Finite Group Databasebeta

← #393718

Subgroups — #393718

Order 512.

1431 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1431 conjugacy classes of subgroups — too many to draw. Only its 282 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×#2643C211yesmaximal, characteristic
2#14349C211yesmaximal, characteristic, complemented
3#13749C211yesmaximal, characteristic, complemented
4#10960C216yesmaximal
5#10912C216yesmaximal, complemented
6C22×#99C2214yes—
7#2643—24no—
8#2643—24no—
9C2×#401C411yescharacteristic, complemented
10C2×#400C413yescomplemented
11C2×#397C413yescomplemented
12C2×#396C411yescharacteristic, complemented
13C2×#386C2213yes—
14C2×#385C2213yes—
15C2×C4×C22⋊C4C2216yes—
16C22×#95C2211yescharacteristic
17C2×#342C22112yes—
18#1443—24no—
19C22⋊#95C2216yes—
20C4×#95—24no—
21C23×D4C2311yescharacteristic
22C24×C4C2313yes—
23C22×C4⋊C4C2×C413yes—
24C22×C4⋊C4C2×C413yes—
25C22×C22⋊C4C2×C413yes—
26C22×C22⋊C4C2×C413yes—
27C22×C22⋊C4C2314yes—
28C22×C22⋊C4C2313yes—
29C22×C22⋊C4C2314yes—
30C22×C42—26no—
31C2×#99—224no—
32#401—24no—
33#400—212no—
34#397—212no—
35#396—24no—
36#386—26no—
37#386—26no—
38#385—26no—
39#385—26no—
40C4×C22⋊C4—224no—
41C2×#95—26no—
42C2×#95—26no—
43C2×#95—26no—
44C2×#95C2×C413yes—
45C2×#95C2×C413yes—
46C2×#95C2×C413yes—
47C2×#95C2×C413yes—
48C2×#95C2×C411yescharacteristic
49C2×#95C2×C413yes—
50C2×#95D414yes—
51C2×#95D414yes—
52#342—224no—
53C25C22×C411yescharacteristic
54C25C2411yescharacteristic
55C22×D4—21no—
56C22×D4—21no—
57C22×D4—212no—
58C23×C4—24no—
59C23×C4—212no—
60C23×C4—24no—
61C23×C4—83no—
62C23×C4—83no—
63C23×C4C22×C411yescharacteristic
64C23×C4C22×C413yes—
65C23×C4C22×C413yes—
66C23×C4C22×C413yes—
67C23×C4C22×C413yes—
68C23×C4C2×D416yes—
69C23×C4C2×D416yes—
70C2×C4⋊C4—26no—
71C2×C4⋊C4—26no—
72C2×C4⋊C4—83no—
73C2×C4⋊C4—83no—
74C2×C22⋊C4—212no—
75C2×C22⋊C4—23no—
76C2×C22⋊C4—23no—
77C2×C22⋊C4—212no—
78C2×C22⋊C4—212no—
79C2×C22⋊C4—212no—
80C2×C22⋊C4—26no—
81C2×C22⋊C4—26no—
82C2×C22⋊C4—83no—
83C2×C22⋊C4—83no—
84C2×C22⋊C4#41124yes—
85C2×C42—224no—
86C2×C42—86no—
87C2×C42—86no—
88#99—416no—
89#95—224no—
90#95—26no—
91#95—26no—
92#95—26no—
93#95—26no—
94#95—86no—
95#95—86no—
96#95C22⋊C414yes—
97#95C22⋊C4112yes—
98C24—21no—
99C24—26no—
100C24—43no—
101C24—41no—
102C24—43no—
103C24—41no—
104C24C23×C411yescharacteristic
105C24C22×D411yescharacteristic
106C24C22×D411yescharacteristic
107C24C2×#41112yes—
108C2×D4—46no—
109C2×D4—44no—
110C2×D4—412no—
111C2×D4—46no—
112C22×C4—26no—
113C22×C4—224no—
114C22×C4—26no—
115C22×C4—26no—
116C22×C4—26no—
117C22×C4—212no—
118C22×C4—21no—
119C22×C4—23no—
120C22×C4—23no—
121C22×C4—26no—
122C22×C4—26no—
123C22×C4—26no—
124C22×C4—46no—
125C22×C4—46no—
126C22×C4—46no—
127C22×C4—46no—
128C22×C4—43no—
129C22×C4—43no—
130C22×C4—43no—
131C22×C4—43no—
132C22×C4—412no—
133C22×C4—44no—
134C22×C4—43no—
135C22×C4—43no—
136C22×C4—43no—
137C22×C4—43no—
138C22×C4—412no—
139C22×C4—44no—
140C22×C4—412no—
141C22×C4—824no—
142C22×C4C2×C22⋊C416yes—
143C22×C4C2×C22⋊C416yes—
144C4⋊C4—86no—
145C4⋊C4—86no—
146C22⋊C4—224no—
147C22⋊C4—46no—
148C22⋊C4—46no—
149C22⋊C4—44no—
150C22⋊C4—412no—
151C22⋊C4—86no—
152C22⋊C4—86no—
153C22⋊C4#126116yes—
154C42—824no—
155C23—23no—
156C23—23no—
157C23—23no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—412no—
162C23—412no—
163C23—43no—
164C23—41no—
165C23—43no—
166C23—44no—
167C23—86no—
168C23—86no—
169C23C22×C22⋊C411yescharacteristic
170C23C4×#4116yes—
171C23#52516yes—
172C23C2×#12614yes—
173C23#529112yes—
174C23#53216yes—
175D4—84no—
176D4—84no—
177C2×C4—216no—
178C2×C4—26no—
179C2×C4—26no—
180C2×C4—26no—
181C2×C4—26no—
182C2×C4—26no—
183C2×C4—26no—
184C2×C4—424no—
185C2×C4—412no—
186C2×C4—424no—
187C2×C4—412no—
188C2×C4—44no—
189C2×C4—43no—
190C2×C4—43no—
191C2×C4—43no—
192C2×C4—43no—
193C2×C4—43no—
194C2×C4—43no—
195C2×C4—43no—
196C2×C4—812no—
197C2×C4—86no—
198C2×C4—86no—
199C2×C4—86no—
200C2×C4—86no—
201C2×C4—812no—
202C2×C4—812no—
203C2×C4—84no—
204C22—26no—
205C22—26no—
206C22—26no—
207C22—26no—
208C22—26no—
209C22—26no—
210C22—44no—
211C22—412no—
212C22—81no—
213C22—83no—
214C22—83no—
215C22—164no—
216C22#229713yes—
217C22C4×#12614yes—
218C22#285214yes—
219C22#286014yes—
220C22#288313yes—
221C22#288411yescharacteristic
222C4—416no—
223C4—46no—
224C4—46no—
225C4—81no—
226C4—83no—
227C4—83no—
228C4—164no—
229C2—24no—
230C2—24no—
231C2—24no—
232C2—161no—
233C2#1471111yescharacteristic
234C2#2142213yes—
235C2#2142313yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14349C2semidirect
3. #13749C2semidirect
5. #10912C2semidirect
9. C2×#401C4semidirect
10. C2×#400C4semidirect
11. C2×#397C4semidirect
12. C2×#396C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26432
#143492
#137492
#109602
#109122