Finite Group Databasebeta

← #4452850

Subgroups — #4452850

Order 512.

2754 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C22×#397C211yesmaximal, characteristic
3#16859C214yesmaximal, complemented
4#16788C214yesmaximal, complemented
5#16528C214yesmaximal, complemented
6#16422C214yesmaximal, complemented
7#15015C211yesmaximal, characteristic, complemented
8#14936C211yesmaximal, characteristic, complemented
9#14934C211yesmaximal, characteristic, complemented
10#14918C212yesmaximal
11#14914C211yesmaximal, characteristic, complemented
12#14727C212yesmaximal
13#14043C212yesmaximal, complemented
14#13972C212yesmaximal, complemented
15#13775C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21#2626—28no—
22C2×#399C2212yes—
23C2×#397—28no—
24C2×#397C2214yes—
25C2×#397C2214yes—
26C2×#397C2214yes—
27C2×#396C2212yes—
28C2×#396C2212yes—
29C2×#393C2212yes—
30C2×#389C2214yescomplemented
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#381C2212yes—
35C2×#381C2212yes—
36C2×#380C2211yescharacteristic
37C2×#379C2212yes—
38C2×#379C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43#1446C2212yescomplemented
44#1446C2212yescomplemented
45#1445C2212yescomplemented
46#1445C2214yescomplemented
47#1445C2212yescomplemented
48#1444C2212yescomplemented
49#1444C2212yescomplemented
50#1444C2214yescomplemented
51#1443C2214yescomplemented
52#1443C2214yescomplemented
53#1442C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2212yescomplemented
56#1442C2212yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1440C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1437C2214yes—
67#1437C2214yes—
68#1436C2214yescomplemented
69#1436C2214yescomplemented
70#1434C2214yescomplemented
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—44no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4D418yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42C2314yes—
119#399—28no—
120#397—216no—
121#397—216no—
122#397—48no—
123#396—28no—
124#396—28no—
125#393—28no—
126#389—48no—
127#385—28no—
128#385—28no—
129#385—44no—
130#381—28no—
131#381—44no—
132#380—24no—
133#379—28no—
134#379—28no—
135C4×C22⋊C4—44no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25—24no—
167C25—24no—
168C25—22no—
169C25—22no—
170C25—42no—
171C25—42no—
172C25—42no—
173C25—42no—
174C25C2×D414yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C22×Q8—22no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8—42no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2412yes—
220C22⋊Q8—416no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—416no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—416no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—416no—
273C2×C22⋊C4—416no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×D418yes—
283C2×C22⋊C4C2×D418yes—
284C2×C22⋊C4#4118yes—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—44no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—42no—
301#95—44no—
302#95—42no—
303#95—42no—
304#95—44no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—28no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—28no—
331C24—28no—
332C24—44no—
333C24—44no—
334C24—44no—
335C24—44no—
336C24—416no—
337C24—416no—
338C24—48no—
339C24—48no—
340C24—88no—
341C24—88no—
342C24—84no—
343C24—84no—
344C24—84no—
345C24—84no—
346C24C22×D414yes—
347C24C22×D412yes—
348C24C2×#4114yes—
349C24H(5,𝔽2)14yes—
350C242−1+412yes—
351C242−1+412yes—
352C24C2511yescharacteristic
353C2×Q8—44no—
354C2×Q8—82no—
355C2×Q8—82no—
356C2×Q8—88no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—416no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C2×#4118yes—
449C22×C4H(5,𝔽2)14yes—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—84no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—84no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C42—88no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—416no—
494C23—416no—
495C23—416no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—88no—
513C23—88no—
514C23—88no—
515C23—88no—
516C23—832no—
517C23—832no—
518C23#53214yes—
519C23#53414yes—
520C23#53614yes—
521C23#53918yes—
522C23#54218yes—
523C23C23×D411yescharacteristic
524C23C22×#4112yes—
525C23C2×H(5,𝔽2)12yes—
526C23C2×H(5,𝔽2)12yes—
527C23C2×2−1+411yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23#58512yes—
530C23#58512yes—
531C23#58512yes—
532Q8—88no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—416no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—88no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4—88no—
606C22—216no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—216no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—832no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22#266318yes—
627C22C2×#53211yescharacteristic
628C22C2×#53411yescharacteristic
629C22C2×#53611yescharacteristic
630C22C2×#53912yes—
631C22C2×#54212yes—
632C22#348411yescharacteristic
633C22#348511yescharacteristic
634C22#348511yescharacteristic
635C22#348511yescharacteristic
636C22#351314yes—
637C22#352114yes—
638C22#353314yes—
639C22#354214yes—
640C22#356112yes—
641C22#356112yes—
642C22#356412yes—
643C22#356412yes—
644C4—48no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—84no—
657C2—28no—
658C2—84no—
659C2C2×#266311yescharacteristic
660C2#3539411yescharacteristic
661C2#3546711yescharacteristic
662C2#3670411yescharacteristic
663C2#3696511yescharacteristic
664C2#3696611yescharacteristic
665C2#3697011yescharacteristic
666C2#4069114yes—
667C2#4075214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16859C2semidirect
4. #16788C2semidirect
5. #16528C2semidirect
6. #16422C2semidirect
7. #15015C2semidirect
8. #14936C2semidirect
9. #14934C2semidirect
11. #14914C2semidirect
13. #14043C2semidirect
14. #13972C2semidirect
30. C2×#389C22semidirect
43. #1446C22semidirect
44. #1446C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1444C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1443C22semidirect
52. #1443C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C22×#3972
#168592
#167882
#165282
#164222
#150152
#149362
#149342
#149182
#149142
#147272
#140432
#139722
#137752