Finite Group Databasebeta

← #134062

Subgroups — #134062

Order 512.

1004 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1004 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×#2504C211yesmaximal, characteristic
2#11511C214yesmaximal, complemented
3#8549C212yesmaximal
4C22×#121C2211yescharacteristic
5C23×C4⋊C4C411yescharacteristic
6#2504—28no—
7C2×#397—22no—
8C2×#390C412yes—
9C2×#386—22no—
10C2×#382C2212yes—
11C2×C4×C22⋊C4C411yescharacteristic
12#1783—22no—
13#1463C2214yes—
14#1462—22no—
15#1324—28no—
16C23×D4C2×C411yescharacteristic
17C23×D4D412yes—
18C24×C4—21no—
19C24×C4C2×C411yescharacteristic
20C2×#121—24no—
21C2×#121—42no—
22C2×#121—42no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—24no—
25C22×C4⋊C4—42no—
26C22×C4⋊C4—42no—
27C22×C4⋊C4C2×C412yes—
28C22×C4⋊C4C2311yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—24no—
31C22×C22⋊C4D412yes—
32C22×C42C2×C412yes—
33C2×#100—44no—
34#407—24no—
35#397—216no—
36#390—24no—
37#390—24no—
38#386—216no—
39#382—24no—
40#382D418yes—
41C4×C22⋊C4—24no—
42C2×#95—21no—
43C2×#95—24no—
44#324—28no—
45#323—44no—
46C25C22⋊C411yescharacteristic
47C25C22⋊C412yes—
48C22×D4—24no—
49C22×D4—22no—
50C22×D4—24no—
51C22×D4—41no—
52C22×D4—41no—
53C22×D4—42no—
54C22×D4—44no—
55C22×D4—48no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—24no—
62C23×C4—44no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4—84no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D412yes—
69C2×M16—44no—
70#121—416no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—416no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4C22⋊C414yes—
81C2×C4⋊C4C2×D414yes—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—216no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—44no—
88C2×C22⋊C4—88no—
89C2×C42—22no—
90C2×C42—44no—
91C2×C42C22⋊C414yes—
92#100—88no—
93C22⋊C8—44no—
94C22⋊C8—44no—
95#95—24no—
96#95—216no—
97C24—24no—
98C24—21no—
99C24—21no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—42no—
104C24—42no—
105C24—44no—
106C24—44no—
107C24—44no—
108C24#9912yes—
109C24C2×C22⋊C411yescharacteristic
110C2×D4—48no—
111C2×D4—44no—
112C2×D4—44no—
113C2×D4—416no—
114C2×D4—48no—
115C2×D4—44no—
116C2×D4—84no—
117C2×D4—84no—
118C2×D4—84no—
119C2×D4—88no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—28no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—48no—
131C22×C4—48no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—48no—
140C22×C4—48no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—88no—
144C22×C4—88no—
145C22×C4—88no—
146C22×C4#9912yes—
147C22×C4C2×C22⋊C412yes—
148C22×C4#12014yes—
149M16—88no—
150C2×C8—84no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—48no—
155C22⋊C4—48no—
156C22⋊C4—48no—
157C22⋊C4—816no—
158C42—28no—
159C42—44no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—24no—
164C23—24no—
165C23—22no—
166C23—22no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—416no—
172C23—44no—
173C23—44no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—44no—
178C23—44no—
179C23—44no—
180C23—816no—
181C23—82no—
182C23—82no—
183C23—82no—
184C23—82no—
185C23#35414yes—
186C23#37911yescharacteristic
187C23C2×#9911yescharacteristic
188C23C2×#9911yescharacteristic
189D4—88no—
190D4—816no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—416no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—48no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—82no—
209C2×C4—88no—
210C2×C4—84no—
211C2×C4—816no—
212C2×C4—84no—
213C2×C4#41314yes—
214C8—164no—
215C22—28no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—24no—
223C22—44no—
224C22—48no—
225C22—44no—
226C22—48no—
227C22—42no—
228C22—42no—
229C22—88no—
230C22—82no—
231C22—82no—
232C22—82no—
233C22—164no—
234C22—164no—
235C22#178111yescharacteristic
236C22#178512yes—
237C22C2×#35412yes—
238C22#213412yes—
239C4—24no—
240C4—42no—
241C4—44no—
242C4—84no—
243C4—84no—
244C2—22no—
245C2—22no—
246C2—22no—
247C2—44no—
248C2—162no—
249C2#854811yescharacteristic
250C2#1269411yescharacteristic
251C2#1270311yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ 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×#25042
#115112
#85492