Finite Group Databasebeta

← #190141

Subgroups — #190141

Order 512.

938 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 234 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#34177C211yesmaximal, characteristic
2#20694C212yesmaximal
3#8755C218yesmaximal, complemented
4#8114C212yesmaximal
5C2×#1306C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2214yes—
8#2343—24no—
9#2343—24no—
10#2315C412yes—
11#2313C2214yes—
12#2283C414yes—
13C2×#381C2211yescharacteristic
14C2×#381C2211yescharacteristic
15#2273C412yes—
16#1822—24no—
17#1821C2218yes—
18#1742—24no—
19#1306C2218yes—
20#1306C2218yes—
21C24×C4C2311yescharacteristic
22C22×M16—24no—
23C2×#117—22no—
24C2×#117—24no—
25C2×#117—22no—
26C2×#117D414yes—
27C2×#117D414yes—
28C22×C4⋊C4C2312yes—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4C2×C412yes—
32C22×C22⋊C4C2×C412yes—
33C22×C42C2×C412yes—
34#407—24no—
35#407—24no—
36#407—24no—
37#407C2318yes—
38C2×C22⋊C8—24no—
39C2×C22⋊C8—24no—
40#387—216no—
41#385—24no—
42#385—24no—
43#383—24no—
44#382—24no—
45#381—22no—
46#381—22no—
47#381C2×C412yes—
48#381C2×C412yes—
49#381C2312yes—
50#381C2312yes—
51#380C2×C418yes—
52C4×C4⋊C4—24no—
53C4×C22⋊C4—24no—
54C4×C22⋊C4—24no—
55C4×C22⋊C4C2×C418yes—
56#376—24no—
57C2×#95—22no—
58C2×#95—22no—
59#343—216no—
60C25C22×C411yescharacteristic
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—44no—
64C23×C4C22×C412yes—
65C23×C4C22×C412yes—
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D412yes—
68C23×C4C2×D412yes—
69C23×C4#4112yes—
70C23×C4#4114yes—
71C23×C4#4112yes—
72C23×C4C2411yescharacteristic
73C2×M16—28no—
74C2×M16—44no—
75C2×M16—416no—
76C22×C8—44no—
77#117—28no—
78#117—28no—
79#117—28no—
80#117—28no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—28no—
84C2×C4⋊C4C2×D418yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—44no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4C22⋊C412yes—
94C2×C22⋊C4C22⋊C412yes—
95C2×C22⋊C4C22⋊C412yes—
96C2×C22⋊C4C22⋊C412yes—
97C2×C22⋊C4C22×C414yes—
98C2×C22⋊C4C22×C414yes—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—28no—
104C2×C42C22⋊C414yes—
105C2×C42C22⋊C414yes—
106C22⋊C8—28no—
107C22⋊C8—28no—
108C22⋊C8—44no—
109C22⋊C8—44no—
110#95—24no—
111#95—24no—
112#95—28no—
113#95—28no—
114#95—24no—
115#95—24no—
116C24—24no—
117C24—82no—
118C24C2×C22⋊C411yescharacteristic
119C24C2×C22⋊C411yescharacteristic
120C24#11712yes—
121C24#11712yes—
122C24C23×C411yescharacteristic
123C22×C4—28no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—48no—
134C22×C4—44no—
135C22×C4—42no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—82no—
139C22×C4—82no—
140C22×C4—84no—
141C22×C4—84no—
142C22×C4C2×C22⋊C412yes—
143C22×C4C2×C22⋊C414yes—
144C22×C4C2×C22⋊C414yes—
145C22×C4#11711yescharacteristic
146C22×C4#11712yes—
147C22×C4#11711yescharacteristic
148C22×C4#12314yes—
149C22×C4#12314yes—
150C22×C4#12314yes—
151C22×C4#12314yes—
152C22×C4C22×D412yes—
153C22×C4C2×#4112yes—
154C22×C4C2×#4112yes—
155M16—416no—
156M16—88no—
157C2×C8—48no—
158C2×C8—88no—
159C4⋊C4—216no—
160C4⋊C4—88no—
161C22⋊C4—24no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—24no—
165C22⋊C4—44no—
166C22⋊C4—44no—
167C22⋊C4—44no—
168C22⋊C4—44no—
169C42—24no—
170C42—24no—
171C42—24no—
172C42—24no—
173C42—88no—
174C23—22no—
175C23—22no—
176C23—24no—
177C23—24no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—44no—
182C23—44no—
183C23—88no—
184C23#38112yes—
185C23#38112yes—
186C23#41014yes—
187C23C22×C22⋊C411yescharacteristic
188C23C2×#11711yescharacteristic
189C23C2×#11711yescharacteristic
190C2×C4—28no—
191C2×C4—28no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—48no—
206C2×C4—48no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—164no—
214C2×C4—164no—
215C2×C4#38112yes—
216C2×C4#38112yes—
217C2×C4C2×#12314yes—
218C8—88no—
219C22—22no—
220C22—24no—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—24no—
227C22—44no—
228C22—44no—
229C22—44no—
230C22—84no—
231C22—84no—
232C22#182014yes—
233C22C2×#38111yescharacteristic
234C22C2×#41012yes—
235C4—24no—
236C4—22no—
237C4—22no—
238C4—44no—
239C4—44no—
240C4—84no—
241C4—84no—
242C2—22no—
243C2—24no—
244C2—41no—
245C2—41no—
246C2—42no—
247C2C2×#182011yescharacteristic
248C2#2071312yes—
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8755C2semidirect

Maximal subgroups

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

subgroupindex
#341772
#206942
#87552
#81142
C2×#13062