Finite Group Databasebeta

← #124999

Subgroups — #124999

Order 512.

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

normal subgroups only

This group has 662 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#13965C211yesmaximal, characteristic
2#8698C212yesmaximal
3#7123C214yesmaximal, complemented
4C2×#385—22no—
5C2×#380C411yescharacteristic
6C2×C4×C22⋊C4C411yescharacteristic
7C22×#95C2211yescharacteristic
8#1779—22no—
9#1444C412yes—
10#1444C2212yes—
11C4⋊#95—22no—
12#1280—22no—
13#1280C2214yes—
14C24×C4C2×C411yescharacteristic
15C24×C4D412yes—
16C22×C4⋊C4—22no—
17C22×C4⋊C4—22no—
18C22×C4⋊C4C2×C412yes—
19C22×C22⋊C4—21no—
20C22×C22⋊C4—21no—
21C22×C42—21no—
22C2×C22⋊C8—24no—
23C2×C22⋊C8—42no—
24C2×C22⋊C8—42no—
25#385—48no—
26#380—41no—
27#380—41no—
28C4×C22⋊C4—24no—
29C2×#95—21no—
30C2×#95—22no—
31C2×#95—22no—
32C2×#95—41no—
33C2×#95—41no—
34C2×#95C2×C412yes—
35C2×#95C2×C411yescharacteristic
36C2×#95D412yes—
37C2×#95C2311yescharacteristic
38C25C22⋊C411yescharacteristic
39C23×C4—22no—
40C23×C4—21no—
41C23×C4—21no—
42C23×C4—22no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—24no—
47C23×C4—22no—
48C23×C4—42no—
49C23×C4—44no—
50C23×C4—82no—
51C23×C4—82no—
52C23×C4C22⋊C412yes—
53C23×C4C22⋊C411yescharacteristic
54C23×C4C22×C411yescharacteristic
55C23×C4C2×D412yes—
56C22×C8—44no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—44no—
59C2×C4⋊C4—44no—
60C2×C4⋊C4—44no—
61C2×C4⋊C4—44no—
62C2×C4⋊C4—82no—
63C2×C4⋊C4—82no—
64C2×C4⋊C4—82no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—44no—
69C2×C22⋊C4—41no—
70C2×C22⋊C4—41no—
71C2×C42—24no—
72C2×C42—82no—
73C22⋊C8—44no—
74C22⋊C8—44no—
75C22⋊C8—84no—
76C22⋊C8—84no—
77#95—22no—
78#95—41no—
79#95—42no—
80#95—41no—
81#95—42no—
82#95—44no—
83#95—44no—
84#95M1614yes—
85C24—22no—
86C24—22no—
87C24—21no—
88C24—21no—
89C24—22no—
90C24—22no—
91C24—42no—
92C24#9911yescharacteristic
93C24#9911yescharacteristic
94C24C2×C22⋊C411yescharacteristic
95C22×C4—24no—
96C22×C4—24no—
97C22×C4—24no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—44no—
110C22×C4—44no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4—82no—
134C22×C4—84no—
135C22×C4—84no—
136C22×C4—88no—
137C22×C4—88no—
138C22×C4#10112yes—
139C22×C4C2×M1612yes—
140C22×C4#13514yes—
141C2×C8—84no—
142C2×C8—84no—
143C2×C8—84no—
144C4⋊C4—88no—
145C4⋊C4—88no—
146C4⋊C4—84no—
147C22⋊C4—216no—
148C22⋊C4—82no—
149C22⋊C4—82no—
150C42—84no—
151C23—22no—
152C23—21no—
153C23—21no—
154C23—24no—
155C23—24no—
156C23—22no—
157C23—22no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—22no—
163C23—22no—
164C23—44no—
165C23—44no—
166C23—44no—
167C23—44no—
168C23—42no—
169C23—42no—
170C23—41no—
171C23—41no—
172C23—42no—
173C23#35312yes—
174C23#35612yes—
175C23#40711yescharacteristic
176C23C2×#9911yescharacteristic
177C23C2×#10111yescharacteristic
178C23#41612yes—
179C23#42112yes—
180C2×C4—28no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—82no—
204C2×C4—84no—
205C2×C4—84no—
206C2×C4—88no—
207C2×C4—88no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C8—164no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—24no—
226C22—24no—
227C22—22no—
228C22—22no—
229C22—48no—
230C22—44no—
231C22—44no—
232C22—42no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22#146311yescharacteristic
239C22#148812yes—
240C22#151612yes—
241C22#153711yescharacteristic
242C22#154411yescharacteristic
243C22#212112yes—
244C22C2×#35311yescharacteristic
245C22C2×#35611yescharacteristic
246C4—48no—
247C4—44no—
248C4—42no—
249C4—42no—
250C4—82no—
251C4—162no—
252C2—22no—
253C2—21no—
254C2—21no—
255C2—42no—
256C2—42no—
257C2#764611yescharacteristic
258C2#1151211yescharacteristic
259C2#1151911yescharacteristic
260C2#1184312yes—
261C2#1185412yes—
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. #7123C2semidirect

Maximal subgroups

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

subgroupindex
#139652
#86982
#71232