Finite Group Databasebeta

← #4378715

Subgroups — #4378715

Order 512.

2854 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic, complemented
2C2×#2321C211yesmaximal, characteristic
3#16895C212yesmaximal, complemented
4#16679C212yesmaximal, complemented
5#16488C214yesmaximal, complemented
6#15948C212yesmaximal
7#15579C212yesmaximal, complemented
8#15480C212yesmaximal, complemented
9#15463C212yesmaximal, complemented
10#15061C212yesmaximal, complemented
11#14894C212yesmaximal, complemented
12#14820C212yesmaximal, complemented
13#14776C212yesmaximal, complemented
14#14376C214yesmaximal, complemented
15#13959C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#120C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic, complemented
20C23×C22⋊C4C2211yescharacteristic
21#2486—28no—
22C2×#398C2212yescomplemented
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#397C2214yes—
27#2321—28no—
28C2×#390C2212yes—
29C2×#389C2212yescomplemented
30C2×#388C2212yes—
31C2×#388C2212yes—
32C2×#386C2211yescharacteristic
33C2×#386C2212yes—
34C2×#386C2214yes—
35C2×#385C2211yescharacteristic
36C2×#383C2211yescharacteristic, complemented
37C2×#381C2212yescomplemented
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic, complemented
40C2×#379C2211yescharacteristic
41C2×#379C2214yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic, complemented
44#1445C2212yescomplemented
45#1445C2212yescomplemented
46#1445C2212yescomplemented
47#1444C2212yescomplemented
48#1443C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2212yescomplemented
52#1442C2214yescomplemented
53#1441C2214yescomplemented
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yescomplemented
65#1438C2214yes—
66#1438C2212yes—
67#1437C2214yes—
68#1437C2212yes—
69#1437C2212yescomplemented
70#1436C2214yescomplemented
71#1436C2214yescomplemented
72#1434C2214yescomplemented
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yescomplemented
76C22⋊#95C2212yescomplemented
77C26C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—28no—
85C2×#123—48no—
86C2×#120—24no—
87C2×#120—28no—
88C2×C4×D4—42no—
89C2×C4×D4—42no—
90C2×C4×D4D418yes—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4—42no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—24no—
102C22×C22⋊C4—42no—
103C22×C22⋊C4—42no—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125#398—28no—
126#397—28no—
127#397—44no—
128#397—44no—
129#397—48no—
130#390—28no—
131#389—28no—
132#388—28no—
133#388—28no—
134#386—24no—
135#386—216no—
136#386—28no—
137#385—24no—
138#383—42no—
139#381—28no—
140#380—28no—
141#380—24no—
142#379—24no—
143#379—48no—
144C4×C22⋊C4—24no—
145#376—42no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25—22no—
171C25—24no—
172C25—24no—
173C25—41no—
174C25—41no—
175C25—44no—
176C25—44no—
177C25C2412yes—
178C25C2411yescharacteristic
179C25C2411yescharacteristic
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—41no—
185C22×D4—41no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4C2×D414yes—
189C22×D4#4118yes—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—28no—
197C23×C4—44no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219#123—432no—
220#120—232no—
221C4×D4—416no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—416no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C2×D414yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—416no—
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×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D414yes—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322C24—22no—
323C24—22no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—28no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—48no—
344C24—48no—
345C24—44no—
346C24—416no—
347C24—44no—
348C24—84no—
349C24—84no—
350C24—88no—
351C24—88no—
352C24—88no—
353C24—84no—
354C24—84no—
355C24—84no—
356C24—84no—
357C24C22×D412yes—
358C24C22×D412yes—
359C24C2×#4114yes—
360C24C2×#4114yes—
361C24H(5,𝔽2)14yes—
362C24C2511yescharacteristic
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—216no—
366C2×D4—44no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—84no—
371C2×D4—84no—
372C2×D4—88no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
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—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—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4114yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—416no—
458C4⋊C4—48no—
459C4⋊C4—84no—
460C4⋊C4—88no—
461C22⋊C4—216no—
462C22⋊C4—28no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4—88no—
473C22⋊C4—84no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C42—28no—
480C42—48no—
481C42—84no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—416no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—816no—
535C23—816no—
536C23—832no—
537C23—832no—
538C23—88no—
539C23—88no—
540C23#52514yes—
541C23#53514yes—
542C23C23×D411yescharacteristic
543C23C22×#4112yes—
544C23C2×H(5,𝔽2)11yescharacteristic
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×H(5,𝔽2)12yes—
547C23#58514yes—
548C23#58514yes—
549D4—416no—
550D4—816no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—416no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
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—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—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—88no—
616C2×C4—816no—
617C2×C4—88no—
618C2×C4#53414yes—
619C22—24no—
620C22—24no—
621C22—28no—
622C22—28no—
623C22—48no—
624C22—48no—
625C22—42no—
626C22—42no—
627C22—48no—
628C22—48no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—48no—
634C22—44no—
635C22—48no—
636C22—44no—
637C22—44no—
638C22—832no—
639C22—84no—
640C22—82no—
641C22—84no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22C2×#52511yescharacteristic
646C22C2×#53411yescharacteristic
647C22C2×#53511yescharacteristic
648C22#348512yes—
649C22#349114yes—
650C22#350012yes—
651C22#351912yes—
652C22#351912yes—
653C22#352114yes—
654C22#353312yes—
655C22#353314yes—
656C22#353312yes—
657C22#354012yes—
658C22#355112yes—
659C22#356014yes—
660C4—24no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—44no—
666C4—84no—
667C4—84no—
668C4—82no—
669C4—84no—
670C4—84no—
671C4—82no—
672C2—44no—
673C2—44no—
674C2—84no—
675C2#3507611yescharacteristic
676C2#3601012yes—
677C2#3696511yescharacteristic
678C2#3741911yescharacteristic
679C2#4137112yes—
680C2#4736114yes—
681C2#4945214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
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×#2486C2semidirect
3. #16895C2semidirect
4. #16679C2semidirect
5. #16488C2semidirect
7. #15579C2semidirect
8. #15480C2semidirect
9. #15463C2semidirect
10. #15061C2semidirect
11. #14894C2semidirect
12. #14820C2semidirect
13. #14776C2semidirect
14. #14376C2semidirect
15. #13959C2semidirect
19. C23×C4⋊C4C22semidirect
22. C2×#398C22semidirect
29. C2×#389C22semidirect
36. C2×#383C22semidirect
37. C2×#381C22semidirect
39. C2×#380C22semidirect
43. C2×#376C22semidirect
44. #1445C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1444C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
64. #1439C22semidirect
69. #1437C22semidirect
70. #1436C22semidirect
71. #1436C22semidirect
72. #1434C22semidirect
75. C22⋊#95C22semidirect
76. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#23212
#168952
#166792
#164882
#159482
#155792
#154802
#154632
#150612
#148942
#148202
#147762
#143762
#139592