Finite Group Databasebeta

← #97093

Subgroups — #97093

Order 512.

1490 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1490 conjugacy classes of subgroups — too many to draw. Only its 122 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×#397C211yesmaximal, characteristic
2#8698C214yesmaximal, complemented
3#8298C212yesmaximal
4C23×C4⋊C4C411yescharacteristic
5C23×C22⋊C4C412yes—
6C23×C22⋊C4C2211yescharacteristic
7C2×#397—28no—
8C2×#397—24no—
9C22×#95C411yescharacteristic
10C22×#95C2212yes—
11#1779C2214yes—
12C2×#323—24no—
13#1280—28no—
14C26C2×C411yescharacteristic
15C24×C4C2×C412yes—
16C24×C4C2×C411yescharacteristic
17C24×C4C2×C412yes—
18C24×C4C2311yescharacteristic
19C22×C4⋊C4—24no—
20C22×C4⋊C4—22no—
21C22×C4⋊C4—44no—
22C22×C22⋊C4—24no—
23C22×C22⋊C4—24no—
24C22×C22⋊C4—24no—
25C22×C22⋊C4—24no—
26C22×C22⋊C4—44no—
27C22×C22⋊C4—48no—
28C22×C22⋊C4D414yes—
29C2×C22⋊C8—28no—
30C2×C22⋊C8—24no—
31#397—232no—
32C2×#95—24no—
33C2×#95—22no—
34C2×#95—24no—
35C2×#95—24no—
36C2×#95D418yes—
37#323—48no—
38C25—24no—
39C25—22no—
40C25—24no—
41C25—42no—
42C25—44no—
43C25—44no—
44C25C22⋊C412yes—
45C25C22×C411yescharacteristic
46C23×C4—22no—
47C23×C4—24no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—28no—
53C23×C4—42no—
54C23×C4—44no—
55C23×C4—44no—
56C23×C4—44no—
57C23×C4—44no—
58C23×C4—44no—
59C23×C4—44no—
60C23×C4—44no—
61C23×C4C22⋊C414yes—
62C23×C4C22⋊C414yes—
63C23×C4C22⋊C412yes—
64C23×C4C2×D414yes—
65C23×C4C2×D412yes—
66C22×C8—44no—
67C2×C4⋊C4—28no—
68C2×C4⋊C4—416no—
69C2×C4⋊C4—44no—
70C2×C4⋊C4—44no—
71C2×C22⋊C4—216no—
72C2×C22⋊C4—28no—
73C2×C22⋊C4—28no—
74C2×C22⋊C4—44no—
75C2×C22⋊C4—416no—
76C2×C22⋊C4—416no—
77C2×C22⋊C4—416no—
78C2×C22⋊C4—48no—
79C2×C22⋊C4—48no—
80C22⋊C8—48no—
81C22⋊C8—416no—
82#95—28no—
83#95—216no—
84C24—22no—
85C24—24no—
86C24—22no—
87C24—24no—
88C24—28no—
89C24—22no—
90C24—22no—
91C24—24no—
92C24—24no—
93C24—416no—
94C24—44no—
95C24—432no—
96C24—44no—
97C24—44no—
98C24—44no—
99C24—44no—
100C24—44no—
101C24—44no—
102C24—44no—
103C24—816no—
104C24—88no—
105C24—88no—
106C24#10014yes—
107C24C2×C22⋊C411yescharacteristic
108C24C2×C22⋊C412yes—
109C24#13114yes—
110C22×C4—28no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—28no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—28no—
118C22×C4—28no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—416no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4—44no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—44no—
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—88no—
143C22×C4—816no—
144C22×C4—816no—
145C22×C4#9914yes—
146C22×C4#12014yes—
147C2×C8—84no—
148C2×C8—88no—
149C4⋊C4—416no—
150C22⋊C4—416no—
151C22⋊C4—432no—
152C23—28no—
153C23—216no—
154C23—28no—
155C23—28no—
156C23—216no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—24no—
164C23—28no—
165C23—416no—
166C23—48no—
167C23—416no—
168C23—416no—
169C23—48no—
170C23—44no—
171C23—42no—
172C23—42no—
173C23—44no—
174C23—44no—
175C23—84no—
176C23—84no—
177C23—88no—
178C23—864no—
179C23—88no—
180C23—88no—
181C23—88no—
182C23—88no—
183C23#37911yescharacteristic
184C23#40812yes—
185C23#40814yes—
186C23C2×#9912yes—
187C23#41014yes—
188C23C2×#10012yes—
189C23#41314yes—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—28no—
193C2×C4—44no—
194C2×C4—416no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—84no—
204C2×C4—88no—
205C2×C4—88no—
206C2×C4—88no—
207C2×C4—88no—
208C2×C4—88no—
209C2×C4—88no—
210C2×C4—88no—
211C8—164no—
212C22—232no—
213C22—22no—
214C22—24no—
215C22—22no—
216C22—24no—
217C22—24no—
218C22—24no—
219C22—24no—
220C22—24no—
221C22—24no—
222C22—24no—
223C22—48no—
224C22—44no—
225C22—88no—
226C22—88no—
227C22—832no—
228C22—82no—
229C22—82no—
230C22—84no—
231C22#147114yes—
232C22#147318yes—
233C22#178011yescharacteristic
234C22#178111yescharacteristic
235C22#178212yes—
236C22#178311yescharacteristic
237C22#178512yes—
238C4—44no—
239C4—44no—
240C4—84no—
241C4—84no—
242C2—24no—
243C2—28no—
244C2—84no—
245C2#830711yescharacteristic
246C2#830912yes—
247C2#854411yescharacteristic
248C2#854811yescharacteristic
249C2#855312yes—
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. #8698C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3972
#86982
#82982