Finite Group Databasebeta

← #124872

Subgroups — #124872

Order 512.

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

normal subgroups only

This group has 746 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#14159C211yesmaximal, characteristic
2#7909C212yesmaximal
3#7131C212yesmaximal, complemented
4#7104C212yesmaximal, complemented
5C22×C22⋊Q8C411yescharacteristic
6C2×#407—22no—
7C2×C4×C22⋊C4C411yescharacteristic
8C22×#95C2211yescharacteristic
9#1446C411yescharacteristic
10#1446C2211yescharacteristic
11#1445C411yescharacteristic
12#1445C2211yescharacteristic
13#1436—22no—
14C22⋊#95—22no—
15#1280—22no—
16#1280C2214yes—
17C23×Q8C2×C411yescharacteristic
18C24×C4C2×C411yescharacteristic
19C24×C4D412yes—
20C22×M16—42no—
21C2×C22⋊Q8—22no—
22C2×C22⋊Q8—22no—
23C2×C22⋊Q8—42no—
24C2×C22⋊Q8—41no—
25C2×C22⋊Q8—41no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C42—21no—
31#407—28no—
32#407—44no—
33C2×C22⋊C8—24no—
34C2×C22⋊C8—42no—
35C4×C22⋊C4—42no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95—41no—
41C2×#95—41no—
42C2×#95C2×C411yescharacteristic
43C2×#95C2×C411yescharacteristic
44C2×#95C2×C411yescharacteristic
45C2×#95D412yes—
46C2×#95C2311yescharacteristic
47C25C22⋊C411yescharacteristic
48C22×Q8—21no—
49C22×Q8—41no—
50C22×Q8—41no—
51C22×Q8—82no—
52C22×Q8M1612yescomplemented
53C23×C4—22no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—24no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—44no—
68C23×C4—82no—
69C23×C4—82no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C412yes—
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D412yes—
74C2×M16—48no—
75C2×M16—84no—
76C2×M16—84no—
77C22×C8—44no—
78C22⋊Q8—44no—
79C22⋊Q8—44no—
80C22⋊Q8—44no—
81C22⋊Q8—44no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4M1612yescomplemented
89C2×C22⋊C4—24no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C42—42no—
97C2×C42—82no—
98C22⋊C8—28no—
99C22⋊C8—48no—
100C22⋊C8—44no—
101#95—41no—
102#95—42no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—44no—
107#95—42no—
108#95—42no—
109#95—82no—
110#95—82no—
111C24—22no—
112C24—22no—
113C24—21no—
114C24—21no—
115C24—22no—
116C24—22no—
117C24—42no—
118C24#9911yescharacteristic
119C24#9911yescharacteristic
120C24C2×C22⋊C411yescharacteristic
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—82no—
124C2×Q8—82no—
125C2×Q8—84no—
126C2×Q8—84no—
127C22×C4—24no—
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—84no—
172C22×C4—88no—
173C22×C4—84no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—84no—
181C22×C4#10012yes—
182C22×C4#10414yes—
183C22×C4C2×M1611yescharacteristic
184C22×C4C2×M1611yescharacteristic
185C22×C4AGL(1,ℤ/8)12yes—
186C22×C4#13712yes—
187M16—816no—
188C2×C8—48no—
189C2×C8—84no—
190C2×C8—84no—
191C4⋊C4—42no—
192C4⋊C4—48no—
193C4⋊C4—42no—
194C22⋊C4—48no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C42—84no—
198C23—22no—
199C23—21no—
200C23—21no—
201C23—24no—
202C23—24no—
203C23—22no—
204C23—22no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—42no—
219C23—41no—
220C23#35412yes—
221C23#35612yes—
222C23#40711yescharacteristic
223C23C2×#9911yescharacteristic
224C23C2×#10011yescharacteristic
225C23#41712yes—
226C23C2×#10412yes—
227Q8—84no—
228Q8—84no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—48no—
236C2×C4—42no—
237C2×C4—48no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C8—88no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—22no—
290C22—22no—
291C22—24no—
292C22—44no—
293C22—44no—
294C22—48no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22#146211yescharacteristic
302C22#148912yes—
303C22#150412yes—
304C22#152511yescharacteristic
305C22#156011yescharacteristic
306C22#212112yes—
307C22C2×#35411yescharacteristic
308C22C2×#35611yescharacteristic
309C4—42no—
310C4—41no—
311C4—41no—
312C4—42no—
313C4—44no—
314C4—42no—
315C4—82no—
316C4—82no—
317C4—82no—
318C4—82no—
319C2—21no—
320C2—21no—
321C2—22no—
322C2—42no—
323C2—42no—
324C2#763111yescharacteristic
325C2#1150211yescharacteristic
326C2#1150611yescharacteristic
327C2#1175912yes—
328C2#1176812yes—
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. #7131C2semidirect
4. #7104C2semidirect
52. C22×Q8M16semidirect
88. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
#141592
#79092
#71312
#71042