
245C2#725711yescharacteristic
246C2#726411yescharacteristic
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×#12922