Finite Group Databasebeta

← #124979

Subgroups — #124979

Order 512.

552 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 98 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#13925C211yesmaximal, characteristic
2#8560C212yesmaximal
3#7132C212yesmaximal
4#7131C212yesmaximal
5C22×C4×Q8C411yescharacteristic
6C2×#382C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8#1797—22no—
9C2×#324—22no—
10#1446C411yescharacteristic
11#1446C2211yescharacteristic
12#1445C411yescharacteristic
13#1445C2211yescharacteristic
14#1436—22no—
15C4⋊#95—22no—
16#1280C2214yes—
17C23×Q8C2×C411yescharacteristic
18C2×C4×Q8—22no—
19C2×C4×Q8—22no—
20C2×C4×Q8—42no—
21C2×C4×Q8—42no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4D412yes—
26C22×C42—21no—
27C22×C42C2×C411yescharacteristic
28C22×C42D412yes—
29C2×C4⋊C8—42no—
30C2×C22⋊C8—24no—
31#382—42no—
32C4×C4⋊C4—42no—
33C2×#95—22no—
34C2×#95—21no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95C2×C411yescharacteristic
39C2×#95C2×C411yescharacteristic
40C2×#95C2×C411yescharacteristic
41C2×#95C2311yescharacteristic
42#336—42no—
43#324—44no—
44C22×Q8—21no—
45C22×Q8—44no—
46C22×Q8—41no—
47C22×Q8—41no—
48C22×Q8M1612yes—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4C22⋊C411yescharacteristic
54C23×C4C22⋊C412yes—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22×C411yescharacteristic
57C23×C4C2×D412yes—
58C22×C8—44no—
59C4×Q8—48no—
60C4×Q8—48no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—44no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4M1612yes—
72C2×C4⋊C4SD1614yes—
73C2×C4⋊C4Q1614yes—
74C2×C42—22no—
75C2×C42—22no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—24no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—42no—
82C2×C42—42no—
83C2×C42—82no—
84C4⋊C8—84no—
85C22⋊C8—44no—
86C22⋊C8—44no—
87#95—41no—
88#95—41no—
89#95—41no—
90#95—41no—
91#95—44no—
92#95—44no—
93#95—42no—
94#95—82no—
95#95—82no—
96C24C2×C22⋊C411yescharacteristic
97C2×Q8—44no—
98C2×Q8—42no—
99C2×Q8—44no—
100C2×Q8—42no—
101C2×Q8—48no—
102C2×Q8—44no—
103C2×Q8—44no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—42no—
121C22×C4—41no—
122C22×C4—41no—
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×C4—42no—
133C22×C4—42no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4#9912yes—
143C22×C4#10012yes—
144C22×C4#10314yes—
145C22×C4#10314yes—
146C22×C4C2×M1611yescharacteristic
147C22×C4C2×M1611yescharacteristic
148C22×C4C2×SD1612yes—
149C22×C4C2×Q1612yes—
150C2×C8—84no—
151C2×C8—84no—
152C2×C8—84no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—42no—
156C4⋊C4—48no—
157C4⋊C4—42no—
158C4⋊C4—84no—
159C42—24no—
160C42—24no—
161C42—44no—
162C42—48no—
163C42—82no—
164C23—22no—
165C23—21no—
166C23—21no—
167C23#40711yescharacteristic
168C23C2×#9911yescharacteristic
169C23C2×#10011yescharacteristic
170C23C2×#10312yes—
171C23#42112yes—
172Q8—44no—
173Q8—44no—
174Q8—48no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4#33314yes—
223C8—164no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22#146211yescharacteristic
233C22#148712yes—
234C22#152012yes—
235C22#153311yescharacteristic
236C22C2×#33311yescharacteristic
237C22#212012yes—
238C22#212711yescharacteristic
239C22#213311yescharacteristic
240C4—24no—
241C4—42no—
242C4—44no—
243C4—42no—
244C4—44no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—82no—
249C4—161no—
250C4—161no—
251C2—21no—
252C2—21no—
253C2—22no—
254C2#764311yescharacteristic
255C2#1150111yescharacteristic
256C2#1150811yescharacteristic
257C2#1189712yes—
258C2#1190512yes—
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
#139252
#85602
#71322
#71312