Finite Group Databasebeta

← #190023

Subgroups — #190023

Order 512.

1362 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 286 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×#2274C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#8735C218yesmaximal, complemented
4#7929C214yesmaximal
5C22×#117C412yes—
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2214yes—
8#2274—22no—
9#2274—22no—
10#2274C414yes—
11#2274C2214yes—
12C2×C4×C4⋊C4—24no—
13C2×C4×C22⋊C4C412yes—
14C2×C4×C22⋊C4C2212yes—
15#1463—24no—
16#1463—22no—
17#1463—22no—
18C2×#324C2218yes—
19#1282C22116yes—
20C24×C4C2×C411yescharacteristic
21C24×C4C2×C411yescharacteristic
22C24×C4C2311yescharacteristic
23C22×M16—24no—
24C2×#117—22no—
25C2×#117—22no—
26C2×#117—216no—
27C2×#117C2×C414yes—
28C2×#117C2×C414yes—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—24no—
31C22×C4⋊C4—24no—
32C22×C4⋊C4C2×C412yes—
33C22×C4⋊C4D414yes—
34C22×C4⋊C4D412yes—
35C22×C4⋊C4D412yes—
36C22×C4⋊C4C2312yes—
37C22×C22⋊C4C2×C414yes—
38C22×C42—22no—
39C22×C42—22no—
40C22×C42—22no—
41#407—24no—
42#407—24no—
43#407—24no—
44#407—24no—
45C2×C22⋊C8—24no—
46C2×C22⋊C8C2318yes—
47C4×C4⋊C4—28no—
48C4×C4⋊C4—28no—
49C4×C4⋊C4—28no—
50C4×C4⋊C4—28no—
51C4×C22⋊C4—24no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—22no—
54C4×C22⋊C4C2×C418yes—
55C4×C22⋊C4C2×C414yes—
56C4×C22⋊C4C2314yes—
57C2×#95—22no—
58C2×#95—22no—
59#324—28no—
60#324—28no—
61C25C22×C411yescharacteristic
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—28no—
65C23×C4—24no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—82no—
69C23×C4—82no—
70C23×C4C22⋊C412yes—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C412yes—
75C23×C4C22×C412yes—
76C23×C4C22×C412yes—
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D412yes—
79C23×C4C2×D412yes—
80C23×C4#4114yes—
81C23×C4#4114yes—
82C23×C4C2411yescharacteristic
83C2×M16—44no—
84C2×M16—44no—
85C2×M16—88no—
86C22×C8—28no—
87#117—216no—
88#117—216no—
89#117—28no—
90#117—216no—
91#117M16116yescomplemented
92C2×C4⋊C4—216no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4C22⋊C414yes—
104C2×C4⋊C4C22⋊C414yes—
105C2×C4⋊C4C2×D414yes—
106C2×C4⋊C4C2×D414yes—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—216no—
109C2×C22⋊C4C22×C418yes—
110C2×C42—216no—
111C2×C42—24no—
112C2×C42—24no—
113C2×C42—28no—
114C2×C42—24no—
115C2×C42—24no—
116C2×C42—84no—
117C2×C42—84no—
118C22⋊C8—28no—
119C22⋊C8—28no—
120C22⋊C8—44no—
121C22⋊C8—44no—
122#95—28no—
123#95—24no—
124#95—24no—
125C24—24no—
126C24—42no—
127C24—42no—
128C24C2×C22⋊C411yescharacteristic
129C24C2×C22⋊C411yescharacteristic
130C24#11712yes—
131C24#11712yes—
132C24C23×C411yescharacteristic
133C22×C4—216no—
134C22×C4—216no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—216no—
146C22×C4—28no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—44no—
153C22×C4—48no—
154C22×C4—48no—
155C22×C4—48no—
156C22×C4—44no—
157C22×C4—81no—
158C22×C4—82no—
159C22×C4—84no—
160C22×C4—81no—
161C22×C4—84no—
162C22×C4—84no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4C2×C22⋊C412yes—
166C22×C4C2×C22⋊C412yes—
167C22×C4C2×C22⋊C414yes—
168C22×C4C2×C22⋊C412yes—
169C22×C4#11714yes—
170C22×C4#12314yes—
171C22×C4#12314yes—
172C22×C4#12314yes—
173C22×C4#12314yes—
174C22×C4C2×M1618yes—
175C22×C4C22×D412yes—
176C22×C4C2×#4114yes—
177M16—88no—
178M16—88no—
179C2×C8—48no—
180C2×C8—88no—
181C4⋊C4—216no—
182C4⋊C4—28no—
183C4⋊C4—28no—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—216no—
187C22⋊C4—216no—
188C22⋊C4—28no—
189C22⋊C4C2×M16116yes—
190C42—216no—
191C42—216no—
192C42—84no—
193C42—84no—
194C42—88no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—24no—
199C23—24no—
200C23—22no—
201C23—24no—
202C23—24no—
203C23—41no—
204C23—42no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—41no—
209C23—416no—
210C23#38112yes—
211C23#38111yescharacteristic
212C23#38111yescharacteristic
213C23C22×C22⋊C411yescharacteristic
214C23C2×#11712yes—
215C23C22×M1614yes—
216C2×C4—28no—
217C2×C4—28no—
218C2×C4—28no—
219C2×C4—28no—
220C2×C4—28no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—216no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—216no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—48no—
236C2×C4—432no—
237C2×C4—48no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—84no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4#38114yes—
245C2×C4#40714yes—
246C2×C4#40714yes—
247C2×C4C2×#12314yes—
248C8—88no—
249C22—28no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—22no—
255C22—21no—
256C22—22no—
257C22—21no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—44no—
262C22—48no—
263C22—48no—
264C22—44no—
265C22#182812yes—
266C22#183012yes—
267C22C2×#38111yescharacteristic
268C22C2×#40712yes—
269C22#292014yes—
270C4—24no—
271C4—24no—
272C4—24no—
273C4—24no—
274C4—416no—
275C4—44no—
276C2—22no—
277C2—22no—
278C2—22no—
279C2—24no—
280C2—42no—
281C2#2068211yescharacteristic
282C2C2×#182811yescharacteristic
283C2C2×#183011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8735C2semidirect
91. #117M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22742
C2×#14632
#87352
#79292