Finite Group Databasebeta

← #480746

Subgroups — #480746

Order 512.

1112 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2563C211yesmaximal, characteristic
2#17362C212yesmaximal
3#11569C218yesmaximal, complemented
4#11511C214yesmaximal, complemented
5C22×C4⋊Q8C411yescharacteristic
6C22×C4×D4C2211yescharacteristic
7C2×#407—22no—
8#2563—22no—
9#2563—22no—
10#2563C414yes—
11#2563C2214yes—
12C2×#393—22no—
13C2×#390C412yes—
14C2×#382C2212yes—
15C2×#380—22no—
16C2×C4×C4⋊C4C411yescharacteristic
17#1465—22no—
18#1465C2214yes—
19#1463C2214yes—
20#1462—22no—
21#1459—22no—
22#1459C2214yes—
23#1324C22116yescomplemented
24C23×Q8—21no—
25C23×D4C2×C411yescharacteristic
26C24×C4D412yes—
27C22×M16—42no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—41no—
30C2×C4⋊Q8—42no—
31C2×C4⋊Q8—41no—
32C2×C4⋊Q8C2×C414yes—
33C2×C4×D4—41no—
34C2×C4×D4—42no—
35C2×C4×D4—41no—
36C2×C4×D4C2×C412yes—
37C2×C4×D4D412yes—
38C2×C4×D4D412yes—
39C2×C4×D4C2312yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2311yescharacteristic
44C22×C22⋊C4—24no—
45C22×C22⋊C4D412yes—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C412yes—
48#407—24no—
49#407—24no—
50#407—42no—
51#407—42no—
52C2×C22⋊C8—24no—
53C2×C22⋊C8—42no—
54#393—24no—
55#393—24no—
56#393—24no—
57#393—24no—
58#390—22no—
59#390—22no—
60#390C2×C418yes—
61#382—24no—
62#382C2×C414yes—
63#382C2314yes—
64#380—24no—
65#380—24no—
66#380—24no—
67#380—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4C2×C414yes—
70C2×#95—21no—
71C2×#95—24no—
72#324—24no—
73#324C2318yes—
74#323—24no—
75#323—24no—
76C25C22⋊C412yes—
77C22×Q8—24no—
78C22×Q8—42no—
79C22×Q8—42no—
80C22×Q8—84no—
81C22×D4—24no—
82C22×D4—42no—
83C22×D4—41no—
84C22×D4—41no—
85C22×D4C22⋊C411yescharacteristic
86C22×D4C22⋊C411yescharacteristic
87C22×D4C22×C412yes—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—48no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C2×M16—44no—
104C2×M16—44no—
105C2×M16—88no—
106C22×C8—44no—
107C4⋊Q8—44no—
108C4⋊Q8—44no—
109C4⋊Q8—44no—
110C4⋊Q8—44no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117C4×D4—42no—
118C4×D4—42no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4C22×C414yes—
131C2×C4⋊C4C22×C411yescharacteristic
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2411yescharacteristic
134C2×C22⋊C4—28no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—88no—
139C2×C22⋊C4C22⋊C412yes—
140C2×C22⋊C4C22⋊C412yes—
141C2×C22⋊C4C2×D412yes—
142C2×C22⋊C4C2×D412yes—
143C2×C42—21no—
144C2×C42—22no—
145C2×C42—21no—
146C2×C42—22no—
147C2×C42—44no—
148C2×C42C22⋊C412yes—
149C2×C42C22×C414yes—
150C2×C42C22×C412yes—
151C22⋊C8—28no—
152C22⋊C8—44no—
153C22⋊C8—44no—
154C22⋊C8—44no—
155#95—24no—
156#95—28no—
157#95—28no—
158C24—24no—
159C24—22no—
160C24—22no—
161C24—42no—
162C24—44no—
163C24—44no—
164C24C2×C22⋊C411yescharacteristic
165C24C2×C22⋊C412yes—
166C24C2×C22⋊C412yes—
167C2×Q8—48no—
168C2×Q8—88no—
169C2×Q8—88no—
170C2×Q8—88no—
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—42no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—48no—
180C2×D4—48no—
181C2×D4#10114yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—48no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—48no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4#9911yescharacteristic
218C22×C4#9912yes—
219C22×C4#9911yescharacteristic
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C23×C411yescharacteristic
225C22×C4C22×D412yes—
226M16—88no—
227M16—88no—
228C2×C8—48no—
229C2×C8—88no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—88no—
243C22⋊C4—88no—
244C42—22no—
245C42—22no—
246C42—22no—
247C42—22no—
248C42—28no—
249C42—44no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—44no—
260C23—44no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—416no—
266C23—44no—
267C23—44no—
268C23—82no—
269C23—82no—
270C23—88no—
271C23C2×#9911yescharacteristic
272C23C2×#9911yescharacteristic
273C23C2×#10114yes—
274C23C22×C22⋊C411yescharacteristic
275Q8—816no—
276D4—44no—
277D4—48no—
278D4—44no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—48no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—88no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4#35414yes—
315C2×C4C2×#9912yes—
316C2×C4C2×#9912yes—
317C2×C4C2×#10112yes—
318C2×C4#56812yes—
319C8—88no—
320C22—24no—
321C22—24no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—24no—
329C22—44no—
330C22—44no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—82no—
337C22—88no—
338C22—88no—
339C22—82no—
340C22C2×#35412yes—
341C22#213412yes—
342C22#287911yescharacteristic
343C22C22×#9911yescharacteristic
344C22C22×#10111yescharacteristic
345C4—22no—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—84no—
352C4—84no—
353C4C2×#35414yes—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—22no—
358C2—44no—
359C2—82no—
360C2#1739711yescharacteristic
361C2C22×#35411yescharacteristic
362C2C2×#213411yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11569C2semidirect
4. #11511C2semidirect
23. #1324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
#173622
#115692
#115112