Finite Group Databasebeta

← #151836

Subgroups — #151836

Order 512.

690 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C2×#2126C212yesmaximal
2#21200C211yesmaximal, characteristic
3#7498C214yesmaximal, complemented
4C2×#532C2211yescharacteristic
5C2×#417—22no—
6C2×#410—22no—
7C2×C4×M16C411yescharacteristic, complemented
8C2×#384C411yescharacteristic, complemented
9#2126—24no—
10#2126C418yescomplemented
11C2×#354—22no—
12C2×#353C2214yes—
13#1959—22no—
14#1957C2212yes—
15#1956C412yescomplemented
16C22×#41D412yes—
17C22×M16—21no—
18#532—42no—
19#532C2×C414yes—
20#532D412yes—
21#532Q812yes—
22C2×C4⋊Q8C2×C411yescharacteristic
23C2×#127C2311yescharacteristic
24C2×#124D412yes—
25C2×#121—41no—
26C2×C4×D4—41no—
27C2×#117—42no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—41no—
30C22×C42C2×C411yescharacteristic
31#417—44no—
32#417—44no—
33C2×#103—24no—
34C2×#102—24no—
35#410—28no—
36#410—44no—
37C2×#99—24no—
38C2×#99—42no—
39C4×M16—42no—
40C2×#97—21no—
41C2×C4×C8C2×C412yes—
42#384—21no—
43#384—41no—
44#384C2×C412yes—
45#354—28no—
46#353—24no—
47#353C2×C418yescomplemented
48C2×#41—24no—
49C2×#41—44no—
50C2×#41—42no—
51C2×#41—42no—
52C2×#41C22⋊C414yes—
53C2×#41C4⋊C414yes—
54C22×Q8C22⋊C412yes—
55C22×D4—22no—
56C22×D4—41no—
57C22×D4C2×D412yes—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—41no—
61C23×C4C22⋊C411yescharacteristic
62C2×M16—24no—
63C2×M16—42no—
64C2×M16—42no—
65C22×C8—22no—
66C4⋊Q8—41no—
67C4⋊Q8C4212yes—
68C4⋊Q8C4⋊C412yes—
69#127—41no—
70#127C22×C412yes—
71#127C2×D411yescharacteristic
72#127C2×Q811yescharacteristic
73#124—42no—
74#124C22⋊C414yes—
75#124C4⋊C414yes—
76#121—44no—
77#121—42no—
78#121—42no—
79C4×D4—44no—
80C4×D4—42no—
81C4×D4—42no—
82#117—48no—
83#117—84no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—84no—
92C2×C4⋊C4—82no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—44no—
95C2×C42—22no—
96C2×C42—21no—
97C2×C42—82no—
98C2×C42C4212yes—
99C2×C42C22⋊C411yescharacteristic
100C2×C42C4⋊C412yes—
101C2×C42C22×C411yescharacteristic
102#103—48no—
103#102—48no—
104#99—28no—
105#99—44no—
106#99—48no—
107#97—42no—
108C4×C8—42no—
109C24—22no—
110C24#9911yescharacteristic
111#41—44no—
112#41—44no—
113#41—44no—
114#41—44no—
115#41—84no—
116#41—84no—
117C2×Q8—22no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8#9514yes—
121C2×D4—24no—
122C2×D4—24no—
123C2×D4—22no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—82no—
130C2×D4—82no—
131C2×D4—82no—
132C2×D4—82no—
133C2×D4C2×C22⋊C412yes—
134C2×D4C2×C4⋊C412yes—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4—84no—
153C22×C4—84no—
154C22×C4#9512yes—
155C22×C4#9912yes—
156C22×C4#9911yescharacteristic
157C22×C4C2×C22⋊C411yescharacteristic
158M16—48no—
159C2×C8—24no—
160C2×C8—42no—
161C2×C8—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—44no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C4⋊C4—88no—
168C4⋊C4—82no—
169C22⋊C4—44no—
170C22⋊C4—44no—
171C22⋊C4—48no—
172C22⋊C4—84no—
173C42—22no—
174C42—22no—
175C42—21no—
176C42—21no—
177C42—88no—
178C42#9512yes—
179C42C2×C4211yescharacteristic
180C42C2×C4⋊C411yescharacteristic
181C23—21no—
182C23—21no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—41no—
187C23—42no—
188C23—42no—
189C23—82no—
190C23—82no—
191C23#34212yes—
192C23C2×#9911yescharacteristic
193Q8—44no—
194Q8—82no—
195D4—42no—
196D4—42no—
197D4—82no—
198D4—84no—
199D4—82no—
200D4—82no—
201D4—84no—
202D4—84no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—84no—
224C2×C4—88no—
225C2×C4—82no—
226C2×C4—162no—
227C2×C4—162no—
228C2×C4—164no—
229C2×C4—164no—
230C2×C4#34212yes—
231C2×C4#35312yes—
232C2×C4#35412yes—
233C2×C4C2×#9511yescharacteristic
234C2×C4C2×#9911yescharacteristic
235C8—44no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—42no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22—82no—
247C22—82no—
248C22—82no—
249C22—162no—
250C22C2×#34211yescharacteristic
251C22C2×#35311yescharacteristic
252C22C2×#35411yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—44no—
259C4—168no—
260C4—162no—
261C4#139314yes—
262C2—21no—
263C2—21no—
264C2—42no—
265C2—82no—
266C2—82no—
267C2C2×#139311yescharacteristic
268C2#1358511yescharacteristic
269C2#1358611yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC42C2×C42
Frattini, cosocleC2×#127C23
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G
LECSG ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7498C2semidirect
7. C2×C4×M16C4semidirect
8. C2×#384C4semidirect
10. #2126C4semidirect
15. #1956C4semidirect
47. #353C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21262
#212002
#74982