Finite Group Databasebeta

← #134110

Subgroups — #134110

Order 512.

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

normal subgroups only

This group has 750 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
1C2×#2565C211yesmaximal, characteristic
2#11512C212yesmaximal
3#11506C212yesmaximal
4#8554C212yesmaximal
5C22×#125C2211yescharacteristic
6C22×C4×Q8C411yescharacteristic
7#2565—24no—
8#2565—24no—
9C2×#398—22no—
10C2×#391C411yescharacteristic
11C2×#391C2211yescharacteristic
12C2×#387—22no—
13C2×#384C2211yescharacteristic
14C2×#382C411yescharacteristic
15C2×C4×C4⋊C4C411yescharacteristic
16#1821—22no—
17#1463—22no—
18#1463C2214yes—
19#1325—24no—
20#1319—24no—
21C23×Q8C2×C411yescharacteristic
22C2×#125—22no—
23C2×#125—22no—
24C2×#125—42no—
25C2×#125—42no—
26C2×C4×Q8—22no—
27C2×C4×Q8—22no—
28C2×C4×Q8—42no—
29C2×C4×Q8—42no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4D412yes—
36C22×C4⋊C4D412yes—
37C22×C4⋊C4C2311yescharacteristic
38C22×C42—21no—
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42#407—24no—
43#398—216no—
44#391—22no—
45#391—22no—
46#391—22no—
47#391D414yes—
48#387—216no—
49#384—22no—
50#384D414yes—
51#382—22no—
52#382—22no—
53C4×C4⋊C4—22no—
54C4×C4⋊C4—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—21no—
58C2×#95—22no—
59#343—44no—
60#324—24no—
61#324—24no—
62#324—44no—
63C22×Q8—24no—
64C22×Q8—21no—
65C22×Q8—42no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8C22⋊C412yes—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D412yes—
78C2×M16—44no—
79#125—48no—
80#125—48no—
81C4×Q8—48no—
82C4×Q8—48no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—48no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4C22⋊C412yes—
106C2×C4⋊C4C2×D412yes—
107C2×C4⋊C4C2×D412yes—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—24no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42C22⋊C412yes—
120C2×C42C22⋊C412yes—
121C22⋊C8—44no—
122C22⋊C8—44no—
123#95—24no—
124#95—28no—
125#95—28no—
126#95—28no—
127C24C2×C22⋊C411yescharacteristic
128C2×Q8—24no—
129C2×Q8—22no—
130C2×Q8—42no—
131C2×Q8—48no—
132C2×Q8—48no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8#10114yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4#9912yes—
172C22×C4#9912yes—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4#12014yes—
176M16—88no—
177C2×C8—84no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—48no—
181C4⋊C4—48no—
182C4⋊C4—42no—
183C4⋊C4—48no—
184C4⋊C4—416no—
185C4⋊C4—42no—
186C4⋊C4—48no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C42—24no—
190C42—24no—
191C42—24no—
192C42—42no—
193C42—42no—
194C42—48no—
195C42—42no—
196C23—22no—
197C23—22no—
198C23—21no—
199C23—21no—
200C23#37911yescharacteristic
201C23C2×#9911yescharacteristic
202C23C2×#9911yescharacteristic
203Q8—44no—
204Q8—44no—
205Q8—48no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—42no—
218C2×C4—48no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—82no—
241C2×C4—84no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4C2×#10112yes—
246C2×C4#41312yes—
247C8—164no—
248C22—24no—
249C22—24no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22#178111yescharacteristic
257C22#178411yescharacteristic
258C22#178511yescharacteristic
259C22#212612yes—
260C22#213512yes—
261C4—22no—
262C4—22no—
263C4—44no—
264C4—44no—
265C4—44no—
266C4—42no—
267C4—42no—
268C4—82no—
269C4—82no—
270C4—82no—
271C2—21no—
272C2—22no—
273C2—22no—
274C2—21no—
275C2#855011yescharacteristic
276C2#1269511yescharacteristic
277C2#1270411yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#125 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25652
#115122
#115062
#85542