Finite Group Databasebeta

← #391696

Subgroups — #391696

Order 512.

2684 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 2684 conjugacy classes of subgroups — too many to draw. Only its 258 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×#2847C211yesmaximal, characteristic
2#20579C212yesmaximal, complemented
3#11198C216yesmaximal, complemented
4#10967C216yesmaximal, complemented
5C25×C4C2211yescharacteristic
6C22×#121C2213yes—
7C22×#120C412yes—
8C22×#100C2214yescomplemented
9C2×#407C22112yes—
10#2847—26no—
11#2847—22no—
12C2×#386C416yes—
13C2×#380C2213yes—
14#1779—26no—
15#1779—22no—
16C2×#343C22112yescomplemented
17C26C2×C411yescharacteristic
18C23×D4C2×C416yes—
19C23×D4C2311yescharacteristic
20C24×C4—412no—
21C24×C4C2×C413yes—
22C24×C4D416yes—
23C24×C4D412yes—
24C24×C4C2313yes—
25C22×M16C2318yes—
26C2×#121—26no—
27C2×#121—26no—
28C2×#121—412no—
29C2×#120—26no—
30C2×#120—22no—
31C2×#120—48no—
32C22×C4⋊C4C2313yes—
33C22×C22⋊C4C2×C416yes—
34C22×C22⋊C4C2×C416yes—
35C2×#100—412no—
36#407—424no—
37C2×C22⋊C8—212no—
38C2×C22⋊C8—212no—
39#386—26no—
40#386—26no—
41#386—212no—
42#380—26no—
43#380—26no—
44C2×#95C2×C416yes—
45#343—412no—
46C25—24no—
47C25—46no—
48C25—46no—
49C25C22⋊C413yes—
50C25C22⋊C411yescharacteristic
51C25C22×C413yes—
52C25C22×C412yes—
53C22×D4—26no—
54C22×D4—26no—
55C22×D4—22no—
56C22×D4—212no—
57C22×D4—412no—
58C22×D4—412no—
59C22×D4—412no—
60C23×C4—212no—
61C23×C4—224no—
62C23×C4—412no—
63C23×C4—412no—
64C23×C4—412no—
65C23×C4—412no—
66C23×C4—412no—
67C23×C4—46no—
68C23×C4—448no—
69C23×C4—416no—
70C23×C4C22⋊C416yes—
71C23×C4C22⋊C413yes—
72C23×C4C22⋊C413yes—
73C23×C4C22×C413yes—
74C23×C4C22×C416yes—
75C23×C4C2×D416yes—
76C23×C4C2×D416yes—
77C23×C4C2411yescharacteristic
78C2×M16—224no—
79C2×M16—224no—
80C22×C8—28no—
81#121—424no—
82#121—424no—
83#120—424no—
84#120—48no—
85C2×C4⋊C4—26no—
86C2×C4⋊C4—212no—
87C2×C22⋊C4—26no—
88C2×C22⋊C4—26no—
89C2×C22⋊C4—212no—
90C2×C22⋊C4—412no—
91C2×C22⋊C4—412no—
92#100—416no—
93C22⋊C8—424no—
94C22⋊C8—424no—
95#95—212no—
96C24—26no—
97C24—26no—
98C24—26no—
99C24—212no—
100C24—26no—
101C24—212no—
102C24—22no—
103C24—212no—
104C24—43no—
105C24—43no—
106C24—46no—
107C24—43no—
108C24—43no—
109C24—46no—
110C24—448no—
111C24—42no—
112C24—46no—
113C24—416no—
114C24—448no—
115C24#10014yescomplemented
116C24C2×C22⋊C413yes—
117C24C2×C22⋊C413yes—
118C24C23×C411yescharacteristic
119C2×D4—412no—
120C2×D4—412no—
121C2×D4—412no—
122C2×D4—412no—
123C2×D4—412no—
124C2×D4—412no—
125C2×D4—412no—
126C2×D4—412no—
127C2×D4—44no—
128C2×D4—848no—
129C22×C4—26no—
130C22×C4—212no—
131C22×C4—212no—
132C22×C4—212no—
133C22×C4—212no—
134C22×C4—212no—
135C22×C4—212no—
136C22×C4—212no—
137C22×C4—26no—
138C22×C4—212no—
139C22×C4—46no—
140C22×C4—46no—
141C22×C4—46no—
142C22×C4—46no—
143C22×C4—46no—
144C22×C4—43no—
145C22×C4—43no—
146C22×C4—448no—
147C22×C4—432no—
148C22×C4—496no—
149C22×C4—436no—
150C22×C4—412no—
151C22×C4—46no—
152C22×C4—46no—
153C22×C4C2×C22⋊C416yes—
154C22×C4C22×D412yes—
155C22×C4H(5,𝔽2)112yes—
156M16—424no—
157M16—48no—
158C2×C8—424no—
159C4⋊C4—412no—
160C22⋊C4—424no—
161C22⋊C4—412no—
162C22⋊C4—412no—
163C23—21no—
164C23—23no—
165C23—224no—
166C23—23no—
167C23—23no—
168C23—26no—
169C23—26no—
170C23—23no—
171C23—23no—
172C23—26no—
173C23—26no—
174C23—412no—
175C23—424no—
176C23—424no—
177C23—424no—
178C23—412no—
179C23—412no—
180C23—412no—
181C23—424no—
182C23—44no—
183C23—412no—
184C23—4128no—
185C23—412no—
186C23—412no—
187C23—46no—
188C23—42no—
189C23—46no—
190C23—48no—
191C23—812no—
192C23—812no—
193C23C2×#10016yes—
194C23#41316yes—
195C23C22×C22⋊C411yescharacteristic
196C23#51816yes—
197D4—824no—
198D4—824no—
199D4—88no—
200C2×C4—26no—
201C2×C4—26no—
202C2×C4—26no—
203C2×C4—424no—
204C2×C4—412no—
205C2×C4—412no—
206C2×C4—412no—
207C2×C4—424no—
208C2×C4—424no—
209C2×C4—46no—
210C2×C4—46no—
211C2×C4—46no—
212C2×C4—812no—
213C2×C4#534112yes—
214C2×C4#560112yes—
215C2×C4H(3,𝔽4)14yes—
216C8—88no—
217C22—26no—
218C22—26no—
219C22—26no—
220C22—26no—
221C22—212no—
222C22—212no—
223C22—448no—
224C22—416no—
225C22—448no—
226C22—43no—
227C22—43no—
228C22—46no—
229C22—43no—
230C22—43no—
231C22—46no—
232C22—82no—
233C22—86no—
234C22—86no—
235C22—168no—
236C22#230813yes—
237C22#237514yes—
238C22C22×#10011yescharacteristic
239C22C2×#41313yes—
240C4—46no—
241C4—86no—
242C4#281218yes—
243C2—24no—
244C2—46no—
245C2—46no—
246C2—162no—
247C2#1465711yescharacteristic
248C2#2175213yes—
249C2#2175413yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2847C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20579C2semidirect
3. #11198C2semidirect
4. #10967C2semidirect
8. C22×#100C22semidirect
16. C2×#343C22semidirect
115. C24#100semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
#205792
#111982
#109672