Finite Group Databasebeta

← #124977

Subgroups — #124977

Order 512.

1166 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 102 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#13922C211yesmaximal, characteristic
2#8559C212yesmaximal
3#7131C214yesmaximal, complemented
4C24×Q8C411yescharacteristic
5C2×#391—22no—
6C2×#380C411yescharacteristic
7C2×C4×C22⋊C4C2211yescharacteristic
8#1797—22no—
9C2×#323—22no—
10#1446C412yes—
11#1446C2212yes—
12C22⋊#95—22no—
13#1280C2214yes—
14C23×Q8—24no—
15C23×Q8—21no—
16C23×Q8—42no—
17C23×Q8—42no—
18C23×Q8—48no—
19C23×Q8C2×C412yes—
20C24×C4—21no—
21C24×C4C2×C411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C22⋊C4—21no—
24C22×C22⋊C4D412yes—
25C22×C42D412yes—
26C2×C4⋊C8—42no—
27C2×C22⋊C8—24no—
28#391—48no—
29#380—42no—
30C4×C22⋊C4—42no—
31C2×#95—24no—
32C2×#95—24no—
33C2×#95C2×C412yes—
34C2×#95C2×C411yescharacteristic
35C2×#95C2311yescharacteristic
36#336—42no—
37#323—44no—
38C25C22⋊C411yescharacteristic
39C22×Q8—24no—
40C22×Q8—28no—
41C22×Q8—22no—
42C22×Q8—48no—
43C22×Q8—42no—
44C22×Q8—44no—
45C22×Q8—416no—
46C22×Q8—42no—
47C22×Q8—42no—
48C22×Q8—432no—
49C22×Q8—48no—
50C22×Q8—432no—
51C22×Q8—416no—
52C22×Q8—48no—
53C22×Q8—42no—
54C22×Q8M1614yes—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—24no—
60C23×C4—22no—
61C23×C4—48no—
62C23×C4—44no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D412yes—
69C22×C8—44no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—44no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—44no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4SD1614yes—
77C2×C22⋊C4Q1614yes—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42—44no—
81C4⋊C8—84no—
82C22⋊C8—44no—
83C22⋊C8—44no—
84#95—41no—
85#95—41no—
86#95—42no—
87#95—48no—
88#95—48no—
89C24—21no—
90C24—21no—
91C24—22no—
92C24—42no—
93C24—41no—
94C24—41no—
95C24#10012yes—
96C24#10314yes—
97C24C2×C22⋊C411yescharacteristic
98C2×Q8—216no—
99C2×Q8—48no—
100C2×Q8—416no—
101C2×Q8—48no—
102C2×Q8—48no—
103C2×Q8—48no—
104C2×Q8—48no—
105C2×Q8—44no—
106C2×Q8—44no—
107C2×Q8—464no—
108C2×Q8—464no—
109C2×Q8—416no—
110C2×Q8—48no—
111C2×Q8—48no—
112C2×Q8—48no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—24no—
116C22×C4—28no—
117C22×C4—28no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—48no—
126C22×C4—48no—
127C22×C4—44no—
128C22×C4—416no—
129C22×C4—44no—
130C22×C4—48no—
131C22×C4—416no—
132C22×C4—416no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—44no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4#9912yes—
147C22×C4#10314yes—
148C22×C4C2×M1612yes—
149C22×C4C2×SD1612yes—
150C22×C4C2×Q1612yes—
151C2×C8—84no—
152C2×C8—84no—
153C2×C8—84no—
154C4⋊C4—84no—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C22⋊C4—84no—
158C42—44no—
159C42—82no—
160C23—22no—
161C23—21no—
162C23—21no—
163C23—22no—
164C23—24no—
165C23—24no—
166C23—22no—
167C23—42no—
168C23—42no—
169C23—44no—
170C23—44no—
171C23—42no—
172C23—44no—
173C23—44no—
174C23—42no—
175C23—42no—
176C23—41no—
177C23—41no—
178C23#33314yes—
179C23#40711yescharacteristic
180C23C2×#9911yescharacteristic
181C23C2×#10011yescharacteristic
182C23C2×#10312yes—
183C23#42112yes—
184Q8—432no—
185Q8—48no—
186Q8—48no—
187Q8—416no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—416no—
198C2×C4—44no—
199C2×C4—48no—
200C2×C4—42no—
201C2×C4—432no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—48no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—84no—
217C2×C4—84no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4AΣL(1,𝔽16)14yes—
221C8—164no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—24no—
231C22—44no—
232C22—44no—
233C22—48no—
234C22—44no—
235C22—44no—
236C22—42no—
237C22—42no—
238C22#146211yescharacteristic
239C22#148712yes—
240C22#152012yes—
241C22#153311yescharacteristic
242C22C2×#33311yescharacteristic
243C22C2×AΣL(1,𝔽16)12yes—
244C22#212911yescharacteristic
245C22#213511yescharacteristic
246C4—24no—
247C4—42no—
248C4—42no—
249C4—44no—
250C4—48no—
251C4—42no—
252C4—162no—
253C2—21no—
254C2—21no—
255C2—22no—
256C2—44no—
257C2#764311yescharacteristic
258C2#1149911yescharacteristic
259C2#1151011yescharacteristic
260C2#1189812yes—
261C2#1190412yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7131C2semidirect

Maximal subgroups

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

subgroupindex
#139222
#85592
#71312