Finite Group Databasebeta

← #6497518

Subgroups — #6497518

Order 512.

2060 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C212yesmaximal
2#36645C211yesmaximal, characteristic, complemented
3C4×#532C211yesmaximal, characteristic
4#33199C214yesmaximal, complemented
5#29870C214yesmaximal, complemented
6#28718C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#28716C211yesmaximal, characteristic, complemented
9#28689C211yesmaximal, characteristic, complemented
10#28688C212yesmaximal, complemented
11#28687C211yesmaximal, characteristic, complemented
12#27650C214yesmaximal, complemented
13#27646C214yesmaximal, complemented
14#21835C212yesmaximal, complemented
15#20808C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#532C2211yescharacteristic
18C2×C4×#41—21no—
19#3228—216no—
20C2×#500C2212yes—
21C2×#498C2212yes—
22C2×#486C2212yes—
23C2×#486C2212yes—
24C2×#482C2218yes—
25C2×#478C2218yes—
26C2×#431C2212yes—
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2862C2212yes—
30#2859C2212yes—
31#2859C2212yes—
32#2857—21no—
33#2857—21no—
34#2857—21no—
35#2857—21no—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8—21no—
38C4×C4⋊Q8—21no—
39C4×C4⋊Q8—21no—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#124—21no—
45C4×#124—21no—
46C4×#121—21no—
47C4×#121—21no—
48C4×#121C2212yes—
49C4×#121C2212yes—
50#2297—21no—
51#2293—21no—
52#2293C2212yes—
53#2285C2211yescharacteristic
54#2285C2212yes—
55#2285C2212yes—
56C42×D4—21no—
57C42×D4C2212yes—
58C2×C43C2211yescharacteristic
59#2085C2218yes—
60#2084C2214yes—
61#2083C2214yes—
62#2073C2218yes—
63#2071C2214yes—
64#2071C2214yes—
65#1990C2212yes—
66#1990C2212yes—
67#1990C2212yes—
68#1990C2212yes—
69#1989C2218yes—
70#1949C2218yes—
71#1948C2214yes—
72#1947C2214yes—
73#1842C2212yes—
74#1831C2212yes—
75#1825C2218yes—
76#1804C2218yes—
77#1802C2214yes—
78#1802C2214yes—
79#1797C2212yes—
80#1797C2212yes—
81C4×C4⋊C8C2211yescharacteristic
82C4×C4⋊C8C2212yes—
83C4×C4⋊C8C2211yescharacteristic
84C22×#41—21no—
85#533—28no—
86#533—28no—
87#532—24no—
88#532D418yes—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#125C2314yes—
98C2×#124—21no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×#121—21no—
102C2×#121C2312yes—
103C4×#41—48no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117—22no—
120C2×#117C2314yes—
121C22×C42—22no—
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#500—28no—
130#498—28no—
131#486—28no—
132#486—28no—
133#482—232no—
134#478—232no—
135#431—28no—
136C2×#107C2318yes—
137C2×#106C2318yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#102C2318yes—
143C2×#102C2318yes—
144C2×C22⋊C8C2314yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—21no—
152#391—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#386—22no—
176#386—22no—
177#384—21no—
178#384—21no—
179#384—21no—
180#384—21no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#384C2312yes—
186#384C2312yes—
187#382—21no—
188#382C2314yes—
189#382C2314yes—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—21no—
195#380C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C43D411yescharacteristic
213C43D411yescharacteristic
214C43D411yescharacteristic
215C43D411yescharacteristic
216C43C2311yescharacteristic
217C43C2311yescharacteristic
218C43C2311yescharacteristic
219C43C2311yescharacteristic
220#340C2318yes—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2312yes—
225#336C2312yes—
226#336C2314yes—
227C2×#41—28no—
228C2×#41—44no—
229C2×#41—44no—
230C22×Q8—21no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C2412yes—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8C2×D414yes—
259#127—22no—
260#127C2×D414yes—
261#125—28no—
262#125—28no—
263#124—28no—
264C22⋊Q8—48no—
265C22⋊Q8—48no—
266#121—28no—
267#121—24no—
268#121C2×D418yes—
269C4×Q8—44no—
270C4×Q8—48no—
271C4×Q8—48no—
272C4×Q8—44no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—48no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4C2×D418yes—
284#117—28no—
285#117—28no—
286#117—48no—
287#117—44no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—48no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D414yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42#4114yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—216no—
364#106—216no—
365C4⋊C8—28no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—28no—
369#102—216no—
370#102—216no—
371C22⋊C8—28no—
372C4×C8—24no—
373C4×C8—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384C24—22no—
385C24C22×D411yescharacteristic
386#41—432no—
387C2×Q8—24no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4C22×D414yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4#12114yes—
442C22×C4#12114yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4C2511yescharacteristic
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4C22×D414yes—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—44no—
523C42—44no—
524C42—48no—
525C42—44no—
526C42#12114yes—
527C42#12114yes—
528C42#13514yes—
529C42AGL(1,ℤ/8)14yes—
530C42C22×D412yes—
531C42C22×D412yes—
532C42C2×#4112yes—
533C42C2×#4112yes—
534C422−1+414yes—
535C422−1+414yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23C2×#12112yes—
550C23C23×D411yescharacteristic
551Q8—48no—
552Q8—88no—
553Q8—88no—
554D4—28no—
555D4—416no—
556D4—44no—
557D4—48no—
558D4—48no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—88no—
606C2×C4—88no—
607C2×C4C2×#12112yes—
608C2×C4C2×#12112yes—
609C2×C4C2×#12112yes—
610C2×C4C2×#12112yes—
611C2×C4C2×#12112yes—
612C2×C4#53612yes—
613C2×C4#53612yes—
614C2×C4#53612yes—
615C2×C4#53612yes—
616C2×C4C2×#13512yes—
617C2×C4C2×#13512yes—
618C2×C4C2×#13512yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4#57512yes—
623C2×C4#57512yes—
624C2×C4C23×D411yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×2−1+412yes—
627C2×C4C2×2−1+412yes—
628C8—44no—
629C8—44no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22C22×#12111yescharacteristic
645C22C2×#53611yescharacteristic
646C22C2×#53611yescharacteristic
647C22C22×#13511yescharacteristic
648C22C22×AGL(1,ℤ/8)11yescharacteristic
649C22C2×#57511yescharacteristic
650C22C2×#57511yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—41no—
660C4—41no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4—84no—
666C4#303514yes—
667C4#303614yes—
668C4#322814yescomplemented
669C4#349912yes—
670C4#350912yes—
671C2—42no—
672C2—84no—
673C2#3531611yescharacteristic
674C2C2×#303511yescharacteristic
675C2C2×#303611yescharacteristic
676C2C2×#322811yescharacteristic
677C2#6084611yescharacteristic
678C2#6086211yescharacteristic
679C2#6086411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36645C2semidirect
4. #33199C2semidirect
5. #29870C2semidirect
6. #28718C2semidirect
7. #28716C2semidirect
8. #28716C2semidirect
9. #28689C2semidirect
10. #28688C2semidirect
11. #28687C2semidirect
12. #27650C2semidirect
13. #27646C2semidirect
14. #21835C2semidirect
15. #20808C2semidirect
668. C4#3228semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#366452
C4×#5322
#331992
#298702
#287182
#287162
#286892
#286882
#286872
#276502
#276462
#218352
#208082