Finite Group Databasebeta

← #4929053

Subgroups — #4929053

Order 512.

5926 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#518C211yesmaximal, characteristic
2C2×#2486C212yesmaximal, complemented
3C2×#2405C212yesmaximal
4C22×#397C211yesmaximal, characteristic, complemented
5#14907C214yesmaximal, complemented
6#14868C214yesmaximal, complemented
7#14824C214yesmaximal, complemented
8#14820C214yesmaximal, complemented
9#14338C211yesmaximal, characteristic, complemented
10#13968C214yesmaximal, complemented
11#13960C212yesmaximal, complemented
12#13869C212yesmaximal, complemented
13C24×D4C2211yescharacteristic
14C22×#123C2214yes—
15C2×#518—216no—
16C2×#518—28no—
17C22×C4×D4C2212yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2212yes—
21C22×#117C2212yes—
22C23×C4⋊C4C2211yescharacteristic, complemented
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2211yescharacteristic
27#2486—216no—
28#2405—216no—
29C2×#398C2214yescomplemented
30C2×#397—28no—
31C2×#397C2214yes—
32C2×#397C2214yescomplemented
33C2×#397C2214yescomplemented
34C2×#394C2218yes—
35C2×#389C2214yescomplemented
36C2×#386C2214yes—
37C2×#385C2214yes—
38C2×#381C2214yescomplemented
39C2×#380C2212yescomplemented
40C2×#380C2214yescomplemented
41C2×#379C2212yes—
42C2×#379C2214yes—
43C22×#95C2211yescharacteristic, complemented
44C22×#95C2212yescomplemented
45#1441C2218yescomplemented
46#1441C2214yescomplemented
47#1440C2218yes—
48#1440C2218yescomplemented
49#1439C2214yes—
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1436C2218yescomplemented
53#1434C2214yescomplemented
54C22⋊#95C2212yescomplemented
55C22⋊#95C2214yescomplemented
56C22⋊#95C2218yescomplemented
57C22⋊#95C2214yescomplemented
58C22⋊#95C2214yescomplemented
59C22⋊#95C2214yescomplemented
60C26C2311yescharacteristic
61C26C2311yescharacteristic
62C23×D4—24no—
63C23×D4—24no—
64C23×D4—28no—
65C23×D4—28no—
66C23×D4D418yes—
67C23×D4C2312yes—
68C23×D4C2312yes—
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C2×#123—216no—
79C2×#123—216no—
80C2×#123—216no—
81#518—2128no—
82C2×C4×D4—28no—
83C2×C4×D4—28no—
84C2×C4×D4—28no—
85C2×C4×D4—28no—
86C2×C4×D4—28no—
87C2×C4×D4—28no—
88C2×C4×D4—28no—
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4D4116yes—
94C2×#117—28no—
95C2×#117—28no—
96C2×#117—28no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4C2318yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—28no—
112C22×C22⋊C4—28no—
113C22×C22⋊C4D418yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2318yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2318yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129#398—216no—
130#397—216no—
131#397—216no—
132#397—216no—
133#394—232no—
134#389—216no—
135#386—216no—
136#385—216no—
137#381—216no—
138#380—216no—
139#380—28no—
140#379—216no—
141#379—28no—
142C2×#95—28no—
143C2×#95—24no—
144C2×#95C2318yes—
145C2×#95C2314yes—
146C2×#95C2318yes—
147C2×#95C2314yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C25—24no—
154C25—22no—
155C25—24no—
156C25—24no—
157C25—22no—
158C25—24no—
159C25—44no—
160C25—44no—
161C25—44no—
162C25—44no—
163C25C2×D414yes—
164C25C2×D414yes—
165C25C2×D414yes—
166C25#4114yes—
167C25C2411yescharacteristic
168C25C2412yes—
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×D4—232no—
172C22×D4—264no—
173C22×D4—232no—
174C22×D4—232no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—264no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4C2×D418yes—
187C22×D4C2×D418yes—
188C22×D4C2×D418yes—
189C22×D4#4118yes—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—44no—
213C23×C4—48no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—48no—
219C23×C4—44no—
220C23×C4C2×D414yes—
221C23×C4C2×D418yes—
222C23×C4C2×D414yes—
223C23×C4C2×D418yes—
224C23×C4C2×D414yes—
225C23×C4C2418yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233#123—464no—
234C4×D4—264no—
235C4×D4—264no—
236C4×D4—264no—
237C4×D4—264no—
238#117—264no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—416no—
253C2×C4⋊C4C2×D418yes—
254C2×C22⋊C4—232no—
255C2×C22⋊C4—264no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—232no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—432no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4C2×D418yes—
294C2×C22⋊C4C2×D418yes—
295C2×C22⋊C4C2×D418yes—
296C2×C22⋊C4C2×D418yes—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42C2×D418yes—
303#95—28no—
304#95—216no—
305#95—216no—
306#95—28no—
307#95—216no—
308#95—216no—
309#95—48no—
310#95—48no—
311#95—48no—
312C24—24no—
313C24—24no—
314C24—28no—
315C24—28no—
316C24—216no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—28no—
326C24—24no—
327C24—24no—
328C24—28no—
329C24—24no—
330C24—24no—
331C24—28no—
332C24—216no—
333C24—28no—
334C24—216no—
335C24—28no—
336C24—216no—
337C24—48no—
338C24—48no—
339C24—416no—
340C24—416no—
341C24—416no—
342C24—88no—
343C24—88no—
344C24—816no—
345C24—88no—
346C24—84no—
347C24—84no—
348C24—88no—
349C24—88no—
350C24—816no—
351C24#121116yes—
352C24C22×D412yes—
353C24C22×D412yes—
354C24C22×D414yes—
355C24C22×D412yes—
356C24C22×D414yes—
357C24C22×D412yes—
358C24C22×D414yes—
359C24C22×D412yes—
360C24C22×D412yes—
361C24C2×#4112yes—
362C24C2×#4112yes—
363C24C2×#4114yes—
364C24C2×#4114yes—
365C24C2×#4112yes—
366C24C2511yescharacteristic
367C2×D4—216no—
368C2×D4—2256no—
369C2×D4—216no—
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—216no—
373C2×D4—216no—
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4—216no—
380C2×D4—216no—
381C2×D4#121132yes—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—216no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—216no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—416no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4#120116yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C2×#4114yes—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4—432no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—216no—
467C22⋊C4—216no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—432no—
473C22⋊C4—88no—
474C22⋊C4—816no—
475C42—216no—
476C42—216no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—216no—
484C23—232no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—216no—
502C23—28no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—216no—
511C23—232no—
512C23—432no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—48no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—88no—
526C23—88no—
527C23—816no—
528C23—832no—
529C23—832no—
530C23—816no—
531C23—832no—
532C23—88no—
533C23—88no—
534C23—816no—
535C23—832no—
536C23C2×#12014yes—
537C23C2×#12114yes—
538C23C2×#12114yes—
539C23C2×#12118yes—
540C23C2×#12118yes—
541C23C2×#12114yes—
542C23#52514yes—
543C23#52518yes—
544C23C23×D411yescharacteristic
545C23C23×D412yes—
546C23C23×D411yescharacteristic
547C23C22×#4111yescharacteristic
548C23C22×#4112yes—
549C23#58514yes—
550C23#58514yes—
551D4—264no—
552D4—264no—
553C2×C4—232no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—416no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—816no—
594C2×C4—88no—
595C2×C4—816no—
596C2×C4—88no—
597C2×C4—88no—
598C2×C4—816no—
599C2×C4—816no—
600C2×C4C2×#12018yes—
601C2×C4C2×#12118yes—
602C22—24no—
603C22—24no—
604C22—28no—
605C22—28no—
606C22—216no—
607C22—216no—
608C22—24no—
609C22—24no—
610C22—28no—
611C22—28no—
612C22—28no—
613C22—216no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22—44no—
618C22—42no—
619C22—42no—
620C22—48no—
621C22—44no—
622C22—48no—
623C22—44no—
624C22—48no—
625C22—48no—
626C22—48no—
627C22—48no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—832no—
632C22—832no—
633C22#284718yes—
634C22C22×#12011yescharacteristic
635C22C22×#12111yescharacteristic
636C22C22×#12112yes—
637C22C2×#52511yescharacteristic
638C22C2×#52512yes—
639C22#348512yes—
640C22#348512yes—
641C22#348514yes—
642C22#348514yes—
643C22#349114yes—
644C22#350014yes—
645C22#352118yes—
646C4—28no—
647C4—28no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—84no—
652C4—84no—
653C4—88no—
654C2—28no—
655C2—44no—
656C2—48no—
657C2—44no—
658C2#3544611yescharacteristic
659C2#3544712yes—
660C2#3545212yes—
661C2#3561611yescharacteristic
662C2#3600914yes—
663C2#3601014yes—
664C2C2×#284711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2486C2semidirect
4. C22×#397C2semidirect
5. #14907C2semidirect
6. #14868C2semidirect
7. #14824C2semidirect
8. #14820C2semidirect
9. #14338C2semidirect
10. #13968C2semidirect
11. #13960C2semidirect
12. #13869C2semidirect
22. C23×C4⋊C4C22semidirect
29. C2×#398C22semidirect
32. C2×#397C22semidirect
33. C2×#397C22semidirect
35. C2×#389C22semidirect
38. C2×#381C22semidirect
39. C2×#380C22semidirect
40. C2×#380C22semidirect
43. C22×#95C22semidirect
44. C22×#95C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
48. #1440C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1436C22semidirect
53. #1434C22semidirect
54. C22⋊#95C22semidirect
55. C22⋊#95C22semidirect
56. C22⋊#95C22semidirect
57. C22⋊#95C22semidirect
58. C22⋊#95C22semidirect
59. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5182
C2×#24862
C2×#24052
C22×#3972
#149072
#148682
#148242
#148202
#143382
#139682
#139602
#138692