Finite Group Databasebeta

← #355121

Subgroups — #355121

Order 512.

2118 conjugacy classes of subgroups; only the 402 normal subgroups are shown.

normal subgroups only

This group has 2118 conjugacy classes of subgroups — too many to draw. Only its 402 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×#2315C211yesmaximal, characteristic
2C2×#1819C212yesmaximal
3#13775C212yesmaximal
4#10912C218yesmaximal, complemented
5#10725C212yesmaximal
6C23×C22⋊C4C2211yescharacteristic
7C23×C22⋊C4C2211yescharacteristic
8C23×C22⋊C4C2211yescharacteristic
9#2315—28no—
10C2×#385C414yescomplemented
11C2×#385C2214yes—
12C2×#380C414yescomplemented
13C2×#379C2214yes—
14C2×C4×C22⋊C4C2214yes—
15#1819C4116yescomplemented
16C2×#342C2218yes—
17C2×#342C2218yes—
18#1437—28no—
19C22⋊#95C2214yes—
20C26C2311yescharacteristic
21C24×C4C2×C412yes—
22C24×C4C2×C412yes—
23C24×C4C2312yes—
24C22×C4⋊C4C2×C418yes—
25C22×C22⋊C4—24no—
26C22×C22⋊C4—24no—
27C22×C22⋊C4—24no—
28C22×C22⋊C4—28no—
29C22×C22⋊C4—44no—
30C22×C22⋊C4—44no—
31C22×C22⋊C4C2×C412yes—
32C22×C22⋊C4C2×C418yes—
33C22×C22⋊C4C2×C412yes—
34C22×C22⋊C4C2×C412yes—
35C22×C22⋊C4C2×C412yes—
36C22×C22⋊C4D418yes—
37C22×C22⋊C4C2312yes—
38C22×C22⋊C4C2318yes—
39C22×C22⋊C4C2312yes—
40C22×C42—24no—
41#385—28no—
42#385—28no—
43#385—216no—
44#380—216no—
45#379—28no—
46#379C2×C4116yes—
47C4×C22⋊C4—48no—
48C2×#95—24no—
49C2×#95—28no—
50C2×#95C2×C418yes—
51#342—48no—
52#342C2×C4132yescomplemented
53C25—24no—
54C25—24no—
55C25—84no—
56C25C22×C411yescharacteristic
57C25C22×C411yescharacteristic
58C25C22×C414yes—
59C25C2×D414yes—
60C25#4114yes—
61C25C2411yescharacteristic
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—28no—
65C23×C4—28no—
66C23×C4—24no—
67C23×C4—44no—
68C23×C4—84no—
69C23×C4—84no—
70C23×C4—84no—
71C23×C4C4218yescomplemented
72C23×C4C22×C414yes—
73C23×C4C22×C414yes—
74C23×C4C22×C414yes—
75C23×C4C2×D414yes—
76C23×C4C2×D414yes—
77C2×C4⋊C4—216no—
78C2×C4⋊C4—88no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—232no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—216no—
85C2×C22⋊C4—216no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—28no—
89C2×C22⋊C4—28no—
90C2×C22⋊C4—232no—
91C2×C22⋊C4—44no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4—44no—
94C2×C22⋊C4—48no—
95C2×C22⋊C4—48no—
96C2×C22⋊C4—416no—
97C2×C22⋊C4—88no—
98C2×C22⋊C4—88no—
99C2×C22⋊C4C4218yescomplemented
100C2×C22⋊C4C22⋊C418yes—
101C2×C22⋊C4C22⋊C418yes—
102C2×C22⋊C4C22×C4116yes—
103C2×C22⋊C4C22×C418yes—
104C2×C42—48no—
105C2×C42—88no—
106#95—216no—
107#95—48no—
108#95—88no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—216no—
114C24—28no—
115C24—28no—
116C24—216no—
117C24—44no—
118C24—44no—
119C24—44no—
120C24—48no—
121C24—48no—
122C24—88no—
123C24—88no—
124C24—816no—
125C24—816no—
126C24C2×C4212yes—
127C24C2×C4212yes—
128C24C2×C22⋊C412yes—
129C24C2×C22⋊C412yes—
130C24#11712yes—
131C24#11712yes—
132C24C4×D4116yes—
133C24C23×C411yescharacteristic
134C24C23×C412yes—
135C24C22×D412yes—
136C24C2×#4112yes—
137C24H(5,𝔽2)14yes—
138C242−1+414yes—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—28no—
144C22×C4—28no—
145C22×C4—216no—
146C22×C4—24no—
147C22×C4—216no—
148C22×C4—28no—
149C22×C4—48no—
150C22×C4—48no—
151C22×C4—48no—
152C22×C4—48no—
153C22×C4—416no—
154C22×C4—416no—
155C22×C4—48no—
156C22×C4—48no—
157C22×C4—48no—
158C22×C4—48no—
159C22×C4—88no—
160C22×C4—88no—
161C22×C4—816no—
162C22×C4—88no—
163C22×C4—88no—
164C22×C4—816no—
165C22×C4C2×C4218yes—
166C22×C4C2×C22⋊C418yes—
167C4⋊C4—816no—
168C22⋊C4—216no—
169C22⋊C4—216no—
170C22⋊C4—232no—
171C22⋊C4—48no—
172C22⋊C4—48no—
173C22⋊C4—48no—
174C22⋊C4—48no—
175C22⋊C4—416no—
176C22⋊C4—832no—
177C42—816no—
178C23—24no—
179C23—24no—
180C23—24no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—24no—
186C23—24no—
187C23—28no—
188C23—28no—
189C23—24no—
190C23—24no—
191C23—232no—
192C23—44no—
193C23—44no—
194C23—48no—
195C23—44no—
196C23—44no—
197C23—44no—
198C23—44no—
199C23—48no—
200C23—416no—
201C23—416no—
202C23—48no—
203C23—48no—
204C23—48no—
205C23—48no—
206C23—816no—
207C23—88no—
208C23—88no—
209C23—832no—
210C23—832no—
211C23C4×C22⋊C414yes—
212C23C4×C22⋊C414yes—
213C23C22×C4211yescharacteristic
214C23C22×C22⋊C411yescharacteristic
215C23C2×#11711yescharacteristic
216C23C2×C4×D414yes—
217C23#51812yes—
218C23#51912yes—
219C23#52018yes—
220C23#53414yes—
221C23#53614yes—
222C23#53918yes—
223C2×C4—28no—
224C2×C4—216no—
225C2×C4—28no—
226C2×C4—48no—
227C2×C4—48no—
228C2×C4—48no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—432no—
232C2×C4—44no—
233C2×C4—416no—
234C2×C4—48no—
235C2×C4—88no—
236C2×C4—88no—
237C2×C4—816no—
238C2×C4—88no—
239C2×C4—816no—
240C2×C4—88no—
241C2×C4—816no—
242C2×C4—88no—
243C22—28no—
244C22—24no—
245C22—22no—
246C22—28no—
247C22—22no—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—28no—
252C22—28no—
253C22—24no—
254C22—24no—
255C22—48no—
256C22—48no—
257C22—48no—
258C22—48no—
259C22—48no—
260C22—48no—
261C22—44no—
262C22—44no—
263C22—44no—
264C22—44no—
265C22—832no—
266C22C2×C4×C22⋊C411yescharacteristic
267C22#227412yes—
268C22#230811yescharacteristic
269C22#231011yescharacteristic
270C22#231312yes—
271C22#232218yes—
272C22#288314yes—
273C4—48no—
274C4—48no—
275C4—816no—
276C4—84no—
277C2—28no—
278C2—22no—
279C2—22no—
280C2—44no—
281C2—44no—
282C2#1377611yescharacteristic
283C2#2039212yes—
284C2#2174912yes—
285C2#2182712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10912C2semidirect
10. C2×#385C4semidirect
12. C2×#380C4semidirect
15. #1819C4semidirect
52. #342C2×C4semidirect
71. C23×C4C42semidirect
99. C2×C22⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23152
C2×#18192
#137752
#109122
#107252