Finite Group Databasebeta

← #772055

Subgroups — #772055

Order 512.

3798 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2321C212yesmaximal, complemented
2#16047C212yesmaximal, complemented
3#15978C212yesmaximal, complemented
4#15966C211yesmaximal, characteristic, complemented
5#15947C211yesmaximal, characteristic
6#15330C214yesmaximal, complemented
7#15268C214yesmaximal, complemented
8#15266C212yesmaximal, complemented
9#15163C212yesmaximal, complemented
10#15120C212yesmaximal, complemented
11#14199C214yesmaximal, complemented
12#13995C214yesmaximal, complemented
13#13847C211yesmaximal, characteristic, complemented
14C22×#120C2212yes—
15C22×#120C2214yes—
16C22×C4×D4C2211yescharacteristic
17C23×C22⋊C4C2212yescomplemented
18C2×#398C2212yescomplemented
19C2×#396C2214yescomplemented
20C2×#394C2214yes—
21C2×#394C2212yes—
22#2321—216no—
23C2×#390C2214yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#388C2214yescomplemented
27C2×#388C2214yescomplemented
28C2×#388C2214yescomplemented
29C2×#386C2212yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#385C2212yescomplemented
33C2×#385C2214yescomplemented
34C2×#379C2212yescomplemented
35C2×#379C2214yescomplemented
36C2×C4×C4⋊C4C2211yescharacteristic, complemented
37C2×C4×C22⋊C4C2212yescomplemented
38C2×C4×C22⋊C4C2214yescomplemented
39C2×#376C2211yescharacteristic, complemented
40#1445C2211yescharacteristic, complemented
41#1445C2211yescharacteristic, complemented
42#1445C2212yescomplemented
43#1444C2212yescomplemented
44#1442C2212yescomplemented
45#1440C2214yescomplemented
46#1440C2214yescomplemented
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1438C2212yes—
54#1438C2212yes—
55#1438C2211yescharacteristic
56#1438C2211yescharacteristic
57#1438C2212yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1437C2214yescomplemented
62#1437C2214yescomplemented
63#1437C2214yescomplemented
64#1436C2214yescomplemented
65C4⋊#95C2212yescomplemented
66#1434C2212yescomplemented
67#1434C2214yescomplemented
68C26C2312yes—
69C23×D4C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2314yes—
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×#120—28no—
76C2×#120—216no—
77C2×#120—216no—
78C2×#120—48no—
79C2×#120—48no—
80C2×C4×D4—42no—
81C2×C4×D4—42no—
82C2×C4×D4D418yescomplemented
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—28no—
90C22×C22⋊C4—44no—
91C22×C22⋊C4—44no—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
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⋊C4C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112C22×C42C2312yes—
113#398—44no—
114#396—48no—
115#394—216no—
116#394—44no—
117#390—216no—
118#390—44no—
119#390—44no—
120#388—216no—
121#388—216no—
122#388—48no—
123#386—28no—
124#386—216no—
125#386—48no—
126#385—28no—
127#385—48no—
128#379—28no—
129#379—48no—
130C4×C4⋊C4—42no—
131C4×C22⋊C4—28no—
132C4×C22⋊C4—48no—
133#376—42no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C25—24no—
154C25—28no—
155C25—28no—
156C25—42no—
157C25—42no—
158C25—44no—
159C25—44no—
160C25—44no—
161C25—44no—
162C25C2414yes—
163C25C2412yes—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4C2×D414yes—
184C22×D4#41116yes—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2×D418yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206#120—264no—
207#120—464no—
208C4×D4—416no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C2×D414yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—432no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42C2×D414yes—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—42no—
296#95—42no—
297#95—41no—
298#95—41no—
299#95—44no—
300#95—41no—
301#95—41no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—42no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—22no—
312C24—24no—
313C24—216no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—416no—
329C24—416no—
330C24—48no—
331C24—416no—
332C24—416no—
333C24—48no—
334C24—88no—
335C24—88no—
336C24—88no—
337C24—88no—
338C24—88no—
339C24—88no—
340C24—88no—
341C24—88no—
342C24—88no—
343C24—88no—
344C24—84no—
345C24—84no—
346C24—84no—
347C24—84no—
348C24C22×D414yes—
349C24C2×#4118yes—
350C24C2×#4118yes—
351C24C2511yescharacteristic
352C2×D4—216no—
353C2×D4—232no—
354C2×D4—44no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—416no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—88no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—84no—
372C2×D4—84no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—216no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
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—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—88no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C2×#4118yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—88no—
446C4⋊C4—88no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C22⋊C4—232no—
450C22⋊C4—216no—
451C22⋊C4—232no—
452C22⋊C4—48no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—416no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C42—28no—
467C42—48no—
468C42—84no—
469C42—88no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—432no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
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—832no—
527C23—832no—
528C23—832no—
529C23—832no—
530C23—832no—
531C23—832no—
532C23—816no—
533C23—816no—
534C23#52518yes—
535C23C23×D411yescharacteristic
536C23C22×#4114yes—
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)12yes—
540C23C2×H(5,𝔽2)12yes—
541C23C2×H(5,𝔽2)12yes—
542C23#58512yes—
543D4—416no—
544D4—832no—
545D4—816no—
546D4—816no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
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—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—816no—
607C2×C4—816no—
608C2×C4#53414yes—
609C22—216no—
610C22—28no—
611C22—44no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—48no—
616C22—44no—
617C22—48no—
618C22—48no—
619C22—48no—
620C22—48no—
621C22—48no—
622C22—48no—
623C22—48no—
624C22—48no—
625C22—864no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22C2×#52512yes—
635C22C2×#53411yescharacteristic
636C22#348112yes—
637C22#348112yes—
638C22#349014yes—
639C22#349612yes—
640C22#349812yes—
641C22#350514yes—
642C22#351414yes—
643C22#351914yes—
644C22#353412yes—
645C22#353712yes—
646C22#354311yescharacteristic
647C22#354511yescharacteristic
648C22#354911yescharacteristic
649C22#354911yescharacteristic
650C4—24no—
651C4—44no—
652C4—48no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—44no—
657C4—42no—
658C4—84no—
659C4—84no—
660C4—82no—
661C2—48no—
662C2—88no—
663C2#3495611yescharacteristic
664C2#3536114yes—
665C2#3712911yescharacteristic
666C2#3729711yescharacteristic
667C2#4336112yes—
668C2#4341912yes—
669C2#4347812yes—
670C2#4379112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15947C2
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×#2321C2semidirect
2. #16047C2semidirect
3. #15978C2semidirect
4. #15966C2semidirect
6. #15330C2semidirect
7. #15268C2semidirect
8. #15266C2semidirect
9. #15163C2semidirect
10. #15120C2semidirect
11. #14199C2semidirect
12. #13995C2semidirect
13. #13847C2semidirect
17. C23×C22⋊C4C22semidirect
18. C2×#398C22semidirect
19. C2×#396C22semidirect
26. C2×#388C22semidirect
27. C2×#388C22semidirect
28. C2×#388C22semidirect
32. C2×#385C22semidirect
33. C2×#385C22semidirect
34. C2×#379C22semidirect
35. C2×#379C22semidirect
36. C2×C4×C4⋊C4C22semidirect
37. C2×C4×C22⋊C4C22semidirect
38. C2×C4×C22⋊C4C22semidirect
39. C2×#376C22semidirect
40. #1445C22semidirect
41. #1445C22semidirect
42. #1445C22semidirect
43. #1444C22semidirect
44. #1442C22semidirect
45. #1440C22semidirect
46. #1440C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. #1436C22semidirect
65. C4⋊#95C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect
82. C2×C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23212
#160472
#159782
#159662
#159472
#153302
#152682
#152662
#151632
#151202
#141992
#139952
#138472