Finite Group Databasebeta

← #4598533

Subgroups — #4598533

Order 512.

3114 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2819C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3#17137C214yesmaximal, complemented
4#16684C212yesmaximal, complemented
5#16509C214yesmaximal, complemented
6#16509C212yesmaximal, complemented
7#16260C214yesmaximal, complemented
8#15761C212yesmaximal, complemented
9#15630C212yesmaximal, complemented
10#15476C214yesmaximal, complemented
11#15070C212yesmaximal, complemented
12#14722C212yesmaximal, complemented
13#14476C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2212yes—
16C22×#124C2212yes—
17C22×#121C2212yes—
18C22×C4×D4C2211yescharacteristic
19#2819—28no—
20#2563—28no—
21C2×#395C2212yes—
22C2×#394C2212yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#394C2212yes—
26C2×#393C2214yes—
27C2×#393C2214yes—
28C2×#393C2214yes—
29C2×#393C2214yes—
30C2×#392C2214yes—
31C2×#392C2214yes—
32C2×#392C2214yes—
33C2×#392C2212yes—
34C2×#392C2214yes—
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#390C2214yes—
38C2×#388C2214yes—
39C2×#386C2214yes—
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#380C2214yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44#1446C2212yes—
45#1446C2212yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2212yes—
61#1438C2214yes—
62#1438C2214yes—
63#1437C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66#1434C2214yes—
67#1434C2212yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2312yes—
76C23×D4C2314yes—
77C24×C4C2312yes—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—42no—
81C2×#124—28no—
82C2×#124—48no—
83C2×#124—48no—
84C2×#124—44no—
85C2×#121—48no—
86C2×#121—44no—
87C2×C4×D4—24no—
88C2×C4×D4—42no—
89C2×C4×D4—42no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116#395—44no—
117#394—28no—
118#394—28no—
119#394—44no—
120#394—44no—
121#393—216no—
122#393—216no—
123#393—48no—
124#393—48no—
125#392—216no—
126#392—48no—
127#392—48no—
128#392—48no—
129#392—44no—
130#390—24no—
131#390—24no—
132#390—48no—
133#388—48no—
134#386—48no—
135#382—24no—
136#382D418yescomplemented
137#380—216no—
138C4×C4⋊C4—24no—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C25C2414yes—
158C25C2412yes—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—42no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—28no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—44no—
181C22×D4—42no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—42no—
202C22×D4—42no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—416no—
220#124—432no—
221#124—432no—
222#121—432no—
223C4×D4—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
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⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—42no—
297C2×C42C2×D414yes—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95C2×D4116yes—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C242−1+414yes—
342C24C2511yescharacteristic
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—84no—
347C2×Q8—88no—
348C2×Q8—84no—
349C2×Q8—84no—
350C2×Q8—88no—
351C2×Q8—88no—
352C2×Q8—88no—
353C2×D4—28no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—44no—
357C2×D4—416no—
358C2×D4—48no—
359C2×D4—416no—
360C2×D4—48no—
361C2×D4—416no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—88no—
367C2×D4—84no—
368C2×D4—88no—
369C2×D4—84no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—88no—
373C2×D4—88no—
374C2×D4—88no—
375C2×D4—88no—
376C2×D4—84no—
377C2×D4—88no—
378C2×D4—84no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—88no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—48no—
439C22×C4—416no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4H(5,𝔽2)18yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4H(5,𝔽2)14yes—
451C22×C4H(5,𝔽2)12yes—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—88no—
459C22⋊C4—48no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—28no—
475C42—28no—
476C42—48no—
477C42—44no—
478C23—24no—
479C23—24no—
480C23—28no—
481C23—24no—
482C23—28no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—44no—
497C23—42no—
498C23—44no—
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—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23C23×D411yescharacteristic
537C23C2×H(5,𝔽2)12yes—
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)12yes—
541C23C2×H(5,𝔽2)14yes—
542C23C2×2−1+412yes—
543C23#58512yes—
544Q8—816no—
545Q8—816no—
546D4—416no—
547D4—832no—
548D4—816no—
549D4—816no—
550D4—816no—
551D4—832no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—816no—
611C2×C4#53514yes—
612C2×C4H(3,𝔽4)14yes—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—48no—
617C22—48no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22C2×#53511yescharacteristic
640C22#348411yescharacteristic
641C22#348611yescharacteristic
642C22#348812yes—
643C22#349812yes—
644C22C2×H(3,𝔽4)11yescharacteristic
645C22#353212yes—
646C22#353612yes—
647C22#353614yes—
648C22#353612yes—
649C22#354114yes—
650C22#354214yes—
651C22#354511yescharacteristic
652C22#354912yes—
653C22#354914yes—
654C22#355812yes—
655C4—24no—
656C4—24no—
657C4—44no—
658C4—44no—
659C4—42no—
660C4—48no—
661C4—44no—
662C4—84no—
663C4—82no—
664C4—82no—
665C4—84no—
666C2—84no—
667C2—164no—
668C2#3670711yescharacteristic
669C2#4296312yes—
670C2#4597711yescharacteristic
671C2#5106712yes—
672C2#5190211yescharacteristic
673C2#5301512yes—
674C2#5498414yes—
675C2#5561512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2819C2semidirect
2. C2×#2563C2semidirect
3. #17137C2semidirect
4. #16684C2semidirect
5. #16509C2semidirect
6. #16509C2semidirect
7. #16260C2semidirect
8. #15761C2semidirect
9. #15630C2semidirect
10. #15476C2semidirect
11. #15070C2semidirect
12. #14722C2semidirect
13. #14476C2semidirect
136. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28192
C2×#25632
#171372
#166842
#165092
#162602
#157612
#156302
#154762
#150702
#147222
#144762