Finite Group Databasebeta

← #97087

Subgroups — #97087

Order 512.

1682 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1682 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
1C22×#399C211yesmaximal, characteristic
2#8299C214yesmaximal, complemented
3#8298C212yesmaximal
4C23×C4⋊C4C411yescharacteristic
5C23×C4⋊C4C2212yes—
6C23×C22⋊C4C412yes—
7C23×C22⋊C4C2211yescharacteristic
8C2×#399—28no—
9C2×#399—24no—
10C22×#95C411yescharacteristic
11#1779C2214yes—
12C2×#324—28no—
13C2×#323—24no—
14C26C2×C411yescharacteristic
15C24×C4C2×C412yes—
16C24×C4C2×C411yescharacteristic
17C24×C4C2×C412yes—
18C24×C4C2311yescharacteristic
19C22×C4⋊C4—24no—
20C22×C4⋊C4—22no—
21C22×C4⋊C4—24no—
22C22×C4⋊C4—24no—
23C22×C4⋊C4—48no—
24C22×C4⋊C4—44no—
25C22×C4⋊C4D418yes—
26C22×C22⋊C4—24no—
27C22×C22⋊C4—24no—
28C22×C22⋊C4—24no—
29C22×C22⋊C4—24no—
30C22×C22⋊C4—44no—
31C22×C22⋊C4—48no—
32C22×C22⋊C4D414yes—
33C2×C22⋊C8—24no—
34C2×C22⋊C8—28no—
35#399—232no—
36C2×#95—24no—
37C2×#95—22no—
38#324—416no—
39#323—48no—
40C25—24no—
41C25—22no—
42C25—24no—
43C25—42no—
44C25—44no—
45C25—44no—
46C25C22⋊C412yes—
47C25C22×C411yescharacteristic
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—28no—
52C23×C4—24no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—28no—
56C23×C4—44no—
57C23×C4—42no—
58C23×C4—44no—
59C23×C4—44no—
60C23×C4—44no—
61C23×C4—44no—
62C23×C4—44no—
63C23×C4C22⋊C414yes—
64C23×C4C22⋊C414yes—
65C23×C4C22⋊C412yes—
66C23×C4C2×D414yes—
67C23×C4C2×D412yes—
68C22×C8—44no—
69C2×C4⋊C4—28no—
70C2×C4⋊C4—216no—
71C2×C4⋊C4—216no—
72C2×C4⋊C4—48no—
73C2×C4⋊C4—416no—
74C2×C4⋊C4—416no—
75C2×C4⋊C4—416no—
76C2×C4⋊C4—44no—
77C2×C4⋊C4—44no—
78C2×C22⋊C4—216no—
79C2×C22⋊C4—28no—
80C2×C22⋊C4—28no—
81C2×C22⋊C4—44no—
82C2×C22⋊C4—416no—
83C2×C22⋊C4—416no—
84C2×C22⋊C4—416no—
85C2×C22⋊C4—48no—
86C2×C22⋊C4—48no—
87C22⋊C8—48no—
88C22⋊C8—416no—
89#95—28no—
90C24—22no—
91C24—24no—
92C24—22no—
93C24—24no—
94C24—28no—
95C24—22no—
96C24—22no—
97C24—24no—
98C24—24no—
99C24—416no—
100C24—44no—
101C24—44no—
102C24—432no—
103C24—44no—
104C24—44no—
105C24—44no—
106C24—44no—
107C24—44no—
108C24—44no—
109C24—816no—
110C24—88no—
111C24—88no—
112C24#10114yes—
113C24C2×C22⋊C411yescharacteristic
114C24C2×C22⋊C412yes—
115C24#13114yes—
116C22×C4—28no—
117C22×C4—28no—
118C22×C4—28no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—28no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—28no—
125C22×C4—24no—
126C22×C4—28no—
127C22×C4—24no—
128C22×C4—28no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—432no—
133C22×C4—416no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—88no—
150C22×C4—816no—
151C22×C4#9918yes—
152C22×C4#9914yes—
153C22×C4#12014yes—
154C2×C8—84no—
155C2×C8—88no—
156C4⋊C4—432no—
157C4⋊C4—416no—
158C22⋊C4—416no—
159C22⋊C4—432no—
160C23—28no—
161C23—28no—
162C23—216no—
163C23—216no—
164C23—28no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—24no—
172C23—28no—
173C23—416no—
174C23—48no—
175C23—416no—
176C23—416no—
177C23—48no—
178C23—44no—
179C23—42no—
180C23—42no—
181C23—44no—
182C23—44no—
183C23—84no—
184C23—84no—
185C23—88no—
186C23—864no—
187C23—88no—
188C23—88no—
189C23—88no—
190C23—88no—
191C23#37911yescharacteristic
192C23#40814yes—
193C23#40812yes—
194C23C2×#9914yes—
195C23C2×#9912yes—
196C23C2×#10112yes—
197C23#41314yes—
198C2×C4—216no—
199C2×C4—24no—
200C2×C4—28no—
201C2×C4—24no—
202C2×C4—28no—
203C2×C4—44no—
204C2×C4—416no—
205C2×C4—416no—
206C2×C4—416no—
207C2×C4—48no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—84no—
216C2×C4—88no—
217C2×C4—88no—
218C2×C4—88no—
219C2×C4—88no—
220C8—164no—
221C22—232no—
222C22—24no—
223C22—22no—
224C22—22no—
225C22—24no—
226C22—24no—
227C22—24no—
228C22—24no—
229C22—24no—
230C22—24no—
231C22—24no—
232C22—48no—
233C22—44no—
234C22—88no—
235C22—832no—
236C22—88no—
237C22—82no—
238C22—82no—
239C22—84no—
240C22#146818yes—
241C22#147214yes—
242C22#178011yescharacteristic
243C22#178111yescharacteristic
244C22#178112yes—
245C22#178411yescharacteristic
246C22#178512yes—
247C4—44no—
248C4—44no—
249C4—48no—
250C4—84no—
251C2—28no—
252C2—24no—
253C2—84no—
254C2#830412yes—
255C2#830811yescharacteristic
256C2#854611yescharacteristic
257C2#854811yescharacteristic
258C2#855012yes—
subgroupG/H
center, socleC23#379
commutatorC25C22×C4
Frattini, cosocleC24×C4C23
prime radicalC26C2×C4
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8299C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3992
#82992
#82982