Finite Group Databasebeta

← #190059

Subgroups — #190059

Order 512.

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

normal subgroups only

This group has 944 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#34048C211yesmaximal, characteristic
2#20694C211yesmaximal, characteristic
3#20688C211yesmaximal, characteristic
4#8740C218yesmaximal, complemented
5#7946C214yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8#2314—22no—
9#2314—22no—
10#2313—22no—
11#2312—22no—
12#2298C414yescomplemented
13#2298C2214yes—
14#2285C412yescomplemented
15#2280C2212yes—
16#2280C2212yes—
17C4×#117C412yescomplemented
18#1822C2214yes—
19#1821—22no—
20#1820C2214yes—
21#1819—22no—
22#1742—22no—
23#1740—22no—
24#1284C22116yes—
25C24×C4C2311yescharacteristic
26C22×M16—22no—
27C2×#117—24no—
28C2×#117—24no—
29C2×#117—22no—
30C2×#117C2×C414yes—
31C2×#117C2×C412yes—
32C2×#117D412yes—
33C2×#117D412yes—
34C2×#117C2312yes—
35C22×C4⋊C4D412yes—
36C22×C22⋊C4D412yes—
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39#407—24no—
40#407—22no—
41#407C2314yes—
42C2×C22⋊C8—24no—
43#388—28no—
44#388—28no—
45#387—28no—
46#387—28no—
47#384—22no—
48#382—22no—
49#381—22no—
50#381C2314yes—
51#380—24no—
52#380—22no—
53#380—22no—
54#380—24no—
55#379—22no—
56#379—22no—
57C4×C4⋊C4—24no—
58C4×C4⋊C4C2×C418yes—
59C4×C22⋊C4—24no—
60C4×C22⋊C4—24no—
61C4×C22⋊C4C2×C414yes—
62C4×C22⋊C4C2314yes—
63#376C2×C414yes—
64C43C2×C414yes—
65#343—28no—
66#342—28no—
67C25C2×D411yescharacteristic
68C23×C4—44no—
69C23×C4C22⋊C412yes—
70C23×C4C22⋊C412yes—
71C23×C4C22×C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4#4114yes—
77C23×C4#4114yes—
78C23×C4C2411yescharacteristic
79C2×M16—24no—
80C2×M16—42no—
81C2×M16—48no—
82C22×C8—42no—
83#117—28no—
84#117—24no—
85#117—24no—
86#117—24no—
87#117—24no—
88#117—28no—
89#117—48no—
90#117—48no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4C22⋊C414yes—
97C2×C4⋊C4C2×D414yes—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4C22⋊C414yes—
106C2×C22⋊C4C2×D414yes—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—28no—
112C2×C42—42no—
113C2×C42C22⋊C412yes—
114C2×C42C22⋊C412yes—
115C2×C42C22×C412yes—
116C2×C42C22×C412yes—
117C2×C42C22×C418yes—
118C22⋊C8—28no—
119C22⋊C8—44no—
120#95—28no—
121#95—24no—
122#95—28no—
123#95—24no—
124#95—28no—
125C24—42no—
126C24—82no—
127C24C2×C22⋊C411yescharacteristic
128C24C2×C22⋊C411yescharacteristic
129C24#12314yes—
130C24C22×D411yescharacteristic
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—28no—
139C22×C4—48no—
140C22×C4—48no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—82no—
146C22×C4—82no—
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C2×C22⋊C414yes—
149C22×C4C2×C22⋊C414yes—
150C22×C4C2×C22⋊C411yescharacteristic
151C22×C4#11712yes—
152C22×C4#11712yes—
153C22×C4#11714yes—
154C22×C4#12312yes—
155C22×C4#12312yes—
156C22×C4#12312yes—
157C22×C4#12312yes—
158C22×C4#12314yes—
159C22×C4C23×C411yescharacteristic
160C22×C4C22×D411yescharacteristic
161C22×C4C2×#4114yes—
162M16—48no—
163M16—84no—
164C2×C8—44no—
165C2×C8—84no—
166C4⋊C4—28no—
167C4⋊C4—28no—
168C4⋊C4—48no—
169C4⋊C4—84no—
170C4⋊C4—88no—
171C22⋊C4—28no—
172C22⋊C4—28no—
173C22⋊C4—48no—
174C22⋊C4—48no—
175C22⋊C4—84no—
176C22⋊C4—84no—
177C22⋊C4—88no—
178C42—24no—
179C42—24no—
180C42—22no—
181C42—22no—
182C42—22no—
183C42—22no—
184C42—48no—
185C42—84no—
186C42—88no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—44no—
193C23—42no—
194C23—44no—
195C23—44no—
196C23—88no—
197C23#38112yes—
198C23#38112yes—
199C23C22×C22⋊C411yescharacteristic
200C23C2×#12312yes—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—48no—
218C2×C4—48no—
219C2×C4—48no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—84no—
225C2×C4—88no—
226C2×C4#38112yes—
227C2×C4#38111yescharacteristic
228C2×C4#38111yescharacteristic
229C2×C4#41014yes—
230C2×C4C2×#11712yes—
231C2×C4C2×#12312yes—
232C8—84no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—24no—
238C22—22no—
239C22—22no—
240C22—24no—
241C22—44no—
242C22—82no—
243C22—82no—
244C22—88no—
245C22C2×#38111yescharacteristic
246C22C2×#41012yes—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—44no—
251C4—44no—
252C4—88no—
253C4—164no—
254C4#182012yes—
255C4#182012yes—
256C2—22no—
257C2—42no—
258C2—42no—
259C2—42no—
260C2C2×#182011yescharacteristic
261C2#2070812yes—
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ 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
4. #8740C2semidirect
12. #2298C4semidirect
14. #2285C4semidirect
17. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
#340482
#206942
#206882
#87402
#79462