Finite Group Databasebeta

← #190087

Subgroups — #190087

Order 512.

1046 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 250 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#34052C211yesmaximal, characteristic
2C2×#1464C212yesmaximal
3#8743C218yesmaximal, complemented
4#7952C214yesmaximal, complemented
5C22×#117C412yes—
6C22×#117C2211yescharacteristic
7C2×#432C2214yes—
8#2317—24no—
9#2316—22no—
10#2315—22no—
11#2296C414yes—
12#2296C2214yes—
13#2285C412yes—
14#2285C2212yes—
15#1464—24no—
16#1464C2218yes—
17C2×#324—22no—
18C2×#323—22no—
19#1285C22116yescomplemented
20C24×C4C2×C411yescharacteristic
21C2×#117—22no—
22C2×#117—22no—
23C2×#117—24no—
24C2×#117—24no—
25C2×#117—48no—
26C2×#117C2×C414yes—
27C2×#117C2×C414yes—
28C2×#117C2×C412yes—
29C2×#117D414yes—
30C2×#117C2312yes—
31C22×C4⋊C4—22no—
32C22×C4⋊C4D412yes—
33C22×C22⋊C4—22no—
34C22×C22⋊C4D412yes—
35C22×C42C2×C411yescharacteristic
36C22×C42C2×C414yes—
37C22×C42C2311yescharacteristic
38#432—24no—
39#432—24no—
40#432—24no—
41#432C2318yes—
42C2×C4⋊C8—24no—
43C2×C22⋊C8—24no—
44C2×#97—24no—
45#389—28no—
46#389—28no—
47#385—28no—
48#385—28no—
49#384—22no—
50#384—22no—
51#382—22no—
52#382—22no—
53#380—24no—
54#380—22no—
55#380—22no—
56C4×C4⋊C4C2×C418yes—
57C4×C4⋊C4C2×C414yes—
58C4×C4⋊C4C2314yes—
59#324—28no—
60#323—28no—
61C25C22⋊C411yescharacteristic
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—24no—
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C412yes—
70C23×C4C22×C412yes—
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D412yes—
73C22×C8—44no—
74#117—28no—
75#117—28no—
76#117—28no—
77#117—28no—
78#117—48no—
79#117—48no—
80#117—48no—
81#117—48no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—28no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—48no—
90C2×C4⋊C4C22⋊C414yes—
91C2×C4⋊C4C2×D414yes—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—48no—
97C2×C22⋊C4C22⋊C414yes—
98C2×C22⋊C4C2×D414yes—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—24no—
102C2×C42—48no—
103C2×C42C22⋊C412yes—
104C2×C42C22×C412yes—
105C2×C42C22×C418yes—
106C2×C42C22×C411yescharacteristic
107C2×C42C2×D412yes—
108C2×C42#4114yes—
109C2×C42#4114yes—
110C2×C42C2411yescharacteristic
111C4⋊C8—28no—
112C4⋊C8—44no—
113C22⋊C8—28no—
114C22⋊C8—44no—
115#97—28no—
116#97—44no—
117#95—28no—
118#95—24no—
119#95—24no—
120C24—22no—
121C24—22no—
122C24—82no—
123C24#10114yes—
124C24C2×C22⋊C412yes—
125C24C2×C22⋊C411yescharacteristic
126C22×C4—216no—
127C22×C4—216no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—28no—
142C22×C4—48no—
143C22×C4—82no—
144C22×C4—88no—
145C22×C4—82no—
146C22×C4#10014yes—
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C2×C22⋊C412yes—
149C22×C4C2×C22⋊C414yes—
150C22×C4C2×C22⋊C412yes—
151C22×C4#11712yes—
152C22×C4#11712yes—
153C22×C4#12314yes—
154C22×C4#12314yes—
155C22×C4C23×C411yescharacteristic
156C22×C4C22×D412yes—
157C2×C8—48no—
158C2×C8—88no—
159C4⋊C4—28no—
160C4⋊C4—28no—
161C4⋊C4—48no—
162C4⋊C4—48no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—88no—
166C22⋊C4—28no—
167C22⋊C4—28no—
168C22⋊C4—48no—
169C22⋊C4—48no—
170C42—24no—
171C42—22no—
172C42—22no—
173C42—48no—
174C42—48no—
175C42#11714yes—
176C42#12314yes—
177C42#12314yes—
178C42C2×#4114yes—
179C23—22no—
180C23—22no—
181C23—28no—
182C23—42no—
183C23—44no—
184C23—82no—
185C23—82no—
186C23—84no—
187C23—84no—
188C23#38112yes—
189C23C2×#10012yes—
190C23C2×#10114yes—
191C23C2×#10112yes—
192C23C22×C22⋊C411yescharacteristic
193C2×C4—216no—
194C2×C4—28no—
195C2×C4—28no—
196C2×C4—28no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—48no—
212C2×C4—48no—
213C2×C4—48no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—816no—
217C2×C4#38111yescharacteristic
218C2×C4#38111yescharacteristic
219C2×C4#38114yes—
220C2×C4C2×#10014yes—
221C2×C4C2×#11712yes—
222C2×C4C2×#12314yes—
223C8—88no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—24no—
228C22—42no—
229C22—42no—
230C22—44no—
231C22—44no—
232C22—44no—
233C22—88no—
234C22#182114yes—
235C22C2×#38111yescharacteristic
236C22C22×#10011yescharacteristic
237C22C22×#10111yescharacteristic
238C4—22no—
239C4—21no—
240C4—21no—
241C4—24no—
242C4—24no—
243C4—24no—
244C4—44no—
245C4—88no—
246C2—21no—
247C2—21no—
248C2—24no—
249C2—44no—
250C2C2×#182111yescharacteristic
251C2#2070812yes—
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8743C2semidirect
4. #7952C2semidirect
19. #1285C22semidirect

Maximal subgroups

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

subgroupindex
#340522
C2×#14642
#87432
#79522