Finite Group Databasebeta

← #122200

Subgroups — #122200

Order 512.

1786 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1786 conjugacy classes of subgroups — too many to draw. Only its 182 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×#395C211yesmaximal, characteristic
2#10727C212yesmaximal
3#7909C214yesmaximal
4C24×Q8C411yescharacteristic
5C23×C4⋊C4C412yes—
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2214yes—
8C2×#395—28no—
9C2×#395—24no—
10C22×#95C411yescharacteristic
11C22×#95C2212yes—
12C2×#343—24no—
13#1280—28no—
14C23×Q8—22no—
15C23×Q8—24no—
16C23×Q8—44no—
17C23×Q8—48no—
18C24×C4C2×C412yes—
19C24×C4C2×C411yescharacteristic
20C24×C4C2×C412yes—
21C24×C4C2×C411yescharacteristic
22C24×C4C2311yescharacteristic
23C22×M16—24no—
24C22×C4⋊C4—24no—
25C22×C4⋊C4—24no—
26C22×C4⋊C4—24no—
27C22×C4⋊C4—24no—
28C22×C4⋊C4—44no—
29C22×C4⋊C4—48no—
30C22×C4⋊C4D414yes—
31#407—28no—
32#407—24no—
33#407Q818yescomplemented
34C2×C22⋊C8—28no—
35#395—232no—
36C2×#95—24no—
37C2×#95—22no—
38C2×#95—24no—
39C2×#95—24no—
40C2×#95D418yes—
41#343—48no—
42C25C22×C411yescharacteristic
43C22×Q8—28no—
44C22×Q8—416no—
45C22×Q8—416no—
46C22×Q8—416no—
47C22×Q8—44no—
48C22×Q8—48no—
49C22×Q8—464no—
50C22×Q8—44no—
51C22×Q8—48no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—24no—
58C23×C4—28no—
59C23×C4—28no—
60C23×C4—22no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4—44no—
65C23×C4—44no—
66C23×C4—44no—
67C23×C4—44no—
68C23×C4—44no—
69C23×C4—44no—
70C23×C4C22⋊C412yes—
71C23×C4C22⋊C412yes—
72C23×C4C22⋊C414yes—
73C23×C4C22⋊C414yes—
74C23×C4C2×D412yes—
75C23×C4C2×D414yes—
76C23×C4C2×Q814yes—
77C23×C4#4114yes—
78C2×M16—28no—
79C2×M16—44no—
80C2×M16—48no—
81C2×M16—48no—
82C22×C8—44no—
83C2×C4⋊C4—216no—
84C2×C4⋊C4—28no—
85C2×C4⋊C4—28no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—416no—
88C2×C4⋊C4—416no—
89C2×C4⋊C4—416no—
90C2×C4⋊C4—48no—
91C2×C4⋊C4—48no—
92C22⋊C8—216no—
93C22⋊C8—48no—
94#95—28no—
95#95—216no—
96C24—22no—
97C24—22no—
98C24—24no—
99C24—24no—
100C24C2×C22⋊C411yescharacteristic
101C24C2×C22⋊C412yes—
102C24C4×Q814yes—
103C2×Q8—416no—
104C2×Q8—432no—
105C2×Q8—432no—
106C2×Q8—464no—
107C2×Q8—432no—
108C2×Q8—464no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—28no—
112C22×C4—28no—
113C22×C4—28no—
114C22×C4—28no—
115C22×C4—28no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—48no—
127C22×C4—48no—
128C22×C4—44no—
129C22×C4—432no—
130C22×C4—44no—
131C22×C4—416no—
132C22×C4—432no—
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—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—88no—
151C22×C4#9914yes—
152C22×C4#10018yes—
153C22×C4#10118yes—
154C22×C4#12014yes—
155C22×C4C22⋊Q814yes—
156C22×C4C22⋊Q818yes—
157C22×C4#12412yes—
158C22×C4#12414yes—
159C22×C4C4⋊Q812yes—
160C22×C4C4⋊Q814yes—
161M16—416no—
162M16—88no—
163C2×C8—48no—
164C2×C8—84no—
165C2×C8—84no—
166C4⋊C4—416no—
167C4⋊C4—432no—
168C23—28no—
169C23—216no—
170C23—28no—
171C23—28no—
172C23—216no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23#37911yescharacteristic
180C23#39112yes—
181C23#39114yes—
182C23C2×#9912yes—
183C23#41012yes—
184C23C2×#10014yes—
185C23C2×#10114yes—
186Q8—464no—
187C2×C4—28no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—28no—
194C2×C4—28no—
195C2×C4—48no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—416no—
199C2×C4—416no—
200C2×C4—48no—
201C2×C4—416no—
202C2×C4—44no—
203C2×C4—416no—
204C2×C4—416no—
205C2×C4—48no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—48no—
213C2×C4—48no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—88no—
217C2×C4—84no—
218C2×C4#39314yes—
219C2×C4#39514yes—
220C8—88no—
221C22—232no—
222C22—22no—
223C22—24no—
224C22—22no—
225C22—24no—
226C22—24no—
227C22—24no—
228C22—24no—
229C22—24no—
230C22—24no—
231C22—24no—
232C22#144611yescharacteristic
233C22#178111yescharacteristic
234C22#178211yescharacteristic
235C22#178312yes—
236C22#178412yes—
237C22#199614yes—
238C22#199718yes—
239C4—24no—
240C4—24no—
241C4—44no—
242C4—48no—
243C4—48no—
244C4—82no—
245C2—24no—
246C2—28no—
247C2#854711yescharacteristic
248C2#855111yescharacteristic
249C2#855512yes—
250C2#1138111yescharacteristic
251C2#1138212yes—
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

Complements

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

NH ≅ G/Ntype
33. #407Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#3952
#107272
#79092