Finite Group Databasebeta

← #391697

Subgroups — #391697

Order 512.

2414 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 2414 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×#2848C211yesmaximal, characteristic
2#20579C212yesmaximal
3#11198C216yesmaximal, complemented
4#10968C216yesmaximal, complemented
5C25×C4C2211yescharacteristic
6C22×C22⋊Q8C2213yes—
7C22×#101C2214yescomplemented
8C2×#407C22112yes—
9#2848—26no—
10#2848—22no—
11C2×#385C416yes—
12C2×#380C2213yes—
13C2×#379C412yes—
14#1779—26no—
15#1779—22no—
16C2×#343C22112yescomplemented
17C26C2×C411yescharacteristic
18C23×Q8C2311yescharacteristic
19C24×C4—412no—
20C24×C4C2×C413yes—
21C24×C4D416yes—
22C24×C4D412yes—
23C24×C4C2313yes—
24C22×M16C2318yes—
25C2×C22⋊Q8—26no—
26C2×C22⋊Q8—26no—
27C2×C22⋊Q8—412no—
28C22×C4⋊C4C2×C416yes—
29C22×C4⋊C4C2313yes—
30C22×C22⋊C4C2×C416yes—
31C22×C22⋊C4C2×C416yes—
32C2×#101—412no—
33#407—424no—
34C2×C22⋊C8—212no—
35C2×C22⋊C8—212no—
36#385—212no—
37#385—26no—
38#385—26no—
39#380—26no—
40#380—26no—
41#379—26no—
42#379—22no—
43C2×#95C2×C416yes—
44#343—412no—
45C25—24no—
46C25—46no—
47C25—46no—
48C25C22⋊C413yes—
49C25C22⋊C411yescharacteristic
50C25C22×C413yes—
51C22×Q8—22no—
52C22×Q8—212no—
53C23×C4—212no—
54C23×C4—224no—
55C23×C4—412no—
56C23×C4—412no—
57C23×C4—412no—
58C23×C4—412no—
59C23×C4—412no—
60C23×C4—46no—
61C23×C4—448no—
62C23×C4—416no—
63C23×C4C22⋊C416yes—
64C23×C4C22⋊C413yes—
65C23×C4C22⋊C413yes—
66C23×C4C22×C412yes—
67C23×C4C22×C413yes—
68C23×C4C22×C416yes—
69C23×C4C2×D416yes—
70C23×C4C2×D416yes—
71C23×C4C2411yescharacteristic
72C2×M16—224no—
73C2×M16—224no—
74C22×C8—28no—
75C22⋊Q8—424no—
76C22⋊Q8—424no—
77C2×C4⋊C4—212no—
78C2×C4⋊C4—26no—
79C2×C4⋊C4—212no—
80C2×C4⋊C4—412no—
81C2×C22⋊C4—26no—
82C2×C22⋊C4—26no—
83C2×C22⋊C4—26no—
84C2×C22⋊C4—26no—
85C2×C22⋊C4—412no—
86C2×C22⋊C4—412no—
87#101—416no—
88C22⋊C8—424no—
89C22⋊C8—424no—
90#95—212no—
91C24—26no—
92C24—26no—
93C24—26no—
94C24—212no—
95C24—26no—
96C24—212no—
97C24—43no—
98C24—43no—
99C24—46no—
100C24—43no—
101C24—43no—
102C24—46no—
103C24—448no—
104C24—416no—
105C24—448no—
106C24#10114yescomplemented
107C24C2×C22⋊C413yes—
108C24C2×C22⋊C413yes—
109C24C23×C411yescharacteristic
110C2×Q8—412no—
111C2×Q8—412no—
112C2×Q8—44no—
113C22×C4—26no—
114C22×C4—212no—
115C22×C4—212no—
116C22×C4—212no—
117C22×C4—212no—
118C22×C4—212no—
119C22×C4—212no—
120C22×C4—212no—
121C22×C4—26no—
122C22×C4—22no—
123C22×C4—212no—
124C22×C4—212no—
125C22×C4—46no—
126C22×C4—46no—
127C22×C4—46no—
128C22×C4—46no—
129C22×C4—46no—
130C22×C4—43no—
131C22×C4—43no—
132C22×C4—448no—
133C22×C4—432no—
134C22×C4—496no—
135C22×C4—436no—
136C22×C4—412no—
137C22×C4—412no—
138C22×C4C2×C22⋊C416yes—
139C22×C4C22×D412yes—
140C22×C4H(5,𝔽2)112yes—
141M16—424no—
142M16—48no—
143C2×C8—424no—
144C4⋊C4—424no—
145C4⋊C4—412no—
146C22⋊C4—412no—
147C22⋊C4—412no—
148C22⋊C4—412no—
149C22⋊C4—412no—
150C23—21no—
151C23—23no—
152C23—224no—
153C23—23no—
154C23—23no—
155C23—26no—
156C23—26no—
157C23—23no—
158C23—23no—
159C23—26no—
160C23—26no—
161C23—412no—
162C23—424no—
163C23—424no—
164C23—424no—
165C23—412no—
166C23—412no—
167C23—412no—
168C23—424no—
169C23—44no—
170C23—412no—
171C23—4128no—
172C23—412no—
173C23—412no—
174C23C2×#10116yes—
175C23#41316yes—
176C23C22×C22⋊C411yescharacteristic
177C23#51816yes—
178Q8—88no—
179C2×C4—26no—
180C2×C4—26no—
181C2×C4—26no—
182C2×C4—424no—
183C2×C4—412no—
184C2×C4—412no—
185C2×C4—412no—
186C2×C4—424no—
187C2×C4—424no—
188C2×C4—46no—
189C2×C4—46no—
190C2×C4—48no—
191C2×C4—412no—
192C2×C4—812no—
193C2×C4#534112yes—
194C2×C4#560112yes—
195C2×C4H(3,𝔽4)14yes—
196C8—88no—
197C22—26no—
198C22—26no—
199C22—26no—
200C22—26no—
201C22—212no—
202C22—212no—
203C22—448no—
204C22—416no—
205C22—448no—
206C22—43no—
207C22—43no—
208C22—46no—
209C22—43no—
210C22—43no—
211C22—46no—
212C22#230813yes—
213C22#237514yes—
214C22C22×#10111yescharacteristic
215C22C2×#41313yes—
216C4—46no—
217C4—82no—
218C4—86no—
219C4#281218yes—
220C2—24no—
221C2—46no—
222C2—46no—
223C2#1465711yescharacteristic
224C2#2175313yes—
225C2#2175413yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
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
3. #11198C2semidirect
4. #10968C2semidirect
7. C22×#101C22semidirect
16. C2×#343C22semidirect
106. C24#101semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
#205792
#111982
#109682