Finite Group Databasebeta

← #480647

Subgroups — #480647

Order 512.

825 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35669C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#11574C212yesmaximal, complemented
5#11565C212yesmaximal, complemented
6#11554C212yesmaximal
7#11552C212yesmaximal, complemented
8#11480C212yesmaximal, complemented
9#11478C212yesmaximal, complemented
10C2×#542C411yescharacteristic
11C2×#519C2211yescharacteristic
12#2983—21no—
13#2982—21no—
14#2568C411yescharacteristic
15#2567C2211yescharacteristic
16#2545C411yescharacteristic
17#2545C2211yescharacteristic
18#2500C2211yescharacteristic
19#2499C2211yescharacteristic
20#2474—21no—
21#2474—21no—
22#2470—21no—
23#2470—21no—
24#2469—21no—
25#2469—21no—
26#2465—21no—
27#2465—21no—
28#2436C411yescharacteristic
29#2434C411yescharacteristic
30#2327C411yescharacteristic
31#2326C2211yescharacteristic
32C4×C4⋊Q8C2211yescharacteristic
33C4×#124C411yescharacteristic
34#2298C411yescharacteristic
35#1467—21no—
36#1467C2212yes—
37#1464—21no—
38#1464C2212yes—
39#1461—21no—
40#1461C2212yes—
41#1461C2212yes—
42#1461C2212yes—
43#1458—21no—
44#1458—21no—
45#1458—21no—
46#1458C2212yes—
47#1325C2214yes—
48#1325C2214yes—
49#1319C2214yes—
50#1318C2214yes—
51C23×Q8C2×C411yescharacteristic
52#542—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—21no—
57C2×#124—21no—
58C2×#124C2×C411yescharacteristic
59C2×#124C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#123—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×#121C2×C411yescharacteristic
65#519—44no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C2×#117C2311yescharacteristic
82#446—44no—
83C8×Q8—44no—
84#432—22no—
85#432—22no—
86#431—22no—
87#431—22no—
88#431—22no—
89#431—22no—
90#400—21no—
91#400—21no—
92#397—21no—
93#397—21no—
94#396—21no—
95#396—21no—
96#396—21no—
97#396—21no—
98#395—21no—
99#395—21no—
100#395—21no—
101#395—21no—
102#394—21no—
103#394—21no—
104#393—21no—
105#393—21no—
106#391—21no—
107#391—21no—
108#391C2×C411yescharacteristic
109#391C2×C411yescharacteristic
110#391C2×C411yescharacteristic
111#391C2×C411yescharacteristic
112#391C2×C411yescharacteristic
113#391C2311yescharacteristic
114#390C2×C411yescharacteristic
115#390C2311yescharacteristic
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#386—21no—
141#386—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#382C2×C411yescharacteristic
149#382C2×C411yescharacteristic
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4C2×C411yescharacteristic
153C4×C4⋊C4C2×C411yescharacteristic
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4C2×C411yescharacteristic
157C4×C22⋊C4C2×C411yescharacteristic
158#376C2×C411yescharacteristic
159C43C2×C411yescharacteristic
160#324—22no—
161#324—22no—
162#324—22no—
163#324—22no—
164#324C2314yes—
165#324C2314yes—
166#323—22no—
167#323—22no—
168C22×Q8—24no—
169C22×Q8—21no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8C22⋊C411yescharacteristic
173C22×Q8C22⋊C411yescharacteristic
174C22×Q8C22×C411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C23×C4—21no—
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C4⋊Q8—24no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186#126—44no—
187#124—24no—
188#124—42no—
189#124—44no—
190#124—42no—
191#123—44no—
192C22⋊Q8—44no—
193C22⋊Q8—42no—
194#121—42no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×D4—42no—
205#117—24no—
206#117—42no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2411yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C2×D411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C4⋊C8—42no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281C4⋊C8—42no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284#97—42no—
285#97—42no—
286C4×C8—42no—
287C4×C8—42no—
288C4×C8—42no—
289C4×C8—42no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312C24—21no—
313C24C2×C22⋊C411yescharacteristic
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—416no—
330C2×Q8—81no—
331C2×Q8—81no—
332C2×Q8#13114yes—
333C2×D4—22no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—81no—
338C2×D4—81no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C2×C8—44no—
373C2×C8—44no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—81no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C4⋊C4—81no—
399C4⋊C4—81no—
400C4⋊C4#9914yes—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—81no—
418C22⋊C4—81no—
419C22⋊C4—81no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—42no—
437C42—42no—
438C42—81no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C22×C22⋊C411yescharacteristic
452Q8—28no—
453Q8—44no—
454Q8—44no—
455Q8—44no—
456Q8—82no—
457Q8—162no—
458D4—82no—
459D4—162no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—162no—
512C2×C4#40812yes—
513C2×C4#40812yes—
514C2×C4C2×#9912yes—
515C2×C4C2×#9912yes—
516C2×C4C2×#9912yes—
517C2×C4#41312yes—
518C2×C4C2×#13112yes—
519C8—84no—
520C8—84no—
521C22—21no—
522C22—21no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22C2×#40811yescharacteristic
536C22C22×#9911yescharacteristic
537C22C2×#41311yescharacteristic
538C4—22no—
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—42no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4—81no—
559C2—21no—
560C2—21no—
561C2—81no—
562C2—161no—
563C2#1738111yescharacteristic
564C2#3336211yescharacteristic
565C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17371C2semidirect
3. #17352C2semidirect
4. #11574C2semidirect
5. #11565C2semidirect
7. #11552C2semidirect
8. #11480C2semidirect
9. #11478C2semidirect

Maximal subgroups

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

subgroupindex
#356692
#173712
#173522
#115742
#115652
#115542
#115522
#114802
#114782