Finite Group Databasebeta

← #93050

Subgroups — #93050

Order 512.

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

normal subgroups only

This group has 325 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#11231C211yesmaximal, characteristic
3#8002C211yesmaximal, characteristic
4#2920—21no—
5C22×#106C411yescharacteristic
6C22×C4⋊C8C2211yescharacteristic
7#2112—21no—
8C2×#340—21no—
9C2×#340C411yescharacteristic
10C4⋊#95—21no—
11#1322—21no—
12#1322C412yescomplemented
13#1292—21no—
14#1292C412yescomplemented
15C23×C8C2×C411yescharacteristic
16C23×C8D411yescharacteristic
17C22×C4⋊C4—21no—
18C22×C4⋊C4—21no—
19C22×C42D411yescharacteristic
20C2×M32—41no—
21#432—22no—
22#432—41no—
23#431—22no—
24#431—41no—
25C2×#106—41no—
26C2×#106—42no—
27#423—22no—
28C2×C4⋊C8—41no—
29C2×C4⋊C8—41no—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C4⋊C8D411yescharacteristic
33C2×C4⋊C8Q811yescharacteristic
34#407—41no—
35C4×M16—41no—
36C2×#95—41no—
37C2×#95—41no—
38C22⋊C16—22no—
39C22⋊C16—41no—
40#340—42no—
41#340—41no—
42C23×C4—41no—
43C23×C4—41no—
44C23×C4C22⋊C411yescharacteristic
45C23×C4D811yescharacteristic
46C23×C4SD1611yescharacteristic
47C2×M16—42no—
48C22×C8—21no—
49C22×C8—21no—
50C22×C8—21no—
51C22×C8—21no—
52C22×C8—41no—
53C22×C8—41no—
54C22×C8—41no—
55C22×C8—41no—
56C22×C8C4211yescharacteristic
57C22×C8C22⋊C411yescharacteristic
58C22×C8C4⋊C411yescharacteristic
59C22×C8C4⋊C411yescharacteristic
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C22⋊C411yescharacteristic
77C2×C42C4⋊C411yescharacteristic
78C2×C42SD1611yescharacteristic
79C2×C42Q1611yescharacteristic
80M32—81no—
81M32—81no—
82C2×C16—42no—
83#106—42no—
84#106—42no—
85C4⋊C8—24no—
86C4⋊C8—41no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89C4⋊C8—41no—
90C4⋊C8—42no—
91C22⋊C8—42no—
92C22⋊C8—42no—
93#97—42no—
94C4×C8—42no—
95#95—82no—
96#95—82no—
97C24#10211yescharacteristic
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—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—81no—
113C22×C4—81no—
114C22×C4—81no—
115C22×C4—81no—
116C22×C4—81no—
117C22×C4—81no—
118C22×C4—81no—
119C22×C4—81no—
120C22×C4—81no—
121C22×C4—81no—
122C22×C4—81no—
123C22×C4—81no—
124C22×C4#9511yescharacteristic
125C22×C4#9911yescharacteristic
126C22×C4#9911yescharacteristic
127C22×C4#10211yescharacteristic
128C22×C4#10311yescharacteristic
129C22×C4#10311yescharacteristic
130C22×C4#10411yescharacteristic
131C22×C4#10411yescharacteristic
132C22×C4#10611yescharacteristic
133C22×C4#10711yescharacteristic
134M16—82no—
135M16—82no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—42no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—44no—
150C2×C8—81no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—81no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C16—82no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23#32711yescharacteristic
175C23#32711yescharacteristic
176C23#34011yescharacteristic
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—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—81no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—161no—
208C2×C4—161no—
209C2×C4—161no—
210C2×C4—161no—
211C2×C4—161no—
212C2×C4—161no—
213C2×C4—161no—
214C2×C4—161no—
215C2×C4#32811yescharacteristic
216C2×C4#32811yescharacteristic
217C2×C4#33911yescharacteristic
218C2×C4#34211yescharacteristic
219C2×C4AΣL(1,𝔽16)11yescharacteristic
220C2×C4#35211yescharacteristic
221C2×C4#35511yescharacteristic
222C2×C4#35611yescharacteristic
223C2×C4#36012yes—
224C8—41no—
225C8—41no—
226C8—41no—
227C8—41no—
228C8—84no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22#129411yescharacteristic
244C22#134111yescharacteristic
245C22#134211yescharacteristic
246C22#134511yescharacteristic
247C22#134611yescharacteristic
248C22#138411yescharacteristic
249C22#138611yescharacteristic
250C4—21no—
251C4—21no—
252C4—22no—
253C4—41no—
254C4—41no—
255C4—41no—
256C4—162no—
257C4—162no—
258C4#135311yescharacteristic
259C4#135411yescharacteristic
260C4#139011yescharacteristic
261C4#139411yescharacteristic
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—21no—
267C2—21no—
268C2#725111yescharacteristic
269C2#725511yescharacteristic
270C2#726211yescharacteristic
subgroupG/H
center, socleC22#1294
commutatorC22×C8C42
Frattini, cosocleC22×C4⋊C8C22
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C22×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. #1322C4semidirect
14. #1292C4semidirect

Maximal subgroups

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

subgroupindex
#114402
#112312
#80022