Finite Group Databasebeta

← #480934

Subgroups — #480934

Order 512.

907 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36496C211yesmaximal, characteristic
2#17429C212yesmaximal, complemented
3#11574C212yesmaximal, complemented
4#11572C214yesmaximal, complemented
5#11530C211yesmaximal, characteristic, complemented
6#11529C212yesmaximal, complemented
7#11476C211yesmaximal, characteristic, complemented
8#11475C212yesmaximal, complemented
9C2×#559C2211yescharacteristic
10C2×#554C411yescharacteristic
11#2974—22no—
12#2861C412yes—
13#2858C411yescharacteristic
14#2567C2211yescharacteristic
15#2566C2212yes—
16#2544C412yes—
17#2541C411yescharacteristic
18#2461—24no—
19#2426—22no—
20#2361—22no—
21C4×#127C2212yes—
22C4×#124C2211yescharacteristic
23C42×Q8C411yescharacteristic
24#1469—22no—
25#1468—24no—
26#1467C2212yes—
27#1465C2214yes—
28#1461C2212yes—
29#1459C2214yes—
30#1325C2214yes—
31#1325C2214yes—
32#1324C2212yes—
33#1324C2212yes—
34#1318C2211yescharacteristic
35#1318C2212yes—
36#1318C2211yescharacteristic
37#559—41no—
38#559—42no—
39#559—41no—
40#554—42no—
41#554—41no—
42#554—41no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C412yes—
45C2×#127C2×C411yescharacteristic
46C2×#125C2×C412yes—
47C2×#124—22no—
48C2×#124C2×C411yescharacteristic
49C2×#124C2×C412yes—
50C2×#124D412yes—
51C2×#123D414yes—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—22no—
54C2×#121—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4—24no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4C2312yes—
63C22×C4⋊C4—21no—
64#436—42no—
65#435—44no—
66C8×D4—42no—
67#432—22no—
68#431—22no—
69#407—24no—
70C2×C22⋊C8—24no—
71#396—24no—
72#395—22no—
73#394—24no—
74#392—22no—
75#391C2×C412yes—
76#391C2×C411yescharacteristic
77#390—22no—
78#390—24no—
79#390C2×C412yes—
80#390C2×C412yes—
81#390C2×C412yes—
82#390C2311yescharacteristic
83#389—22no—
84#388—22no—
85#388—24no—
86#388—24no—
87#386—24no—
88#385—24no—
89#384C2312yes—
90#382C2311yescharacteristic
91C4×C4⋊C4—22no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C22⋊C4—22no—
96C43C2×C412yes—
97C43C2×C411yescharacteristic
98#324—22no—
99#324—22no—
100#324C2312yes—
101#324C2314yes—
102#324C2312yes—
103#323—24no—
104#323—24no—
105C22×Q8—21no—
106C22×Q8C22⋊C411yescharacteristic
107C22×Q8C22×C411yescharacteristic
108C22×D4—24no—
109C22×D4C22⋊C412yes—
110C22×D4C22⋊C412yes—
111C22×D4C22×C412yes—
112C22×D4C22×C411yescharacteristic
113C23×C4—21no—
114C23×C4—22no—
115C23×C4C2×D414yes—
116C2×M16—44no—
117C22×C8—44no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120#127—24no—
121#127—42no—
122#127—41no—
123#127—41no—
124#125—42no—
125#125—42no—
126#124—22no—
127#124—42no—
128#124—42no—
129#124—41no—
130#124—42no—
131#124—42no—
132#124—84no—
133#123—44no—
134#123—44no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137C22⋊Q8—42no—
138C22⋊Q8—42no—
139#121—48no—
140C4×Q8—22no—
141C4×Q8—21no—
142C4×Q8—21no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—44no—
146C4×D4—22no—
147C4×D4—24no—
148C4×D4—22no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—41no—
154C4×D4—41no—
155C4×D4—88no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2411yescharacteristic
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4C22⋊C414yes—
179C2×C22⋊C4C22⋊C414yes—
180C2×C22⋊C4C2×D414yes—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D412yes—
191C4⋊C8—42no—
192C4⋊C8—42no—
193C22⋊C8—22no—
194C22⋊C8—24no—
195C22⋊C8—22no—
196C22⋊C8—44no—
197C22⋊C8—44no—
198#97—42no—
199C4×C8—42no—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—24no—
204#95—22no—
205C24—21no—
206C24C2×C22⋊C414yes—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—82no—
216C2×D4—22no—
217C2×D4—21no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—44no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—48no—
227C2×D4—42no—
228C2×D4—48no—
229C2×D4—84no—
230C2×D4—82no—
231C2×D4#13112yes—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—44no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—84no—
245C22×C4—84no—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D412yes—
253M16—164no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—44no—
257C2×C8—164no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—84no—
272C4⋊C4#9912yes—
273C4⋊C4#9911yescharacteristic
274C4⋊C4#9911yescharacteristic
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—48no—
284C22⋊C4—48no—
285C22⋊C4—84no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—22no—
294C42—21no—
295C42—22no—
296C42—41no—
297C42—44no—
298C42—42no—
299C42—44no—
300C42—42no—
301C23—22no—
302C23—24no—
303C23—24no—
304C23—24no—
305C23—42no—
306C23—42no—
307C23—44no—
308C23—44no—
309C23—41no—
310C23—41no—
311C23C22×C22⋊C411yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—81no—
315Q8—81no—
316Q8—162no—
317D4—42no—
318D4—42no—
319D4—44no—
320D4—82no—
321D4—82no—
322D4—82no—
323D4—816no—
324D4—162no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—162no—
356C2×C4—162no—
357C2×C4C2×#9911yescharacteristic
358C2×C4C2×#9912yes—
359C2×C4C2×#9912yes—
360C2×C4C2×#9911yescharacteristic
361C2×C4C2×#13111yescharacteristic
362C2×C4#56811yescharacteristic
363C8—82no—
364C8—84no—
365C8—82no—
366C22—21no—
367C22—21no—
368C22—44no—
369C22—42no—
370C22—44no—
371C22—44no—
372C22—44no—
373C22—84no—
374C22—84no—
375C22—82no—
376C22—81no—
377C22—81no—
378C22#288011yescharacteristic
379C22C22×#9911yescharacteristic
380C22#288911yescharacteristic
381C4—21no—
382C4—21no—
383C4—22no—
384C4—22no—
385C4—21no—
386C4—42no—
387C4—44no—
388C4—42no—
389C4—42no—
390C4—41no—
391C4—42no—
392C4—81no—
393C4—82no—
394C4—81no—
395C4#213212yes—
396C2—21no—
397C2—21no—
398C2—84no—
399C2—82no—
400C2#1741411yescharacteristic
401C2#3335911yescharacteristic
402C2C2×#213211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#2541C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#559 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17429C2semidirect
3. #11574C2semidirect
4. #11572C2semidirect
5. #11530C2semidirect
6. #11529C2semidirect
7. #11476C2semidirect
8. #11475C2semidirect

Maximal subgroups

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

subgroupindex
#364962
#174292
#115742
#115722
#115302
#115292
#114762
#114752