Finite Group Databasebeta

← #375659

Subgroups — #375659

Order 512.

1589 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1589 conjugacy classes of subgroups — too many to draw. Only its 266 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×#2853C211yesmaximal, characteristic
2C4×#381C212yesmaximal, complemented
3#11197C212yesmaximal, complemented
4#10966C212yesmaximal
5#10960C214yesmaximal
6#10912C214yesmaximal, complemented
7C22×#123C2211yescharacteristic
8C23×C42C2211yescharacteristic
9C22×#99C2214yes—
10#2853—24no—
11#2853—24no—
12C2×#388C414yescomplemented
13C2×#387C414yescomplemented
14C2×#386C2212yes—
15C2×#385C2212yes—
16C2×#381C2214yes—
17C2×#381C2211yescharacteristic
18C2×C4×C22⋊C4C2214yes—
19C2×#342C2218yes—
20C2×#342C2214yes—
21#1434—24no—
22C22⋊#95C2214yes—
23C4×#95—24no—
24C23×D4C2311yescharacteristic
25C24×C4C2311yescharacteristic
26C24×C4C2312yes—
27C2×#123—24no—
28C2×#123—44no—
29C22×C4⋊C4C2×C412yes—
30C22×C4⋊C4C2×C414yes—
31C22×C22⋊C4C2×C412yes—
32C22×C22⋊C4C2×C414yes—
33C22×C22⋊C4C2311yescharacteristic
34C22×C22⋊C4C2312yes—
35C22×C22⋊C4C2318yes—
36C22×C42—24no—
37C22×C42—44no—
38C22×C42C2×C414yes—
39C22×C42D414yes—
40C22×C42D414yes—
41C2×#99—216no—
42C2×#99—44no—
43#388—24no—
44#388—24no—
45#388—24no—
46#388—24no—
47#387—24no—
48#387—24no—
49#387—24no—
50#387—24no—
51#386—22no—
52#386—24no—
53#386—22no—
54#385—22no—
55#385—24no—
56#385—22no—
57#381—24no—
58#381—216no—
59C4×C22⋊C4—48no—
60C2×#95—28no—
61C2×#95—28no—
62C2×#95—24no—
63C2×#95C2×C414yes—
64C2×#95C2×C414yes—
65C2×#95C2×C414yes—
66#342—216no—
67#342—44no—
68C25C22×C411yescharacteristic
69C25C2411yescharacteristic
70C22×D4—22no—
71C22×D4—28no—
72C22×D4—24no—
73C23×C4—28no—
74C23×C4—24no—
75C23×C4—28no—
76C23×C4—28no—
77C23×C4—48no—
78C23×C4C22×C411yescharacteristic
79C23×C4C22×C412yes—
80C23×C4C22×C414yes—
81C23×C4C22×C412yes—
82C23×C4C22×C414yes—
83C23×C4C2×D414yes—
84C23×C4C2×D412yes—
85C23×C4C2×D414yes—
86C23×C4C2×D412yes—
87#123—416no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—48no—
93C2×C22⋊C4—28no—
94C2×C22⋊C4—28no—
95C2×C22⋊C4—28no—
96C2×C22⋊C4—28no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—28no—
101C2×C22⋊C4—28no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—44no—
107C2×C22⋊C4—48no—
108C2×C22⋊C4#41116yes—
109C2×C42—48no—
110C2×C42—48no—
111C2×C42—48no—
112C2×C42—48no—
113C2×C42—416no—
114C2×C42C22⋊C414yes—
115C2×C42C22⋊C414yes—
116C2×C42C22⋊C414yes—
117C2×C42C22⋊C414yes—
118#99—416no—
119#95—232no—
120#95—24no—
121#95—24no—
122#95—24no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—416no—
127#95—48no—
128C24—21no—
129C24—24no—
130C24—22no—
131C24—44no—
132C24—44no—
133C24C23×C411yescharacteristic
134C24C22×D412yes—
135C24C2×#4118yes—
136C24H(5,𝔽2)14yes—
137C2×D4—44no—
138C2×D4—44no—
139C2×D4—44no—
140C2×D4—44no—
141C2×D4—48no—
142C2×D4—44no—
143C22×C4—24no—
144C22×C4—28no—
145C22×C4—28no—
146C22×C4—24no—
147C22×C4—28no—
148C22×C4—28no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—28no—
154C22×C4—28no—
155C22×C4—24no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—216no—
161C22×C4—28no—
162C22×C4—24no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—432no—
167C22×C4—44no—
168C22×C4—48no—
169C22×C4—44no—
170C22×C4—416no—
171C22×C4—416no—
172C22×C4—48no—
173C22×C4—48no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4C2×C22⋊C414yes—
178C22×C4C2×C22⋊C414yes—
179C22×C4C2×C22⋊C414yes—
180C4⋊C4—48no—
181C4⋊C4—48no—
182C4⋊C4—48no—
183C22⋊C4—216no—
184C22⋊C4—216no—
185C22⋊C4—48no—
186C22⋊C4—48no—
187C22⋊C4—44no—
188C22⋊C4—44no—
189C22⋊C4—44no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—48no—
193C42—44no—
194C42—44no—
195C42—44no—
196C42—44no—
197C42—816no—
198C23—24no—
199C23—24no—
200C23—24no—
201C23—48no—
202C23—416no—
203C23—41no—
204C23—44no—
205C23—42no—
206C23—44no—
207C23—88no—
208C23—84no—
209C23C22×C22⋊C411yescharacteristic
210C23C4×#4114yes—
211C23#51812yes—
212C23#52514yes—
213C23#53214yes—
214C23#53414yes—
215C23#55114yes—
216C23#55318yes—
217C23#55914yes—
218D4—88no—
219C2×C4—216no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—28no—
227C2×C4—416no—
228C2×C4—416no—
229C2×C4—48no—
230C2×C4—416no—
231C2×C4—48no—
232C2×C4—432no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—416no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—88no—
243C2×C4—816no—
244C2×C4—88no—
245C2×C4—88no—
246C2×C4—84no—
247C2×C4#41018yes—
248C22—28no—
249C22—28no—
250C22—28no—
251C22—24no—
252C22—24no—
253C22—24no—
254C22—44no—
255C22—48no—
256C22—44no—
257C22—81no—
258C22—84no—
259C22—82no—
260C22—164no—
261C22#229712yes—
262C22#230811yescharacteristic
263C22#237414yes—
264C22#260218yes—
265C22C2×#41014yes—
266C4—28no—
267C4—44no—
268C4—44no—
269C4—816no—
270C4—81no—
271C4—84no—
272C4—82no—
273C2—28no—
274C2—24no—
275C2—161no—
276C2#1416711yescharacteristic
277C2#2142114yes—
278C2#2174812yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#381C2semidirect
3. #11197C2semidirect
6. #10912C2semidirect
12. C2×#388C4semidirect
13. C2×#387C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C4×#3812
#111972
#109662
#109602
#109122