Finite Group Databasebeta

← #124998

Subgroups — #124998

Order 512.

590 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 82 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#13966C211yesmaximal, characteristic
2#8698C212yesmaximal
3#7122C214yesmaximal, complemented
4C2×#380C411yescharacteristic
5C2×C4×C22⋊C4C411yescharacteristic
6C22×#95C2211yescharacteristic
7#1779—22no—
8#1442C412yes—
9#1442C2212yes—
10#1434—22no—
11C22⋊#95—22no—
12#1280—22no—
13#1280C2214yes—
14C24×C4C2×C411yescharacteristic
15C24×C4D412yes—
16C22×C4⋊C4C2×C412yes—
17C22×C22⋊C4—21no—
18C22×C22⋊C4—21no—
19C22×C42—21no—
20C2×C22⋊C8—24no—
21C2×C22⋊C8—42no—
22C2×C22⋊C8—42no—
23#380—41no—
24#380—41no—
25C4×C22⋊C4—24no—
26C2×#95—22no—
27C2×#95—21no—
28C2×#95—22no—
29C2×#95—22no—
30C2×#95—22no—
31C2×#95—41no—
32C2×#95—41no—
33C2×#95C2×C412yes—
34C2×#95C2×C411yescharacteristic
35C2×#95D412yes—
36C2×#95C2311yescharacteristic
37C25C22⋊C411yescharacteristic
38C23×C4—21no—
39C23×C4—21no—
40C23×C4—22no—
41C23×C4—22no—
42C23×C4—22no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—24no—
46C23×C4—22no—
47C23×C4—42no—
48C23×C4—44no—
49C23×C4—82no—
50C23×C4—82no—
51C23×C4C22⋊C412yes—
52C23×C4C22⋊C411yescharacteristic
53C23×C4C22×C411yescharacteristic
54C23×C4C2×D412yes—
55C22×C8—44no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—82no—
58C2×C22⋊C4—24no—
59C2×C22⋊C4—42no—
60C2×C22⋊C4—42no—
61C2×C22⋊C4—44no—
62C2×C22⋊C4—41no—
63C2×C22⋊C4—41no—
64C2×C42—24no—
65C2×C42—82no—
66C22⋊C8—44no—
67C22⋊C8—44no—
68C22⋊C8—84no—
69C22⋊C8—84no—
70#95—22no—
71#95—41no—
72#95—42no—
73#95—41no—
74#95—44no—
75#95—42no—
76#95—44no—
77#95—82no—
78#95—82no—
79#95M1614yes—
80C24—22no—
81C24—22no—
82C24—21no—
83C24—21no—
84C24—22no—
85C24—22no—
86C24—42no—
87C24#9911yescharacteristic
88C24#9911yescharacteristic
89C24C2×C22⋊C411yescharacteristic
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—24no—
96C22×C4—24no—
97C22×C4—22no—
98C22×C4—24no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—44no—
103C22×C4—44no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—44no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—44no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—82no—
126C22×C4—82no—
127C22×C4—82no—
128C22×C4—82no—
129C22×C4—84no—
130C22×C4—84no—
131C22×C4—88no—
132C22×C4—88no—
133C22×C4#10112yes—
134C22×C4C2×M1612yes—
135C22×C4#13514yes—
136C2×C8—84no—
137C2×C8—84no—
138C2×C8—84no—
139C4⋊C4—84no—
140C22⋊C4—48no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C42—84no—
144C23—22no—
145C23—21no—
146C23—21no—
147C23—24no—
148C23—24no—
149C23—22no—
150C23—22no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—22no—
156C23—22no—
157C23—44no—
158C23—44no—
159C23—44no—
160C23—44no—
161C23—42no—
162C23—42no—
163C23—41no—
164C23—41no—
165C23—42no—
166C23#35412yes—
167C23#35512yes—
168C23#40711yescharacteristic
169C23C2×#9911yescharacteristic
170C23C2×#10111yescharacteristic
171C23#41612yes—
172C23#42112yes—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4—84no—
194C2×C4—84no—
195C2×C4—88no—
196C2×C4—88no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C8—164no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—24no—
219C22—24no—
220C22—22no—
221C22—22no—
222C22—48no—
223C22—44no—
224C22—44no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22#146311yescharacteristic
232C22#148812yes—
233C22#151612yes—
234C22#153711yescharacteristic
235C22#154411yescharacteristic
236C22#212112yes—
237C22C2×#35411yescharacteristic
238C22C2×#35511yescharacteristic
239C4—44no—
240C4—42no—
241C4—42no—
242C4—84no—
243C4—82no—
244C4—162no—
245C2—22no—
246C2—21no—
247C2—21no—
248C2—42no—
249C2—42no—
250C2#764611yescharacteristic
251C2#1151111yescharacteristic
252C2#1151911yescharacteristic
253C2#1184512yes—
254C2#1185212yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#139662
#86982
#71222