Finite Group Databasebeta

← #647297

Subgroups — #647297

Order 512.

2934 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#17020C214yesmaximal, complemented
4#17008C211yesmaximal, characteristic, complemented
5#16990C211yesmaximal, characteristic, complemented
6#16888C211yesmaximal, characteristic, complemented
7#16811C211yesmaximal, characteristic, complemented
8#15016C212yesmaximal, complemented
9#14916C212yesmaximal, complemented
10#14836C212yesmaximal, complemented
11#14766C212yesmaximal, complemented
12#14747C212yesmaximal
13#14735C212yesmaximal, complemented
14#14308C212yesmaximal, complemented
15#14300C212yesmaximal, complemented
16#13975C214yesmaximal, complemented
17C2×#1434C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2851—28no—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#397C2212yes—
27C2×#394C2212yes—
28C4×#121—28no—
29C2×#390C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#388C2212yes—
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#386C2214yes—
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2214yes—
42C2×#382C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C22×#95C2212yes—
50C22×#95C2212yes—
51C22×#95C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1445C2214yes—
55#1445C2212yes—
56#1444C2212yes—
57#1444C2214yes—
58#1444C2212yes—
59#1443C2214yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2212yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1438C2212yes—
76#1436C2214yes—
77#1436C2214yes—
78#1434C2212yes—
79#1434C2211yescharacteristic
80#1434C2211yescharacteristic
81#1434C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2314yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×#125—24no—
93C2×#125—44no—
94C2×#121—28no—
95C2×#121—24no—
96C2×C4×D4—28no—
97C2×C4×D4—24no—
98C2×C4×D4—44no—
99C2×C4×D4—48no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C42—44no—
126C22×C42D418yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#399—44no—
133#398—28no—
134#397—44no—
135#394—28no—
136#390—24no—
137#389—24no—
138#389—24no—
139#389—24no—
140#388—28no—
141#387—28no—
142#387—28no—
143#386—216no—
144#386—28no—
145#386—48no—
146#385—28no—
147#385—28no—
148#385—48no—
149#382—24no—
150#380—24no—
151#380—24no—
152#380—24no—
153#380—48no—
154C4×C4⋊C4—24no—
155C4×C22⋊C4—28no—
156C2×#95—28no—
157C2×#95—44no—
158C2×#95—44no—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—28no—
193C22×D4—28no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—42no—
197C22×D4#4118yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238#125—416no—
239#121—232no—
240C4×D4—416no—
241C4×D4—432no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
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⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—416no—
308C2×C42—44no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—42no—
333#95—44no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95#4118yes—
343C24—22no—
344C24—22no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24C22×D412yes—
359C24C2×#4114yes—
360C24C2×#4114yes—
361C24H(5,𝔽2)14yes—
362C242−1+414yes—
363C24C2511yescharacteristic
364C2×D4—28no—
365C2×D4—216no—
366C2×D4—232no—
367C2×D4—416no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—416no—
372C2×D4—44no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4118yes—
471C22×C4C2×#4114yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+412yes—
475C4⋊C4—28no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C22⋊C4—216no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C42—28no—
501C42—28no—
502C42—44no—
503C42—44no—
504C42—48no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—28no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23#52514yes—
542C23#53514yes—
543C23#53614yes—
544C23#539116yes—
545C23C23×D411yescharacteristic
546C23C22×#4112yes—
547C23C22×#4112yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×2−1+412yes—
552C23C2×2−1+411yescharacteristic
553C23#58512yes—
554D4—416no—
555D4—432no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—416no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
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—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—816no—
634C2×C4—832no—
635C2×C4#53214yes—
636C2×C4#53414yes—
637C2×C4#53714yes—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—216no—
642C22—216no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—44no—
647C22—44no—
648C22—48no—
649C22—48no—
650C22—44no—
651C22—82no—
652C22—82no—
653C22—84no—
654C22—84no—
655C22—82no—
656C22—82no—
657C22#263318yes—
658C22C2×#52511yescharacteristic
659C22C2×#53211yescharacteristic
660C22C2×#53411yescharacteristic
661C22C2×#53511yescharacteristic
662C22C2×#53611yescharacteristic
663C22C2×#53711yescharacteristic
664C22#344911yescharacteristic
665C22C2×#53914yes—
666C22#348114yes—
667C22#349514yes—
668C22#351412yes—
669C22#351512yes—
670C22#351912yes—
671C22#352212yes—
672C22#353414yes—
673C22#353612yes—
674C22#353812yes—
675C4—24no—
676C4—24no—
677C4—24no—
678C4—44no—
679C4—42no—
680C4—42no—
681C4—48no—
682C4—42no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—84no—
687C4—82no—
688C4—82no—
689C4—82no—
690C2—28no—
691C2—84no—
692C2C2×#263311yescharacteristic
693C2#3495711yescharacteristic
694C2#3513011yescharacteristic
695C2#3523811yescharacteristic
696C2#3532611yescharacteristic
697C2#3744211yescharacteristic
698C2#3744611yescharacteristic
699C2#3850414yes—
700C2#3853614yes—
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
1. C2×#2851C2semidirect
3. #17020C2semidirect
4. #17008C2semidirect
5. #16990C2semidirect
6. #16888C2semidirect
7. #16811C2semidirect
8. #15016C2semidirect
9. #14916C2semidirect
10. #14836C2semidirect
11. #14766C2semidirect
13. #14735C2semidirect
14. #14308C2semidirect
15. #14300C2semidirect
16. #13975C2semidirect
17. C2×#1434C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×C4×#1212
#170202
#170082
#169902
#168882
#168112
#150162
#149162
#148362
#147662
#147472
#147352
#143082
#143002
#139752
C2×#14342