Finite Group Databasebeta

← #1007685

Subgroups — #1007685

Order 512.

2682 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C212yesmaximal, complemented
2#17202C212yesmaximal, complemented
3#16808C212yesmaximal
4#16478C214yesmaximal, complemented
5#15978C212yesmaximal, complemented
6#15404C214yesmaximal, complemented
7#15216C212yesmaximal, complemented
8#15181C212yesmaximal, complemented
9#15150C211yesmaximal, characteristic, complemented
10#14577C212yesmaximal, complemented
11#14130C211yesmaximal, characteristic, complemented
12#14054C212yesmaximal, complemented
13#13876C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C4×#392C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2562—216no—
21C2×#400C2214yes—
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#394C2214yes—
25C2×#392C2211yescharacteristic
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#386C2214yes—
31C2×#385C2212yes—
32C2×#384C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42#1445C2212yes—
43#1445C2212yes—
44#1445C2214yes—
45#1444C2211yescharacteristic
46#1444C2212yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2212yes—
55#1439C2211yescharacteristic
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1438C2211yescharacteristic
64#1438C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#125—28no—
82C2×#125—28no—
83C2×#125—44no—
84C2×C4×D4—28no—
85C2×C4×D4—28no—
86C2×C4×D4—44no—
87C2×C4×D4—42no—
88C2×C4×D4—44no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2314yes—
120#400—216no—
121#397—28no—
122#397—44no—
123#394—216no—
124#392—24no—
125#389—44no—
126#388—216no—
127#388—216no—
128#388—48no—
129#386—48no—
130#385—44no—
131#384—42no—
132#380—28no—
133#380—28no—
134#380—28no—
135#380—44no—
136#380—44no—
137#380D418yescomplemented
138C4×C4⋊C4—28no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—42no—
141C2×#95C2311yescharacteristic
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—22no—
175C22×D4—42no—
176C22×D4—41no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—41no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4C2×D414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210#125—432no—
211C4×D4—432no—
212C4×D4—416no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C2×D418yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—42no—
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⋊C4C2×D418yes—
286C2×C22⋊C4#4114yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—28no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42#4118yes—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—41no—
311#95—41no—
312#95—41no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—42no—
320#95—44no—
321#95C2×D418yes—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—21no—
327C24—22no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24C22×D412yes—
345C24C2×#4112yes—
346C24H(5,𝔽2)12yes—
347C242−1+414yes—
348C24C2511yescharacteristic
349C2×D4—216no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—44no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—48no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4—84no—
445C22×C4C22×D412yes—
446C22×C4C22×D418yes—
447C22×C4C22×D412yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C42−1+412yes—
457C4⋊C4—216no—
458C4⋊C4—48no—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—88no—
469C22⋊C4—216no—
470C22⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—416no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—216no—
483C42—44no—
484C42—44no—
485C42—88no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23#55014yes—
544C23C23×D411yescharacteristic
545C23C22×#4112yes—
546C23C22×#4111yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×2−1+411yescharacteristic
552C23C2×2−1+412yes—
553C23#58512yes—
554C23#58511yescharacteristic
555D4—432no—
556D4—416no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—416no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—816no—
633C2×C4—816no—
634C2×C4#53714yes—
635C2×C4#54218yes—
636C2×C4#54618yes—
637C2×C4#55914yes—
638C2×C4H(3,𝔽4)14yes—
639C22—24no—
640C22—28no—
641C22—44no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—42no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22C2×#53711yescharacteristic
666C22C2×#54212yes—
667C22C2×#54612yes—
668C22C2×#55011yescharacteristic
669C22#347512yes—
670C22C2×#55911yescharacteristic
671C22#348912yes—
672C22#349212yes—
673C22#349212yes—
674C22#349611yescharacteristic
675C22#350511yescharacteristic
676C22#351111yescharacteristic
677C22#351212yes—
678C22#352012yes—
679C22C2×H(3,𝔽4)11yescharacteristic
680C22#353711yescharacteristic
681C22#353811yescharacteristic
682C22#354112yes—
683C22#354212yes—
684C22#354612yes—
685C22#355111yescharacteristic
686C22#355911yescharacteristic
687C22#356312yes—
688C4—44no—
689C4—48no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—42no—
695C4—84no—
696C4—84no—
697C4—82no—
698C4—81no—
699C4—84no—
700C4#271218yes—
701C2—44no—
702C2—84no—
703C2#3974211yescharacteristic
704C2#3990212yes—
705C2#4245612yes—
706C2#4298011yescharacteristic
707C2#4510412yes—
708C2#4537511yescharacteristic
709C2#4566711yescharacteristic
710C2C2×#271211yescharacteristic
711C2#5296711yescharacteristic
712C2#5902212yes—
713C2#5986511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2562C2semidirect
2. #17202C2semidirect
4. #16478C2semidirect
5. #15978C2semidirect
6. #15404C2semidirect
7. #15216C2semidirect
8. #15181C2semidirect
9. #15150C2semidirect
10. #14577C2semidirect
11. #14130C2semidirect
12. #14054C2semidirect
13. #13876C2semidirect
14. #13872C2semidirect
137. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
#172022
#168082
#164782
#159782
#154042
#152162
#151812
#151502
#145772
#141302
#140542
#138762
#138722
C4×#3922