Finite Group Databasebeta

← #190004

Subgroups — #190004

Order 512.

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

normal subgroups only

This group has 878 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#17357C212yesmaximal
3#8732C218yesmaximal, complemented
4#7868C212yesmaximal, complemented
5#7856C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7#2920C2214yes—
8#2314—24no—
9#2313—22no—
10#2312—22no—
11#2298C414yes—
12#2298C2214yes—
13#2285C412yes—
14#2280C412yes—
15C4×#117C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1464—24no—
18#1463—22no—
19#1462—22no—
20#1458C2218yes—
21#1276C2218yescomplemented
22#1274C2218yescomplemented
23C24×C4C2×C411yescharacteristic
24C2×#117—22no—
25C2×#117—22no—
26C2×#117—22no—
27C2×#117—22no—
28C2×#117C2×C412yes—
29C2×#117C2×C412yes—
30C2×#117C2×C412yes—
31C2×#117D414yes—
32C2×#117C2312yes—
33C22×C4⋊C4D412yes—
34C22×C22⋊C4D412yes—
35C22×C42C2×C411yescharacteristic
36C22×C42C2311yescharacteristic
37#432—24no—
38#432—24no—
39#431—24no—
40#431C2318yes—
41#423—24no—
42#407—24no—
43C4×M16—24no—
44#388—28no—
45#388—28no—
46#387—28no—
47#387—28no—
48#384—22no—
49#382—22no—
50#381—22no—
51#380—24no—
52#380—24no—
53#379—22no—
54C4×C4⋊C4—22no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4C2×C418yes—
57C4×C22⋊C4—22no—
58C4×C22⋊C4—22no—
59C4×C22⋊C4C2×C418yes—
60#376C2×C412yes—
61#376C2312yes—
62C43C2×C412yes—
63C43C2312yes—
64#324—28no—
65#323—28no—
66C25C22⋊C411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—44no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C412yes—
75C23×C4C22×C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D412yes—
78C2×M16—44no—
79#117—24no—
80#117—24no—
81#117—28no—
82#117—28no—
83#117—24no—
84#117—24no—
85#117—44no—
86#117—44no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—28no—
92C2×C4⋊C4C22⋊C414yes—
93C2×C4⋊C4C2×D414yes—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—28no—
99C2×C22⋊C4C22⋊C414yes—
100C2×C22⋊C4C2×D414yes—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—24no—
106C2×C42—24no—
107C2×C42C22⋊C412yes—
108C2×C42C22×C414yes—
109C2×C42C22×C414yes—
110C2×C42C22×C412yes—
111C2×C42C22×C411yescharacteristic
112C2×C42C2×D412yes—
113C2×C42#4112yes—
114C2×C42#4112yes—
115C2×C42#4112yes—
116C2×C42#4112yes—
117C2×C42C2411yescharacteristic
118C4⋊C8—28no—
119C4⋊C8—44no—
120C22⋊C8—28no—
121C22⋊C8—44no—
122#97—44no—
123C4×C8—28no—
124#95—24no—
125#95—28no—
126#95—28no—
127#95—24no—
128C24—21no—
129C24—21no—
130C24—42no—
131C24—82no—
132C24C2×C22⋊C412yes—
133C24C2×C22⋊C411yescharacteristic
134C22×C4—24no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—28no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—82no—
153C22×C4—82no—
154C22×C4C2×C22⋊C412yes—
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C2×C22⋊C414yes—
157C22×C4C2×C22⋊C412yes—
158C22×C4#11711yescharacteristic
159C22×C4#11711yescharacteristic
160C22×C4#11711yescharacteristic
161C22×C4#11711yescharacteristic
162C22×C4#12314yes—
163C22×C4#12314yes—
164C22×C4C23×C411yescharacteristic
165C22×C4C22×D412yes—
166M16—88no—
167C2×C8—48no—
168C4⋊C4—28no—
169C4⋊C4—28no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—88no—
173C22⋊C4—28no—
174C22⋊C4—28no—
175C22⋊C4—44no—
176C22⋊C4—44no—
177C22⋊C4—88no—
178C42—22no—
179C42—22no—
180C42—22no—
181C42—22no—
182C42—28no—
183C42—28no—
184C42—84no—
185C42—84no—
186C42#11712yes—
187C42#11712yes—
188C42#12314yes—
189C42#12314yes—
190C42C2×#4112yes—
191C42C2×#4112yes—
192C23—22no—
193C23—22no—
194C23—24no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—44no—
199C23—82no—
200C23—82no—
201C23—88no—
202C23#38111yescharacteristic
203C23#38111yescharacteristic
204C23C22×C22⋊C411yescharacteristic
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—48no—
225C2×C4—48no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4#38111yescharacteristic
231C2×C4#38111yescharacteristic
232C2×C4#38114yes—
233C2×C4C2×#11711yescharacteristic
234C2×C4C2×#11711yescharacteristic
235C2×C4C2×#12314yes—
236C2×C4#56814yes—
237C8—88no—
238C22—22no—
239C22—21no—
240C22—21no—
241C22—22no—
242C22—22no—
243C22—44no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—84no—
248C22—84no—
249C22C2×#38111yescharacteristic
250C22#287912yes—
251C4—21no—
252C4—21no—
253C4—24no—
254C4—24no—
255C4—42no—
256C4—44no—
257C4—84no—
258C4—84no—
259C4#292112yes—
260C4#292112yes—
261C2—22no—
262C2—41no—
263C2—41no—
264C2—82no—
265C2#2068311yescharacteristic
266C2#2070812yes—
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#381
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8732C2semidirect
4. #7868C2semidirect
5. #7856C2semidirect
21. #1276C22semidirect
22. #1274C22semidirect

Maximal subgroups

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

subgroupindex
#340482
#173572
#87322
#78682
#78562