Finite Group Databasebeta

← #124856

Subgroups — #124856

Order 512.

674 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 90 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#13924C211yesmaximal, characteristic
2#8152C212yesmaximal
3#7132C214yesmaximal, complemented
4C23×C4⋊C4C411yescharacteristic
5C2×#380C411yescharacteristic
6C2×C4×C22⋊C4C2211yescharacteristic
7#1749—22no—
8C2×#323—22no—
9#1445C412yes—
10#1445C2212yes—
11C4⋊#95—22no—
12C22⋊#95—22no—
13#1280C2214yes—
14C24×C4—21no—
15C24×C4C2×C411yescharacteristic
16C22×C4⋊C4—21no—
17C22×C4⋊C4—24no—
18C22×C4⋊C4—21no—
19C22×C4⋊C4—42no—
20C22×C4⋊C4—42no—
21C22×C4⋊C4C2×C412yes—
22C22×C22⋊C4—21no—
23C22×C22⋊C4D412yes—
24C22×C42D412yes—
25C2×C22⋊C8—24no—
26C2×#97—42no—
27#380—42no—
28C4×C22⋊C4—42no—
29C2×#95—24no—
30C2×#95—24no—
31C2×#95C2×C412yes—
32C2×#95C2×C411yescharacteristic
33C2×#95C2311yescharacteristic
34#336—42no—
35#323—28no—
36C25C22⋊C411yescharacteristic
37C23×C4—22no—
38C23×C4—22no—
39C23×C4—22no—
40C23×C4—22no—
41C23×C4—24no—
42C23×C4—22no—
43C23×C4—44no—
44C23×C4—42no—
45C23×C4—42no—
46C23×C4—82no—
47C23×C4—82no—
48C23×C4C22⋊C412yes—
49C23×C4C22⋊C411yescharacteristic
50C23×C4C22×C411yescharacteristic
51C23×C4C2×D412yes—
52C22×C8—44no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—416no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—44no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4M1614yes—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—42no—
66C2×C22⋊C4—42no—
67C2×C22⋊C4—44no—
68C2×C42—22no—
69C2×C42—22no—
70C2×C42—44no—
71C22⋊C8—28no—
72C22⋊C8—44no—
73#97—84no—
74#95—41no—
75#95—41no—
76#95—42no—
77#95—84no—
78#95—84no—
79C24—21no—
80C24—21no—
81C24—22no—
82C24—42no—
83C24—41no—
84C24—41no—
85C24#10012yes—
86C24#10414yes—
87C24C2×C22⋊C411yescharacteristic
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—24no—
91C22×C4—28no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—24no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—48no—
101C22×C4—48no—
102C22×C4—44no—
103C22×C4—44no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—44no—
110C22×C4—44no—
111C22×C4—48no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—88no—
120C22×C4—88no—
121C22×C4—84no—
122C22×C4—84no—
123C22×C4#9912yes—
124C22×C4C2×M1612yes—
125C22×C4#13714yes—
126C2×C8—48no—
127C2×C8—84no—
128C2×C8—84no—
129C4⋊C4—44no—
130C4⋊C4—48no—
131C4⋊C4—44no—
132C4⋊C4—84no—
133C22⋊C4—48no—
134C22⋊C4—84no—
135C42—44no—
136C42—82no—
137C23—22no—
138C23—21no—
139C23—21no—
140C23—22no—
141C23—22no—
142C23—24no—
143C23—24no—
144C23—42no—
145C23—42no—
146C23—44no—
147C23—44no—
148C23—42no—
149C23—42no—
150C23—44no—
151C23—44no—
152C23—42no—
153C23—41no—
154C23—41no—
155C23#32914yes—
156C23#40711yescharacteristic
157C23C2×#9911yescharacteristic
158C23C2×#10011yescharacteristic
159C23#41912yes—
160C23C2×#10412yes—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—416no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—44no—
174C2×C4—44no—
175C2×C4—48no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—82no—
181C2×C4—84no—
182C2×C4—816no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—84no—
188C2×C4—84no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4#35214yes—
192C8—88no—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—22no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—44no—
203C22—44no—
204C22—48no—
205C22—44no—
206C22—44no—
207C22—42no—
208C22—42no—
209C22#146211yescharacteristic
210C22#149112yes—
211C22#150812yes—
212C22C2×#32911yescharacteristic
213C22#155811yescharacteristic
214C22C2×#35212yes—
215C22#212811yescharacteristic
216C22#213411yescharacteristic
217C4—24no—
218C4—44no—
219C4—42no—
220C4—82no—
221C4—84no—
222C4—162no—
223C2—21no—
224C2—21no—
225C2—22no—
226C2—44no—
227C2#762911yescharacteristic
228C2#1150011yescharacteristic
229C2#1150911yescharacteristic
230C2#1178612yes—
231C2#1179012yes—
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. #7132C2semidirect

Maximal subgroups

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

subgroupindex
#139242
#81522
#71322