Finite Group Databasebeta

← #646734

Subgroups — #646734

Order 512.

2682 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#16964C211yesmaximal, characteristic, complemented
4#16963C211yesmaximal, characteristic, complemented
5#16692C211yesmaximal, characteristic, complemented
6#16691C211yesmaximal, characteristic, complemented
7#16397C214yesmaximal, complemented
8#15103C212yesmaximal, complemented
9#15081C212yesmaximal, complemented
10#15063C212yesmaximal, complemented
11#14996C212yesmaximal
12#14825C212yesmaximal, complemented
13#14824C212yesmaximal, complemented
14#14682C211yesmaximal, characteristic, complemented
15#14345C212yesmaximal, complemented
16#14296C212yesmaximal, complemented
17#14066C214yesmaximal, complemented
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C42C2211yescharacteristic
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#394C2214yes—
27C4×#121—28no—
28C2×#390C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#388C2214yes—
31C2×#388C2212yes—
32C2×#386C2212yes—
33#2283—28no—
34C2×#383C2212yes—
35C2×#382C2212yes—
36C2×#382C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#381C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1445C2214yes—
52#1444C2212yes—
53#1444C2212yes—
54#1443C2214yes—
55#1442C2212yes—
56#1442C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1438C2212yes—
70#1438C2212yes—
71#1438C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77#1434C2211yescharacteristic
78#1434C2211yescharacteristic
79#1434C2212yes—
80#1434C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#121—24no—
92C2×#121—28no—
93C2×C4×D4—28no—
94C2×C4×D4—24no—
95C2×C4×D4—28no—
96C2×C4×D4—44no—
97C2×C4×D4—44no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42—28no—
122C22×C42D418yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#400—44no—
131#399—44no—
132#398—44no—
133#397—44no—
134#394—48no—
135#390—24no—
136#389—24no—
137#388—216no—
138#388—28no—
139#386—28no—
140#383—28no—
141#382—24no—
142#382—24no—
143#382—28no—
144#381—28no—
145#380—24no—
146#380—24no—
147#380—24no—
148#380—44no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—28no—
154#376—28no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—42no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230#121—232no—
231C4×D4—432no—
232C4×D4—416no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
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⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4#4118yes—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—232no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42#4118yes—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24C22×D412yes—
355C24C2×#4114yes—
356C24H(5,𝔽2)12yes—
357C242−1+412yes—
358C24C2511yescharacteristic
359C2×D4—28no—
360C2×D4—216no—
361C2×D4—416no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—44no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
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—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4114yes—
464C22×C4H(5,𝔽2)14yes—
465C4⋊C4—28no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—88no—
474C4⋊C4—88no—
475C4⋊C4—88no—
476C22⋊C4—216no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C42—28no—
489C42—28no—
490C42—28no—
491C42—48no—
492C42—48no—
493C42—48no—
494C42#123116yes—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—24no—
511C23—24no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23#53714yes—
535C23C23×D411yescharacteristic
536C23C22×#4112yes—
537C23C22×#4112yes—
538C23C2×H(5,𝔽2)12yes—
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×2−1+411yescharacteristic
541C23#58512yes—
542C23#58512yes—
543C23#58512yes—
544D4—416no—
545D4—432no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—416no—
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—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
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—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—816no—
622C2×C4C2×#12314yes—
623C2×C4C2×#12314yes—
624C2×C4C2×#12314yes—
625C2×C4#52514yes—
626C2×C4#53514yes—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—48no—
631C22—48no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—48no—
637C22—48no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—84no—
642C22—84no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22C22×#12311yescharacteristic
648C22C2×#52511yescharacteristic
649C22C2×#53511yescharacteristic
650C22C2×#53711yescharacteristic
651C22#344911yescharacteristic
652C22#344911yescharacteristic
653C22#344911yescharacteristic
654C22#345514yes—
655C22#347514yes—
656C22#349414yes—
657C22#349412yes—
658C22#349512yes—
659C22#351912yes—
660C22#352012yes—
661C22#354512yes—
662C22#355312yes—
663C22#356214yes—
664C4—24no—
665C4—24no—
666C4—24no—
667C4—24no—
668C4—24no—
669C4—44no—
670C4—42no—
671C4—84no—
672C4—84no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C2—44no—
680C2—84no—
681C2#3463611yescharacteristic
682C2#3480911yescharacteristic
683C2#3523611yescharacteristic
684C2#3526211yescharacteristic
685C2#3526311yescharacteristic
686C2#3754411yescharacteristic
687C2#3760811yescharacteristic
688C2#3792414yes—
689C2#3792514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #16964C2semidirect
4. #16963C2semidirect
5. #16692C2semidirect
6. #16691C2semidirect
7. #16397C2semidirect
8. #15103C2semidirect
9. #15081C2semidirect
10. #15063C2semidirect
12. #14825C2semidirect
13. #14824C2semidirect
14. #14682C2semidirect
15. #14345C2semidirect
16. #14296C2semidirect
17. #14066C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#22832
#169642
#169632
#166922
#166912
#163972
#151032
#150812
#150632
#149962
#148252
#148242
#146822
#143452
#142962
#140662