Finite Group Databasebeta

← #391707

Subgroups — #391707

Order 512.

2031 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 2031 conjugacy classes of subgroups — too many to draw. Only its 242 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
1#34632C211yesmaximal, characteristic
2#20580C211yesmaximal, characteristic, complemented
3#13851C211yesmaximal, characteristic, complemented
4#11201C216yesmaximal, complemented
5#10971C216yesmaximal, complemented
6C23×#41C2211yescharacteristic
7C2×#525C2213yes—
8C22×#104C2214yes—
9C2×#407C2216yes—
10#2616C414yescomplemented
11#2615C414yescomplemented
12#2614—21no—
13#2613—23no—
14#2388—24no—
15#2280C2213yes—
16C2×C4×C22⋊C4C2216yes—
17#1780—24no—
18C2×#339C22112yes—
19#1446—21no—
20#1438—23no—
21C22×#41—412no—
22C22×#41C2×C414yes—
23C22×#41D414yes—
24C22×#41C2311yescharacteristic
25C23×Q8D411yescharacteristic
26C23×D4D413yes—
27C24×C4C2313yes—
28C22×M16C2314yes—
29#525—412no—
30C2×#123—23no—
31C2×C22⋊Q8C2×C416yes—
32C2×#121C2×C416yes—
33C2×#120—23no—
34C2×C4×D4—26no—
35C2×#117C2313yes—
36C22×C22⋊C4—26no—
37C22×C42C2314yes—
38C2×#104—412no—
39#407—412no—
40C2×C22⋊C8—212no—
41#395—24no—
42#394—212no—
43#393—24no—
44#392—212no—
45#381—23no—
46#380C2×C4112yes—
47#379—23no—
48C4×C22⋊C4—26no—
49C4×C22⋊C4—412no—
50C2×#95—26no—
51#339—412no—
52C25C2×D413yes—
53C2×#41—212no—
54C2×#41—424no—
55C2×#41—424no—
56C2×#41—448no—
57C2×#41—416no—
58C22×Q8—46no—
59C22×Q8C22⋊C414yes—
60C22×Q8C22×C411yescharacteristic
61C22×D4—21no—
62C22×D4—26no—
63C22×D4—46no—
64C22×D4—46no—
65C22×D4—46no—
66C22×D4C22⋊C4112yes—
67C22×D4C22×C411yescharacteristic
68C23×C4—212no—
69C23×C4—212no—
70C23×C4—212no—
71C23×C4—24no—
72C23×C4—21no—
73C23×C4C22×C416yes—
74C23×C4C2×D413yes—
75C23×C4C2×D416yes—
76C23×C4C2411yescharacteristic
77C2×M16—212no—
78C2×M16—212no—
79C22×C8—24no—
80#123—412no—
81C22⋊Q8—412no—
82#121—412no—
83#120—412no—
84C4×D4—424no—
85#117—212no—
86C2×C4⋊C4—26no—
87C2×C4⋊C4C22×C416yes—
88C2×C22⋊C4—26no—
89C2×C22⋊C4—23no—
90C2×C22⋊C4—26no—
91C2×C22⋊C4—212no—
92C2×C22⋊C4—424no—
93C2×C22⋊C4—412no—
94C2×C42—23no—
95C2×C42—212no—
96C2×C42—212no—
97#104—416no—
98C22⋊C8—424no—
99#95—212no—
100#95—412no—
101C24—26no—
102C24—26no—
103C24—26no—
104C24—21no—
105C24—412no—
106C24C2×C22⋊C416yes—
107C24C22×D411yescharacteristic
108#41—464no—
109#41—464no—
110#41—412no—
111#41—44no—
112C2×Q8—26no—
113C2×Q8—412no—
114C2×Q8—416no—
115C2×D4—26no—
116C2×D4—412no—
117C2×D4—412no—
118C2×D4—412no—
119C2×D4—448no—
120C2×D4—412no—
121C2×D4—46no—
122C2×D4—812no—
123C22×C4—26no—
124C22×C4—26no—
125C22×C4—26no—
126C22×C4—212no—
127C22×C4—212no—
128C22×C4—212no—
129C22×C4—212no—
130C22×C4—212no—
131C22×C4—212no—
132C22×C4—28no—
133C22×C4—26no—
134C22×C4—21no—
135C22×C4—436no—
136C22×C4—412no—
137C22×C4—412no—
138C22×C4—412no—
139C22×C4—46no—
140C22×C4C2×C22⋊C416yes—
141C22×C4C23×C411yescharacteristic
142C22×C4C22×D411yescharacteristic
143C22×C4H(5,𝔽2)112yes—
144M16—412no—
145M16—44no—
146C2×C8—412no—
147C4⋊C4—212no—
148C4⋊C4—412no—
149C22⋊C4—412no—
150C22⋊C4—412no—
151C22⋊C4—448no—
152C22⋊C4—46no—
153C22⋊C4—812no—
154C42—46no—
155C42—412no—
156C42—44no—
157C23—24no—
158C23—26no—
159C23—26no—
160C23—26no—
161C23—212no—
162C23—212no—
163C23—412no—
164C23—412no—
165C23—412no—
166C23—43no—
167C23—44no—
168C23—86no—
169C23—86no—
170C23—86no—
171C23—86no—
172C23C22×C22⋊C411yescharacteristic
173C23#53416yes—
174Q8—416no—
175Q8—44no—
176D4—448no—
177D4—44no—
178D4—824no—
179D4—84no—
180C2×C4—26no—
181C2×C4—26no—
182C2×C4—26no—
183C2×C4—412no—
184C2×C4—424no—
185C2×C4—424no—
186C2×C4—424no—
187C2×C4—46no—
188C2×C4—43no—
189C2×C4—412no—
190C2×C4—812no—
191C2×C4—84no—
192C2×C4—84no—
193C2×C4#51816yes—
194C2×C4#53416yes—
195C2×C4#560112yes—
196C2×C4H(3,𝔽4)14yes—
197C8—84no—
198C22—26no—
199C22—26no—
200C22—46no—
201C22—412no—
202C22—46no—
203C22—412no—
204C22—46no—
205C22—824no—
206C22—86no—
207C22—81no—
208C22#230813yes—
209C22#281214yes—
210C22#290114yes—
211C4—46no—
212C4—88no—
213C4—86no—
214C4—81no—
215C4#237514yes—
216C4#281214yes—
217C2—24no—
218C2—86no—
219C2—161no—
220C2#1465711yescharacteristic
221C2#2177516yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radical#34632C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×#41 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20580C2semidirect
3. #13851C2semidirect
4. #11201C2semidirect
5. #10971C2semidirect
10. #2616C4semidirect
11. #2615C4semidirect

Maximal subgroups

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

subgroupindex
#346322
#205802
#138512
#112012
#109712