Finite Group Databasebeta

← #124860

Subgroups — #124860

Order 512.

610 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 90 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#14155C211yesmaximal, characteristic
2#8153C212yesmaximal
3#7132C214yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5C2×#391—22no—
6C2×C4×C4⋊C4C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8#1749—22no—
9C2×#324—22no—
10#1445C412yes—
11#1445C2212yes—
12C4⋊#95—22no—
13#1280C2214yes—
14C23×Q8—21no—
15C2×C4⋊Q8—24no—
16C2×C4⋊Q8—41no—
17C2×C4⋊Q8—41no—
18C2×C4⋊Q8—42no—
19C22×C4⋊C4—21no—
20C22×C4⋊C4—21no—
21C22×C4⋊C4C2×C412yes—
22C22×C4⋊C4D412yes—
23C22×C42—21no—
24C22×C42C2×C411yescharacteristic
25C22×C42D412yes—
26C2×C22⋊C8—24no—
27C2×#97—42no—
28#391—48no—
29C4×C4⋊C4—41no—
30C4×C4⋊C4—41no—
31C4×C4⋊C4—42no—
32C2×#95—24no—
33C2×#95—24no—
34C2×#95C2×C412yes—
35C2×#95C2×C411yescharacteristic
36C2×#95C2311yescharacteristic
37#336—42no—
38#324—28no—
39C22×Q8—24no—
40C22×Q8—42no—
41C22×Q8—44no—
42C22×Q8—44no—
43C22×Q8—41no—
44C22×Q8—41no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4C22⋊C411yescharacteristic
48C23×C4C22⋊C412yes—
49C23×C4C22⋊C411yescharacteristic
50C23×C4C22×C411yescharacteristic
51C23×C4C2×D412yes—
52C22×C8—44no—
53C4⋊Q8—48no—
54C4⋊Q8—48no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4M1614yes—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42—44no—
75C2×C42—42no—
76C2×C42—44no—
77C22⋊C8—28no—
78C22⋊C8—44no—
79#97—84no—
80#95—41no—
81#95—41no—
82#95—42no—
83#95—48no—
84#95—48no—
85C24C2×C22⋊C411yescharacteristic
86C2×Q8—44no—
87C2×Q8—44no—
88C2×Q8—416no—
89C2×Q8—88no—
90C2×Q8—82no—
91C2×Q8—82no—
92C2×Q8—82no—
93C2×Q8—82no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—44no—
116C22×C4—42no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—44no—
123C22×C4#9911yescharacteristic
124C22×C4#9911yescharacteristic
125C22×C4#10012yes—
126C22×C4#10414yes—
127C22×C4C2×M1612yes—
128C22×C4#13714yes—
129C2×C8—48no—
130C2×C8—84no—
131C2×C8—84no—
132C4⋊C4—48no—
133C4⋊C4—44no—
134C4⋊C4—48no—
135C4⋊C4—48no—
136C4⋊C4—44no—
137C42—24no—
138C42—24no—
139C42—44no—
140C42—82no—
141C42—84no—
142C23—22no—
143C23—21no—
144C23—21no—
145C23#40711yescharacteristic
146C23C2×#9911yescharacteristic
147C23C2×#10011yescharacteristic
148C23#41912yes—
149C23C2×#10412yes—
150Q8—88no—
151Q8—88no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—48no—
172C2×C4—44no—
173C2×C4—48no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—82no—
179C2×C4—84no—
180C2×C4—82no—
181C2×C4—84no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4#32914yes—
187C2×C4#35312yes—
188C2×C4#35512yes—
189C8—88no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22#146211yescharacteristic
199C22#149112yes—
200C22#150812yes—
201C22C2×#32911yescharacteristic
202C22#155811yescharacteristic
203C22#212312yes—
204C22C2×#35311yescharacteristic
205C22C2×#35511yescharacteristic
206C4—24no—
207C4—22no—
208C4—22no—
209C4—44no—
210C4—42no—
211C4—84no—
212C4—82no—
213C4—84no—
214C2—21no—
215C2—21no—
216C2—22no—
217C2#762911yescharacteristic
218C2#1150411yescharacteristic
219C2#1150511yescharacteristic
220C2#1178812yes—
221C2#1179412yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ 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
#141552
#81532
#71322