Finite Group Databasebeta

← #189712

Subgroups — #189712

Order 512.

1798 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1798 conjugacy classes of subgroups — too many to draw. Only its 322 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×C4×C4⋊C4C211yesmaximal, characteristic
2#20682C212yesmaximal
3#8701C218yesmaximal, complemented
4#7909C214yesmaximal
5C23×C4⋊C4C412yes—
6C23×C42C412yes—
7C23×C42C2211yescharacteristic
8C2×#407C2214yes—
9C2×C4×C4⋊C4—22no—
10C2×C4×C4⋊C4—22no—
11C2×C4×C4⋊C4—24no—
12C2×C4×C4⋊C4C414yes—
13C2×C4×C4⋊C4C2214yes—
14C22×#95C2212yes—
15#1817—22no—
16#1817—22no—
17#1817C2218yes—
18#1728—24no—
19#1280C22116yes—
20C24×C4C2×C411yescharacteristic
21C24×C4C2×C411yescharacteristic
22C24×C4C2×C414yes—
23C24×C4C2311yescharacteristic
24C22×M16—24no—
25C22×C4⋊C4—24no—
26C22×C4⋊C4—24no—
27C22×C4⋊C4—216no—
28C22×C4⋊C4C2×C414yes—
29C22×C4⋊C4C2×C414yes—
30C22×C42—28no—
31C22×C42—24no—
32C22×C42—22no—
33C22×C42—22no—
34C22×C42—48no—
35C22×C42C2×C412yes—
36C22×C42C2×C418yes—
37C22×C42D412yes—
38C22×C42D412yes—
39C22×C42D414yes—
40C22×C42C2312yes—
41#407—216no—
42C2×C22⋊C8—24no—
43C2×C22⋊C8C2318yes—
44C4×C4⋊C4—28no—
45C4×C4⋊C4—28no—
46C4×C4⋊C4—28no—
47C4×C4⋊C4—28no—
48C2×#95—24no—
49C2×#95—24no—
50C2×#95C2×C414yes—
51C2×#95C2314yes—
52#336—28no—
53#336—28no—
54C25C22×C411yescharacteristic
55C23×C4—24no—
56C23×C4—24no—
57C23×C4—24no—
58C23×C4—24no—
59C23×C4—28no—
60C23×C4—24no—
61C23×C4—24no—
62C23×C4—216no—
63C23×C4—44no—
64C23×C4—48no—
65C23×C4—48no—
66C23×C4C22⋊C412yes—
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C412yes—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22×C412yes—
72C23×C4C22×C412yes—
73C23×C4C22×C418yes—
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D412yes—
76C23×C4C2×D412yes—
77C23×C4#4118yes—
78C23×C4C2411yescharacteristic
79C2×M16—416no—
80C2×M16—48no—
81C22×C8—28no—
82C2×C4⋊C4—232no—
83C2×C4⋊C4—232no—
84C2×C4⋊C4—28no—
85C2×C4⋊C4—28no—
86C2×C4⋊C4—216no—
87C2×C4⋊C4—28no—
88C2×C4⋊C4M16116yescomplemented
89C2×C42—28no—
90C2×C42—216no—
91C2×C42—28no—
92C2×C42—28no—
93C2×C42—28no—
94C2×C42—28no—
95C2×C42—28no—
96C2×C42—48no—
97C2×C42—416no—
98C2×C42—48no—
99C2×C42—416no—
100C2×C42C22⋊C414yes—
101C2×C42C22⋊C414yes—
102C2×C42C2×D414yes—
103C2×C42C2×D414yes—
104C22⋊C8—216no—
105C22⋊C8—48no—
106#95—28no—
107#95—28no—
108C24—24no—
109C24—24no—
110C24—22no—
111C24—22no—
112C24C2×C22⋊C411yescharacteristic
113C24C2×C22⋊C411yescharacteristic
114C24#11714yes—
115C24C23×C411yescharacteristic
116C22×C4—232no—
117C22×C4—232no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—28no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—28no—
130C22×C4—28no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—216no—
136C22×C4—216no—
137C22×C4—28no—
138C22×C4—28no—
139C22×C4—42no—
140C22×C4—44no—
141C22×C4—416no—
142C22×C4—42no—
143C22×C4—48no—
144C22×C4—464no—
145C22×C4—48no—
146C22×C4—48no—
147C22×C4—48no—
148C22×C4—48no—
149C22×C4—48no—
150C22×C4#10014yes—
151C22×C4#10114yes—
152C22×C4C2×C22⋊C412yes—
153C22×C4C2×C22⋊C414yes—
154C22×C4C2×C22⋊C412yes—
155C22×C4C2×C22⋊C412yes—
156C22×C4#11714yes—
157C22×C4#12318yes—
158C22×C4#12318yes—
159C22×C4C2×M1618yes—
160C22×C4C2×M16116yes—
161C22×C4C22×D412yes—
162C22×C4C2×#4114yes—
163M16—816no—
164C2×C8—416no—
165C2×C8—48no—
166C4⋊C4—232no—
167C4⋊C4—232no—
168C42—28no—
169C42—28no—
170C42—28no—
171C42—28no—
172C42—416no—
173C42—416no—
174C23—216no—
175C23—216no—
176C23—28no—
177C23—28no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—28no—
187C23#38112yes—
188C23#38112yes—
189C23#41014yes—
190C23C2×#10012yes—
191C23C2×#10112yes—
192C23C22×C22⋊C411yescharacteristic
193C23C2×#11712yes—
194C23C22×M1614yes—
195C2×C4—216no—
196C2×C4—216no—
197C2×C4—216no—
198C2×C4—216no—
199C2×C4—216no—
200C2×C4—28no—
201C2×C4—28no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—216no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—216no—
212C2×C4—44no—
213C2×C4—48no—
214C2×C4—44no—
215C2×C4—416no—
216C2×C4—432no—
217C2×C4—432no—
218C2×C4—48no—
219C2×C4—48no—
220C2×C4—416no—
221C2×C4#38114yes—
222C2×C4#40718yes—
223C2×C4C2×#10014yes—
224C2×C4C2×#10114yes—
225C2×C4C2×#12314yes—
226C8—88no—
227C22—232no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—24no—
233C22—24no—
234C22—24no—
235C22—22no—
236C22—22no—
237C22—24no—
238C22—24no—
239C22—24no—
240C22#182014yes—
241C22#182114yes—
242C22C2×#38111yescharacteristic
243C22C2×#40712yes—
244C22C2×#41012yes—
245C22C22×#10011yescharacteristic
246C22C22×#10111yescharacteristic
247C22#292014yes—
248C4—28no—
249C4—24no—
250C4—24no—
251C4—24no—
252C4—24no—
253C4—416no—
254C2—24no—
255C2—24no—
256C2—24no—
257C2#1736412yes—
258C2#1736512yes—
259C2#2068211yescharacteristic
260C2C2×#182011yescharacteristic
261C2C2×#182111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8701C2semidirect
88. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C4⋊C42
#206822
#87012
#79092