Finite Group Databasebeta

← #122201

Subgroups — #122201

Order 512.

1162 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 166 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×#400C211yesmaximal, characteristic
2#10727C212yesmaximal
3#7909C214yesmaximal
4C23×C4⋊C4C411yescharacteristic
5C23×C4⋊C4C2211yescharacteristic
6C2×#407C2214yes—
7C2×#400—28no—
8C2×#400—24no—
9C22×#95C411yescharacteristic
10C22×#95C412yes—
11C22×#95C2212yes—
12C2×#343—24no—
13#1280—28no—
14C24×C4C2×C411yescharacteristic
15C24×C4C2×C412yes—
16C24×C4C2×C411yescharacteristic
17C24×C4C2×C412yes—
18C24×C4C2311yescharacteristic
19C22×M16—24no—
20C22×C4⋊C4—24no—
21C22×C4⋊C4—22no—
22C22×C4⋊C4—24no—
23C22×C4⋊C4—44no—
24C22×C4⋊C4—44no—
25C22×C4⋊C4D414yes—
26#407—28no—
27#407—24no—
28#407Q818yes—
29C2×C22⋊C8—28no—
30#400—232no—
31C2×#95—24no—
32C2×#95—24no—
33C2×#95—24no—
34C2×#95—24no—
35C2×#95—22no—
36C2×#95—24no—
37C2×#95—24no—
38C2×#95D418yes—
39#343—48no—
40C25C22×C411yescharacteristic
41C23×C4—22no—
42C23×C4—24no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4—42no—
51C23×C4—44no—
52C23×C4—44no—
53C23×C4—44no—
54C23×C4—44no—
55C23×C4—44no—
56C23×C4—44no—
57C23×C4—44no—
58C23×C4—44no—
59C23×C4C22⋊C414yes—
60C23×C4C22⋊C412yes—
61C23×C4C22⋊C414yes—
62C23×C4C22⋊C412yes—
63C23×C4C2×D412yes—
64C23×C4C2×D414yes—
65C23×C4C2×Q814yes—
66C23×C4#4114yes—
67C2×M16—28no—
68C2×M16—44no—
69C2×M16—48no—
70C2×M16—48no—
71C22×C8—44no—
72C2×C4⋊C4—28no—
73C2×C4⋊C4—28no—
74C2×C4⋊C4—28no—
75C2×C4⋊C4—44no—
76C2×C4⋊C4—416no—
77C2×C4⋊C4—416no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4—44no—
80C22⋊C8—216no—
81C22⋊C8—48no—
82#95—28no—
83#95—216no—
84#95—216no—
85C24—22no—
86C24—22no—
87C24—24no—
88C24—24no—
89C24C2×C22⋊C411yescharacteristic
90C24C2×C22⋊C412yes—
91C24C4×Q814yes—
92C22×C4—24no—
93C22×C4—24no—
94C22×C4—28no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—24no—
98C22×C4—22no—
99C22×C4—24no—
100C22×C4—24no—
101C22×C4—28no—
102C22×C4—28no—
103C22×C4—44no—
104C22×C4—44no—
105C22×C4—48no—
106C22×C4—48no—
107C22×C4—416no—
108C22×C4—44no—
109C22×C4—44no—
110C22×C4—44no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—44no—
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—88no—
132C22×C4—816no—
133C22×C4—816no—
134C22×C4#9914yes—
135C22×C4#12014yes—
136C22×C4C22⋊Q814yes—
137C22×C4C22⋊Q818yes—
138C22×C4#12412yes—
139C22×C4#12414yes—
140C22×C4C4⋊Q812yes—
141C22×C4C4⋊Q814yes—
142M16—416no—
143M16—88no—
144C2×C8—48no—
145C2×C8—84no—
146C2×C8—84no—
147C4⋊C4—416no—
148C4⋊C4—416no—
149C23—28no—
150C23—28no—
151C23—216no—
152C23—28no—
153C23—216no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23#37911yescharacteristic
161C23#39112yes—
162C23#39114yes—
163C23C2×#9912yes—
164C23#41012yes—
165C23#41314yes—
166C23#41314yes—
167C2×C4—28no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—48no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—416no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—88no—
189C2×C4—88no—
190C2×C4—88no—
191C2×C4—88no—
192C2×C4—88no—
193C2×C4—84no—
194C2×C4—88no—
195C2×C4—88no—
196C2×C4#39314yes—
197C2×C4#39514yes—
198C8—88no—
199C22—232no—
200C22—22no—
201C22—24no—
202C22—22no—
203C22—24no—
204C22—24no—
205C22—24no—
206C22—24no—
207C22—24no—
208C22—24no—
209C22—24no—
210C22#144611yescharacteristic
211C22#178111yescharacteristic
212C22#178211yescharacteristic
213C22#178512yes—
214C22#178512yes—
215C22#199614yes—
216C22#199818yes—
217C4—24no—
218C4—24no—
219C4—44no—
220C4—82no—
221C4—84no—
222C4—84no—
223C2—24no—
224C2—28no—
225C2#854811yescharacteristic
226C2#855211yescharacteristic
227C2#855612yes—
228C2#1138111yescharacteristic
229C2#1138312yes—
subgroupG/H
center, socleC23#379
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C22×#4002
#107272
#79092