Finite Group Databasebeta

← #119559

Subgroups — #119559

Order 512.

757 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 118 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
1#17392C212yesmaximal
2#14658C211yesmaximal, characteristic
3#8107C214yesmaximal, complemented
4C22×#123C2211yescharacteristic
5C2×#407C2214yes—
6C2×#380C2212yes—
7#1468—28no—
8#1463—22no—
9#1462—22no—
10#1441—22no—
11#1440C412yes—
12#1439—22no—
13#1438C412yes—
14#1305—28no—
15C23×D4C2×C411yescharacteristic
16C24×C4C2311yescharacteristic
17C22×M16—24no—
18C2×#123—24no—
19C2×#123—24no—
20C2×#123—24no—
21C22×C4⋊C4—22no—
22C22×C4⋊C4D412yes—
23C22×C22⋊C4—22no—
24C22×C22⋊C4C2×C411yescharacteristic
25C22×C22⋊C4C2×C414yes—
26C22×C22⋊C4D412yes—
27#407—24no—
28#407—24no—
29#407—24no—
30#407D418yes—
31C2×C22⋊C8—24no—
32C2×C22⋊C8—24no—
33#380—24no—
34#380—24no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—24no—
38#324—28no—
39#323—28no—
40C25C22⋊C411yescharacteristic
41C25C22×C411yescharacteristic
42C22×D4—22no—
43C22×D4—24no—
44C22×D4—82no—
45C23×C4—24no—
46C23×C4—82no—
47C23×C4C22⋊C412yes—
48C23×C4C22⋊C411yescharacteristic
49C23×C4C2×D412yes—
50C23×C4C2×D414yes—
51C23×C4#4114yes—
52C23×C4#4114yes—
53C2×M16—28no—
54C2×M16—44no—
55C2×M16—88no—
56C22×C8—44no—
57#123—44no—
58#123—44no—
59#123—44no—
60#123—44no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—44no—
66C2×C4⋊C4—84no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—28no—
74C2×C22⋊C4—84no—
75C2×C22⋊C4—82no—
76C2×C22⋊C4—84no—
77C22⋊C8—28no—
78C22⋊C8—28no—
79C22⋊C8—44no—
80C22⋊C8—44no—
81#95—42no—
82#95—42no—
83#95—84no—
84#95—82no—
85C24—21no—
86C24—22no—
87C24—42no—
88C24—42no—
89C24—42no—
90C24—42no—
91C24—42no—
92C24—42no—
93C24C2×C22⋊C411yescharacteristic
94C24#11712yes—
95C24C4×D414yes—
96C2×D4—44no—
97C2×D4—84no—
98C2×D4—88no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—44no—
105C22×C4—48no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—44no—
110C22×C4—44no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—84no—
117C22×C4—84no—
118C22×C4#12012yes—
119C22×C4#12114yes—
120C22×C4#12114yes—
121C22×C4#12314yes—
122C22×C4#12314yes—
123C22×C4#12312yes—
124C22×C4#12414yes—
125C22×C4#12614yes—
126M16—88no—
127M16—88no—
128C2×C8—48no—
129C2×C8—88no—
130C4⋊C4—44no—
131C4⋊C4—44no—
132C4⋊C4—88no—
133C22⋊C4—44no—
134C22⋊C4—44no—
135C22⋊C4—44no—
136C22⋊C4—88no—
137C22⋊C4—88no—
138C23—22no—
139C23—22no—
140C23—22no—
141C23—24no—
142C23—24no—
143C23—24no—
144C23—44no—
145C23—44no—
146C23—44no—
147C23—44no—
148C23—42no—
149C23—41no—
150C23—42no—
151C23—42no—
152C23—82no—
153C23—84no—
154C23—82no—
155C23—82no—
156C23—84no—
157C23—82no—
158C23—82no—
159C23—82no—
160C23#38111yescharacteristic
161C23#38612yes—
162C23#38814yes—
163D4—88no—
164C2×C4—28no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—84no—
181C2×C4—84no—
182C2×C4—84no—
183C2×C4—84no—
184C2×C4—84no—
185C2×C4—84no—
186C2×C4—84no—
187C2×C4#39414yes—
188C2×C4#39714yes—
189C8—88no—
190C22—22no—
191C22—24no—
192C22—22no—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—24no—
197C22—28no—
198C22—84no—
199C22—84no—
200C22—84no—
201C22—84no—
202C22—81no—
203C22—82no—
204C22—162no—
205C22—162no—
206C22#144011yescharacteristic
207C22#182812yes—
208C22#191412yes—
209C22#191512yes—
210C4—24no—
211C4—24no—
212C4—81no—
213C4—82no—
214C4—164no—
215C2—22no—
216C2—22no—
217C2—22no—
218C2—44no—
219C2—161no—
220C2#1116911yescharacteristic
221C2#1117312yes—
subgroupG/H
center, socleC22#1440
commutatorC25C22×C4
Frattini, cosocleC24×C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8107C2semidirect

Maximal subgroups

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

subgroupindex
#173922
#146582
#81072