Finite Group Databasebeta

← #124868

Subgroups — #124868

Order 512.

724 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 86 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#13958C211yesmaximal, characteristic
2#7909C212yesmaximal
3#7132C212yesmaximal, complemented
4#7104C212yesmaximal, complemented
5C22×#117C411yescharacteristic
6C2×#407—22no—
7C2×#381C411yescharacteristic
8C22×#95C2211yescharacteristic
9#1445C411yescharacteristic
10#1445C411yescharacteristic
11#1445C2211yescharacteristic
12#1445C2211yescharacteristic
13#1434—22no—
14C22⋊#95—22no—
15#1280—22no—
16#1280C2214yes—
17C24×C4C2×C411yescharacteristic
18C24×C4D412yes—
19C22×M16—42no—
20C2×#117—22no—
21C2×#117—22no—
22C2×#117—42no—
23C2×#117—42no—
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C22⋊C4—21no—
27C22×C22⋊C4—21no—
28C22×C42—21no—
29#407—28no—
30#407—44no—
31C2×C22⋊C8—24no—
32C2×C22⋊C8—42no—
33#381—42no—
34C2×#95—22no—
35C2×#95—21no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—42no—
40C2×#95C2×C411yescharacteristic
41C2×#95C2×C411yescharacteristic
42C2×#95C2×C411yescharacteristic
43C2×#95D412yes—
44C2×#95C2311yescharacteristic
45C25C22⋊C411yescharacteristic
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—21no—
51C23×C4—21no—
52C23×C4—24no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—44no—
60C23×C4—44no—
61C23×C4—82no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C412yes—
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D412yes—
66C2×M16—48no—
67C2×M16—84no—
68C2×M16—84no—
69C22×C8—44no—
70#117—48no—
71#117—48no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4M1612yescomplemented
79C2×C4⋊C4M1612yescomplemented
80C2×C22⋊C4—24no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—44no—
84C2×C22⋊C4—42no—
85C2×C42—24no—
86C2×C42—42no—
87C2×C42—42no—
88C22⋊C8—28no—
89C22⋊C8—48no—
90C22⋊C8—44no—
91#95—41no—
92#95—42no—
93#95—41no—
94#95—41no—
95#95—41no—
96#95—42no—
97#95—44no—
98#95—42no—
99#95—42no—
100#95—82no—
101#95—82no—
102C24—24no—
103C24—21no—
104C24—21no—
105C24—22no—
106C24—22no—
107C24—41no—
108C24—41no—
109C24#9912yes—
110C24C2×C22⋊C411yescharacteristic
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—44no—
128C22×C4—48no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—84no—
159C22×C4—88no—
160C22×C4—84no—
161C22×C4—82no—
162C22×C4—82no—
163C22×C4#10012yes—
164C22×C4#10414yes—
165C22×C4C2×M1611yescharacteristic
166C22×C4C2×M1611yescharacteristic
167C22×C4AGL(1,ℤ/8)12yes—
168C22×C4#13712yes—
169M16—816no—
170C2×C8—48no—
171C2×C8—84no—
172C2×C8—84no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C22⋊C4—48no—
178C22⋊C4—84no—
179C42—48no—
180C23—22no—
181C23—21no—
182C23—21no—
183C23—24no—
184C23—24no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—44no—
192C23—44no—
193C23—44no—
194C23—44no—
195C23—42no—
196C23—42no—
197C23—41no—
198C23—42no—
199C23—41no—
200C23#35214yes—
201C23#40711yescharacteristic
202C23C2×#9911yescharacteristic
203C23C2×#10011yescharacteristic
204C23#41712yes—
205C23C2×#10412yes—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—48no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—48no—
216C2×C4—48no—
217C2×C4—48no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—81no—
254C8—88no—
255C22—22no—
256C22—22no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—24no—
265C22—24no—
266C22—44no—
267C22—44no—
268C22—48no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22#146211yescharacteristic
274C22#148912yes—
275C22#150412yes—
276C22#152511yescharacteristic
277C22#156011yescharacteristic
278C22C2×#35212yes—
279C22#212711yescharacteristic
280C22#213311yescharacteristic
281C4—42no—
282C4—44no—
283C4—42no—
284C4—42no—
285C4—41no—
286C4—41no—
287C4—82no—
288C4—82no—
289C4—161no—
290C4—161no—
291C2—21no—
292C2—21no—
293C2—22no—
294C2—44no—
295C2#763111yescharacteristic
296C2#1150011yescharacteristic
297C2#1150811yescharacteristic
298C2#1175512yes—
299C2#1176412yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7132C2semidirect
4. #7104C2semidirect
78. C2×C4⋊C4M16semidirect
79. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
#139582
#79092
#71322
#71042