Finite Group Databasebeta

← #134066

Subgroups — #134066

Order 512.

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

normal subgroups only

This group has 858 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
1C2×#2503C211yesmaximal, characteristic
2#11511C214yesmaximal, complemented
3#8555C212yesmaximal
4C22×C22⋊Q8C2211yescharacteristic
5C23×C4⋊C4C411yescharacteristic
6#2503—28no—
7C2×#400—22no—
8C2×#391C412yes—
9C2×#382C2212yes—
10C2×#381—22no—
11C2×C4×C22⋊C4C411yescharacteristic
12#1821—22no—
13#1463C2214yes—
14#1462—22no—
15#1324—28no—
16C23×Q8C2×C411yescharacteristic
17C24×C4—21no—
18C24×C4C2×C411yescharacteristic
19C2×C22⋊Q8—24no—
20C2×C22⋊Q8—42no—
21C2×C22⋊Q8—42no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—24no—
24C22×C4⋊C4—42no—
25C22×C4⋊C4—42no—
26C22×C4⋊C4C2×C412yes—
27C22×C4⋊C4D412yes—
28C22×C4⋊C4C2311yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4D412yes—
31C22×C42C2×C412yes—
32#407—24no—
33#400—216no—
34#391—24no—
35#391—24no—
36#382—24no—
37#382D418yes—
38#381—216no—
39C4×C22⋊C4—24no—
40C2×#95—21no—
41C2×#95—24no—
42C2×#95—24no—
43#343—44no—
44#324—28no—
45#323—44no—
46C25C22⋊C411yescharacteristic
47C22×Q8—22no—
48C22×Q8—41no—
49C22×Q8—41no—
50C22×Q8—44no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—24no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—44no—
58C23×C4—42no—
59C23×C4—48no—
60C23×C4—42no—
61C23×C4C22⋊C412yes—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22×C411yescharacteristic
64C23×C4C2×D412yes—
65C2×M16—44no—
66C22⋊Q8—416no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—416no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4C22⋊C414yes—
80C2×C4⋊C4C2×D414yes—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—44no—
86C2×C42—22no—
87C2×C42—44no—
88C2×C42C22⋊C414yes—
89C22⋊C8—44no—
90C22⋊C8—44no—
91#95—24no—
92#95—216no—
93#95—216no—
94C24—24no—
95C24—21no—
96C24—21no—
97C24—42no—
98C24—42no—
99C24#9912yes—
100C24C2×C22⋊C411yescharacteristic
101C2×Q8—44no—
102C2×Q8—44no—
103C2×Q8—44no—
104C2×Q8—84no—
105C2×Q8—84no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—24no—
109C22×C4—28no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—48no—
120C22×C4—48no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4—42no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—48no—
128C22×C4—44no—
129C22×C4—416no—
130C22×C4—48no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—44no—
134C22×C4—88no—
135C22×C4—84no—
136C22×C4—84no—
137C22×C4#9912yes—
138C22×C4C2×C22⋊C412yes—
139C22×C4#12014yes—
140M16—88no—
141C2×C8—84no—
142C4⋊C4—44no—
143C4⋊C4—48no—
144C4⋊C4—44no—
145C4⋊C4—44no—
146C4⋊C4—48no—
147C22⋊C4—48no—
148C22⋊C4—48no—
149C42—28no—
150C42—44no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—24no—
155C23—24no—
156C23—22no—
157C23—22no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—416no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23#35514yes—
166C23#37911yescharacteristic
167C23C2×#9911yescharacteristic
168C23C2×#9911yescharacteristic
169Q8—88no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—416no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—48no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—82no—
190C2×C4—88no—
191C2×C4—84no—
192C2×C4—816no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4#41314yes—
197C8—164no—
198C22—28no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—24no—
206C22—44no—
207C22—48no—
208C22—48no—
209C22—44no—
210C22—42no—
211C22—42no—
212C22#178111yescharacteristic
213C22#178512yes—
214C22#212912yes—
215C22C2×#35512yes—
216C4—24no—
217C4—42no—
218C4—44no—
219C4—84no—
220C4—82no—
221C4—84no—
222C2—22no—
223C2—22no—
224C2—22no—
225C2—44no—
226C2#854811yescharacteristic
227C2#1269811yescharacteristic
228C2#1269911yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11511C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
#115112
#85552