Finite Group Databasebeta

← #4389955

Subgroups — #4389955

Order 512.

2614 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2822C211yesmaximal, characteristic, complemented
2C2×#2562C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic, complemented
4C2×#2580C211yesmaximal, characteristic
5C2×#2634C211yesmaximal, characteristic, complemented
6C2×#2633C211yesmaximal, characteristic
7#17064C212yesmaximal, complemented
8#16812C212yesmaximal, complemented
9#15814C214yesmaximal
10#15804C214yesmaximal, complemented
11#15513C214yesmaximal, complemented
12#15513C214yesmaximal, complemented
13#15065C212yesmaximal, complemented
14#15020C212yesmaximal, complemented
15#13781C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2822—28no—
21#2634—28no—
22#2633—28no—
23#2580—28no—
24#2565—28no—
25#2562—28no—
26C2×#400C2212yes—
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#400C2212yes—
30C2×#400C2212yes—
31C2×#399C2212yes—
32C2×#399C2212yes—
33C2×#398C2212yes—
34C2×#398C2212yes—
35C2×#398C2212yes—
36C2×#398C2212yes—
37C2×#398C2214yes—
38C2×#398C2214yes—
39C2×#397C2212yes—
40C2×#395C2212yes—
41C2×#394C2212yes—
42C2×#394C2212yes—
43C2×#393C2212yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#390C2212yes—
47C2×#389C2212yes—
48C2×#388C2212yes—
49C2×#388C2212yes—
50C2×#387C2214yes—
51C2×#387C2214yes—
52C2×#385C2214yes—
53C2×#385C2214yes—
54C2×#384C2211yescharacteristic
55C2×#383C2211yescharacteristic
56C2×#382C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×#380C2212yes—
59C2×#380C2212yes—
60C2×C4×C4⋊C4C2211yescharacteristic
61#1445C2214yes—
62#1445C2214yes—
63#1445C2214yes—
64#1445C2214yes—
65#1444C2214yes—
66#1444C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1437C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2214yes—
78C4⋊#95C2214yes—
79#1434C2212yes—
80C4×#95C2212yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4C2312yes—
84C2×#125—24no—
85C2×#125—24no—
86C2×#125—24no—
87C2×#125—24no—
88C2×#125—24no—
89C2×#125—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#400—28no—
129#400—28no—
130#400—28no—
131#400—28no—
132#400—28no—
133#399—28no—
134#399—28no—
135#398—216no—
136#398—216no—
137#398—28no—
138#398—28no—
139#398—28no—
140#398—28no—
141#397—28no—
142#395—28no—
143#394—28no—
144#394—28no—
145#393—28no—
146#391—24no—
147#391—24no—
148#390—28no—
149#389—28no—
150#388—28no—
151#388—28no—
152#387—216no—
153#387—216no—
154#385—216no—
155#385—216no—
156#384—24no—
157#383D418yes—
158#382—24no—
159#382D418yes—
160#380—28no—
161#380—28no—
162C4×C4⋊C4D418yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C25C2412yes—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—42no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—42no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217#125—416no—
218#125—416no—
219C4×Q8—416no—
220C4×D4—416no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95C2×D418yes—
330#95C2×D418yes—
331#95C2×D418yes—
332#95C2×D418yes—
333#95C2×D418yes—
334#95C2×D418yes—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—22no—
345C242−1+414yes—
346C24C2511yescharacteristic
347C2×Q8—28no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×D4—28no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—44no—
356C2×D4—44no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4C22×D414yes—
426C22×C4C22×D418yes—
427C22×C4C22×D418yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C42−1+414yes—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—416no—
442C4⋊C4—416no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C4⋊C4—88no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C42—44no—
467C42—44no—
468C42—48no—
469C42—88no—
470C42#127116yes—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—28no—
475C23—28no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23C23×D411yescharacteristic
495C23C23×D411yescharacteristic
496C23C23×D411yescharacteristic
497C23C2×2−1+412yes—
498C23C2×2−1+412yes—
499C23#58514yes—
500C23#58512yes—
501C23#58512yes—
502Q8—416no—
503D4—416no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—816no—
579C2×C4C2×#12714yes—
580C2×C4C2×#12714yes—
581C2×C4C2×#12714yes—
582C22—44no—
583C22—48no—
584C22—44no—
585C22—44no—
586C22—48no—
587C22—82no—
588C22—84no—
589C22—82no—
590C22—84no—
591C22—82no—
592C22—82no—
593C22C22×#12711yescharacteristic
594C22#344912yes—
595C22#348511yescharacteristic
596C22#348511yescharacteristic
597C22#348611yescharacteristic
598C22#348611yescharacteristic
599C22#348912yes—
600C22#348912yes—
601C22#348912yes—
602C22#348912yes—
603C22#350012yes—
604C22#350012yes—
605C22#355314yes—
606C22#355714yes—
607C22#356314yes—
608C22#356614yes—
609C4—24no—
610C4—24no—
611C4—24no—
612C4—48no—
613C4—44no—
614C4—48no—
615C4—44no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C2—84no—
624C2#3532911yescharacteristic
625C2#3563411yescharacteristic
626C2#3563411yescharacteristic
627C2#4142512yes—
628C2#4217112yes—
629C2#4776214yes—
630C2#5089812yes—
631C2#5121212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2822C2semidirect
3. C2×#2565C2semidirect
5. C2×#2634C2semidirect
7. #17064C2semidirect
8. #16812C2semidirect
10. #15804C2semidirect
11. #15513C2semidirect
12. #15513C2semidirect
13. #15065C2semidirect
14. #15020C2semidirect
15. #13781C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#25622
C2×#25652
C2×#25802
C2×#26342
C2×#26332
#170642
#168122
#158142
#158042
#155132
#150652
#150202
#137812