Finite Group Databasebeta

← #105813

Subgroups — #105813

Order 512.

1166 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2316C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3C2×#1305C214yesmaximal
4C22×#117C412yes—
5C23×C4⋊C4C2211yescharacteristic
6C2×#407C2214yes—
7#2316—28no—
8C2×#389—24no—
9C2×#380C412yes—
10C2×#380C2212yes—
11#1463—24no—
12#1463C418yescomplemented
13C2×#324—24no—
14#1305—28no—
15C24×C4C2×C411yescharacteristic
16C24×C4C2×C411yescharacteristic
17C24×C4C2311yescharacteristic
18C22×M16—24no—
19C2×#117—28no—
20C2×#117—44no—
21C2×#117—44no—
22C22×C4⋊C4—24no—
23C22×C4⋊C4—24no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4C2×C414yes—
28C22×C4⋊C4D412yes—
29C22×C4⋊C4D414yes—
30C22×C4⋊C4Q812yes—
31C22×C22⋊C4C2×C414yes—
32C22×C42—22no—
33#407—24no—
34#407—24no—
35#407—24no—
36#407C2×C418yes—
37C2×C22⋊C8—24no—
38C2×C22⋊C8—24no—
39#389—232no—
40#380—24no—
41#380—28no—
42#380D418yes—
43C2×#95—22no—
44C2×#95—22no—
45#324—216no—
46C25C22×C411yescharacteristic
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—24no—
50C23×C4—24no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—24no—
54C23×C4—82no—
55C23×C4—82no—
56C23×C4C4212yes—
57C23×C4C4212yes—
58C23×C4C22⋊C412yes—
59C23×C4C22⋊C412yes—
60C23×C4C4⋊C412yes—
61C23×C4C4⋊C412yes—
62C23×C4C22×C412yes—
63C23×C4C2×D412yes—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D414yes—
66C23×C4C2×Q811yescharacteristic
67C23×C4#4114yes—
68C2×M16—28no—
69C2×M16—44no—
70C2×M16—88no—
71C22×C8—44no—
72#117—432no—
73C2×C4⋊C4—28no—
74C2×C4⋊C4—28no—
75C2×C4⋊C4—28no—
76C2×C4⋊C4—28no—
77C2×C4⋊C4—28no—
78C2×C4⋊C4—28no—
79C2×C4⋊C4—28no—
80C2×C4⋊C4—28no—
81C2×C4⋊C4—28no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—84no—
85C2×C4⋊C4—84no—
86C2×C4⋊C4C22⋊C418yes—
87C2×C4⋊C4C4⋊C418yes—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—44no—
90C2×C22⋊C4—44no—
91C2×C22⋊C4C22⋊C418yes—
92C2×C42—28no—
93C2×C42—44no—
94C2×C42—44no—
95C22⋊C8—28no—
96C22⋊C8—28no—
97C22⋊C8—44no—
98C22⋊C8—44no—
99#95—28no—
100#95—28no—
101C24—22no—
102C24—22no—
103C24—42no—
104C24—42no—
105C24C2×C4211yescharacteristic
106C24C2×C22⋊C411yescharacteristic
107C24C2×C22⋊C412yes—
108C24C2×C4⋊C411yescharacteristic
109C24#11712yes—
110C22×C4—28no—
111C22×C4—216no—
112C22×C4—28no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—28no—
120C22×C4—28no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—28no—
126C22×C4—44no—
127C22×C4—48no—
128C22×C4—44no—
129C22×C4—48no—
130C22×C4—48no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—82no—
134C22×C4—84no—
135C22×C4—81no—
136C22×C4—81no—
137C22×C4—84no—
138C22×C4—84no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4#9514yes—
142C22×C4#9514yes—
143C22×C4#10014yes—
144C22×C4#10114yes—
145C22×C4C2×C22⋊C412yes—
146C22×C4C2×C4⋊C412yes—
147C22×C4C4×D414yes—
148C22×C4C4×D414yes—
149C22×C4#12014yes—
150C22×C4#12114yes—
151C22×C4C22⋊Q814yes—
152C22×C4#12314yes—
153M16—88no—
154M16—88no—
155C2×C8—48no—
156C2×C8—88no—
157C4⋊C4—216no—
158C4⋊C4—216no—
159C4⋊C4—416no—
160C4⋊C4—88no—
161C4⋊C4—88no—
162C22⋊C4—48no—
163C22⋊C4—48no—
164C42—416no—
165C23—22no—
166C23—22no—
167C23—24no—
168C23—24no—
169C23—24no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—41no—
175C23—42no—
176C23—41no—
177C23—41no—
178C23—42no—
179C23—41no—
180C23—48no—
181C23—48no—
182C23C2×#9511yescharacteristic
183C23C4×C22⋊C412yes—
184C23#37911yescharacteristic
185C23#38012yes—
186C23#38111yescharacteristic
187C23C2×#10012yes—
188C23C2×#10112yes—
189C2×C4—216no—
190C2×C4—216no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—28no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—48no—
210C2×C4—48no—
211C2×C4—48no—
212C2×C4—84no—
213C2×C4—84no—
214C2×C4—88no—
215C2×C4—88no—
216C2×C4#34318yes—
217C2×C4#38514yes—
218C2×C4#38614yes—
219C8—88no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—24no—
225C22—24no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—44no—
233C22—44no—
234C22—44no—
235C22—44no—
236C22—48no—
237C22C22⋊#9511yescharacteristic
238C22C2×#34312yes—
239C22#178311yescharacteristic
240C22#178411yescharacteristic
241C22#182112yes—
242C22#182812yes—
243C22#183012yes—
244C4—24no—
245C4—24no—
246C4—28no—
247C4—44no—
248C4—88no—
249C2—21no—
250C2—22no—
251C2—21no—
252C2—42no—
253C2—42no—
254C2#875714yes—
255C2#1072711yescharacteristic
256C2#1076811yescharacteristic
257C2#1077011yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1463C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C2×#14632
C2×#13052