Finite Group Databasebeta

← #4862583

Subgroups — #4862583

Order 512.

2670 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic, complemented
2C2×#2635C211yesmaximal, characteristic
3#17123C214yesmaximal, complemented
4#16824C212yesmaximal, complemented
5#16602C212yesmaximal, complemented
6#16570C212yesmaximal, complemented
7#16514C212yesmaximal, complemented
8#16364C212yesmaximal, complemented
9#16338C212yesmaximal, complemented
10#16265C212yesmaximal, complemented
11#16253C212yesmaximal, complemented
12#15450C214yesmaximal, complemented
13#15078C212yesmaximal, complemented
14#14759C212yesmaximal, complemented
15#13960C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2635—28no—
22#2542—28no—
23C2×#400C2211yescharacteristic
24C2×#399C2211yescharacteristic
25C2×#399C2212yes—
26C2×#399C2212yes—
27C2×#397C2214yes—
28C2×#396C2214yes—
29C2×#394C2211yescharacteristic
30C2×#394C2212yes—
31C2×#394C2214yes—
32C2×#393C2212yes—
33C2×#393C2212yes—
34C2×#392C2212yes—
35C2×#392C2211yescharacteristic
36C2×#390C2212yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#384C2212yes—
43C2×#381C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2212yes—
48#1446C2212yes—
49#1444C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2212yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1438C2214yes—
70#1438C2214yes—
71#1438C2214yes—
72#1434C2214yes—
73#1434C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C23×D4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8—42no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8—48no—
92C2×#121—24no—
93C2×#121—42no—
94C2×#121—42no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#400—24no—
127#399—24no—
128#399—28no—
129#399—28no—
130#397—48no—
131#396—48no—
132#394—24no—
133#394—28no—
134#394—48no—
135#393—44no—
136#393—44no—
137#392—28no—
138#392—24no—
139#390—28no—
140#388—28no—
141#388—48no—
142#386—28no—
143#386—44no—
144#386—48no—
145#384—28no—
146#381—24no—
147#380—42no—
148C4×C4⋊C4—24no—
149C2×#95—42no—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4C2×D414yes—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C4⋊Q8—416no—
228C22⋊Q8—432no—
229C22⋊Q8—432no—
230#121—416no—
231C4×D4—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D414yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
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—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D414yes—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42C2×D414yes—
309#95—24no—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332C24—24no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—24no—
338C24—24no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24C22×D412yes—
358C24C22×D412yes—
359C242−1+412yes—
360C242−1+412yes—
361C24C2511yescharacteristic
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—88no—
366C2×Q8—88no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—48no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—88no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—88no—
387C2×D4—88no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
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—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—84no—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4H(5,𝔽2)18yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C42−1+412yes—
467C4⋊C4—216no—
468C4⋊C4—44no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—84no—
475C4⋊C4—88no—
476C4⋊C4—84no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C22⋊C4—88no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C42—28no—
494C42—48no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—28no—
504C23—28no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—42no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23#53614yes—
552C23C23×D411yescharacteristic
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×H(5,𝔽2)11yescharacteristic
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)14yes—
557C23C2×2−1+411yescharacteristic
558C23C2×2−1+411yescharacteristic
559C23C2×2−1+411yescharacteristic
560C23#58514yes—
561Q8—816no—
562D4—416no—
563D4—88no—
564D4—816no—
565D4—816no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
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—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—88no—
631C2×C4—88no—
632C2×C4—816no—
633C2×C4#53414yes—
634C2×C4#53714yes—
635C22—28no—
636C22—24no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—42no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22C2×#53411yescharacteristic
665C22C2×#53611yescharacteristic
666C22C2×#53711yescharacteristic
667C22#348412yes—
668C22#350012yes—
669C22#353611yescharacteristic
670C22#353611yescharacteristic
671C22#353811yescharacteristic
672C22#353811yescharacteristic
673C22#354012yes—
674C22#354114yes—
675C22#354214yes—
676C22#354312yes—
677C22#354312yes—
678C22#355112yes—
679C22#355914yes—
680C22#356114yes—
681C4—24no—
682C4—24no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—48no—
687C4—82no—
688C4—82no—
689C4—84no—
690C4—84no—
691C4—84no—
692C4—82no—
693C2—44no—
694C2—82no—
695C2—162no—
696C2#3723211yescharacteristic
697C2#3741911yescharacteristic
698C2#3744211yescharacteristic
699C2#4049112yes—
700C2#4242412yes—
701C2#5349412yes—
702C2#5485312yes—
703C2#5895812yes—
704C2#5896312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2635C2
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×#2542C2semidirect
3. #17123C2semidirect
4. #16824C2semidirect
5. #16602C2semidirect
6. #16570C2semidirect
7. #16514C2semidirect
8. #16364C2semidirect
9. #16338C2semidirect
10. #16265C2semidirect
11. #16253C2semidirect
12. #15450C2semidirect
13. #15078C2semidirect
14. #14759C2semidirect
15. #13960C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#26352
#171232
#168242
#166022
#165702
#165142
#163642
#163382
#162652
#162532
#154502
#150782
#147592
#139602