Finite Group Databasebeta

← #126017

Subgroups — #126017

Order 512.

423 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 60 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#16244C211yesmaximal, characteristic
2#10568C212yesmaximal
3#9972C211yesmaximal, characteristic
4#7132C211yesmaximal, characteristic
5#7122C212yesmaximal
6C2×#484—21no—
7#2659C2211yescharacteristic
8#1853—22no—
9C2×#328—21no—
10#1445—21no—
11#1445C2211yescharacteristic
12#1442—22no—
13#1442C2212yes—
14#1436—21no—
15#1436—22no—
16#1294—22no—
17#1280C2212yes—
18#1280C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C22×C4⋊C4—21no—
21C22×C4⋊C4D411yescharacteristic
22C22×C4⋊C4D412yes—
23C22×C42—21no—
24#484—42no—
25#484—42no—
26C2×#107—41no—
27C2×#106—42no—
28C2×#103—41no—
29C2×C22⋊C8—22no—
30C2×C22⋊C8—21no—
31#398—41no—
32#395—41no—
33#380—22no—
34C2×#95—22no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95D412yes—
42C2×#95D411yescharacteristic
43C2×#95C2311yescharacteristic
44#340—42no—
45#328—42no—
46C22×Q8—21no—
47C23×C4—21no—
48C23×C4—22no—
49C23×C4—21no—
50C23×C4—22no—
51C23×C4C2×D411yescharacteristic
52C23×C4C2×D412yes—
53C22×C8—42no—
54C22×C8—41no—
55C22⋊Q8—41no—
56C22⋊Q8—41no—
57C22⋊Q8—41no—
58C22⋊Q8—41no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—44no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—81no—
70C2×C4⋊C4—81no—
71C2×C4⋊C4SD1612yes—
72C2×C22⋊C4—41no—
73C2×C42—82no—
74C2×C42—81no—
75#107—44no—
76#106—84no—
77#103—44no—
78C22⋊C8—22no—
79C22⋊C8—24no—
80C22⋊C8—41no—
81C22⋊C8—42no—
82#95—22no—
83#95—22no—
84#95—41no—
85#95—41no—
86#95—44no—
87#95—44no—
88#95—44no—
89#95—42no—
90#95—42no—
91#95—42no—
92#95—81no—
93#95—82no—
94#95—81no—
95C24#12011yescharacteristic
96C2×Q8—41no—
97C2×Q8—41no—
98C2×Q8—82no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—41no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—41no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4C2×SD1611yescharacteristic
133C22×C4#13512yes—
134C22×C4#13711yescharacteristic
135C22×C4#13712yes—
136C2×C8—42no—
137C2×C8—44no—
138C2×C8—82no—
139C2×C8—82no—
140C2×C8—82no—
141C4⋊C4—42no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—44no—
145C4⋊C4—81no—
146C4⋊C4—81no—
147C4⋊C4—81no—
148C4⋊C4—82no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C4⋊C4—84no—
152C22⋊C4—81no—
153C22⋊C4—81no—
154C42—84no—
155C23—22no—
156C23—21no—
157C23—21no—
158C23#44811yescharacteristic
159C23#45212yes—
160C23#45311yescharacteristic
161C23#45612yes—
162C23B(4,𝔽2)11yescharacteristic
163Q8—82no—
164Q8—161no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—81no—
184C2×C4—82no—
185C2×C4—81no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—81no—
206C2×C4—81no—
207C8—82no—
208C8—84no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22#159611yescharacteristic
218C22#160312yes—
219C22#163411yescharacteristic
220C22#165312yes—
221C22#166011yescharacteristic
222C22#219712yes—
223C22#219811yescharacteristic
224C22#220211yescharacteristic
225C4—41no—
226C4—41no—
227C4—42no—
228C4—42no—
229C4—82no—
230C4—82no—
231C4—84no—
232C4—162no—
233C4—161no—
234C4—161no—
235C2—21no—
236C2—21no—
237C2—22no—
238C2#776011yescharacteristic
239C2#1212111yescharacteristic
240C2#1212711yescharacteristic
241C2#1222912yes—
242C2#1223512yes—
subgroupG/H
center, socleC23B(4,𝔽2)
commutator, Frattini, cosocleC2×#95C23
prime radicalC24#120
LCSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#162442
#105682
#99722
#71322
#71222