Finite Group Databasebeta

← #6606573

Subgroups — #6606573

Order 512.

1895 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3340C212yesmaximal
2#36188C211yesmaximal, characteristic, complemented
3C4×#555C211yesmaximal, characteristic
4#33195C211yesmaximal, characteristic, complemented
5#33114C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28284C211yesmaximal, characteristic, complemented
8#27646C212yesmaximal, complemented
9#25454C212yesmaximal, complemented
10#25205C214yesmaximal, complemented
11#24751C212yesmaximal, complemented
12#24681C212yesmaximal, complemented
13#24586C214yesmaximal, complemented
14D4×C4⋊C8C211yesmaximal, characteristic, complemented
15#22490C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17#21169C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#555C2211yescharacteristic
20#3340—216no—
21C2×#498C2212yes—
22C2×#485C2214yes—
23C2×#482C2214yes—
24C2×#477C2212yes—
25C2×#474C2212yes—
26C2×#465C2214yes—
27C2×#461C2211yescharacteristic
28C2×C4×SD16C2212yes—
29C2×C8×D4C2212yes—
30C2×#427C2214yes—
31C22×C4⋊C8C2212yes—
32#2868C2211yescharacteristic
33#2857C2212yes—
34#2568C2212yes—
35#2547—21no—
36#2547—21no—
37#2499C2214yes—
38#2496—22no—
39#2496—22no—
40#2372—22no—
41#2369C2211yescharacteristic
42#2365—21no—
43#2365—21no—
44#2363—21no—
45#2363C2212yes—
46#2354C2211yescharacteristic
47#2354C2211yescharacteristic
48#2346—21no—
49#2346C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#124—22no—
53C4×#121—21no—
54C4×#121C2212yes—
55#2285C2212yes—
56C42×D4C2211yescharacteristic
57C42×D4C2211yescharacteristic
58C4×#117—21no—
59C4×#117C2212yes—
60#2085C2214yes—
61#2084C2214yes—
62#2073C2214yes—
63#2068C2214yes—
64#2066C2214yes—
65#2055C2214yes—
66#1990C2212yes—
67#1988C2212yes—
68#1981C2211yescharacteristic
69#1981C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2211yescharacteristic
75#1947C2212yes—
76#1947C2212yes—
77#1947C2212yes—
78#1939C2212yes—
79#1928C2214yes—
80#1927C2212yes—
81#1927C2214yes—
82#1892C2214yes—
83#1877C2212yes—
84#1873C2214yes—
85#1850C2214yes—
86#1842C2211yescharacteristic
87#1825C2214yes—
88#1803C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#1735C2214yes—
96C22×SD16C2312yes—
97C23×C8C2314yes—
98#556—28no—
99#556—28no—
100#555—24no—
101#555D418yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—24no—
108C2×#124—22no—
109C2×#124—22no—
110C2×C22⋊Q8C2314yes—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117—21no—
125C2×#117C2314yes—
126C2×#117C2312yes—
127C2×#117C2312yes—
128C22×C42—21no—
129C22×C42C2312yes—
130#498—28no—
131#485—216no—
132#482—216no—
133#477—28no—
134#474—28no—
135#465—216no—
136#461—24no—
137C4×SD16—28no—
138C8×D4—28no—
139#427—216no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×C4⋊C8—28no—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2311yescharacteristic
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161#400—24no—
162#397—24no—
163#397—24no—
164#391—22no—
165#391—22no—
166#391—21no—
167#391—22no—
168#391C2312yes—
169#391C2314yes—
170#390—21no—
171#390—22no—
172#390C2312yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—24no—
188#387—24no—
189#386—22no—
190#386—22no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382—22no—
198#382C2314yes—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—21no—
204#380C2312yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4D412yes—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4C2312yes—
231#376—22no—
232#376—22no—
233C43—22no—
234C43C2311yescharacteristic
235#340C2314yes—
236#340C2314yes—
237#340C2312yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×D4—22no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—42no—
259C22×C8—28no—
260C22×C8—28no—
261C22×C8—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—24no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8C2×D414yes—
273#126—416no—
274#124—216no—
275#124—48no—
276C22⋊Q8—28no—
277C22⋊Q8—28no—
278#121—28no—
279#121—24no—
280#121C2×D418yes—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—48no—
284C4×Q8—82no—
285C4×D4—22no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—42no—
291C4×D4—41no—
292C4×D4—84no—
293C4×D4C2×D414yes—
294C4×D4C2×D414yes—
295C4×D4#4114yes—
296#117—24no—
297#117—28no—
298#117—28no—
299#117—28no—
300#117—24no—
301#117—42no—
302#117—44no—
303#117C2×D418yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—28no—
368#107—28no—
369#106—24no—
370#106—28no—
371#106—28no—
372C4⋊C8—28no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375C4⋊C8—22no—
376C4⋊C8—42no—
377C4⋊C8#4114yes—
378#103—24no—
379#103—24no—
380#103—28no—
381#103—24no—
382#103—44no—
383#102—24no—
384#102—28no—
385C22⋊C8—28no—
386C4×C8—24no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24—21no—
406C24C22×D412yes—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—81no—
418C2×Q8—81no—
419C2×D4—28no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4#12114yes—
430C2×D4C22×D412yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D414yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4114yes—
469C22×C4C2511yescharacteristic
470SD16—48no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—416no—
480C2×C8C2×#4114yes—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—82no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4#12114yes—
511C4⋊C4C22×D414yes—
512C4⋊C4C22×D412yes—
513C4⋊C42−1+412yes—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—42no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4#12118yes—
528C42—28no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—41no—
538C42—44no—
539C42—44no—
540C42—48no—
541C42—44no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—82no—
547C42#13514yes—
548C42C22×D412yes—
549C42C2×#4112yes—
550C42C2×#4112yes—
551C422−1+412yes—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C2×#12114yes—
566C23C23×D411yescharacteristic
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—84no—
572Q8—88no—
573D4—22no—
574D4—22no—
575D4—28no—
576D4—41no—
577D4—41no—
578D4—88no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4C2×#12114yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4#54712yes—
633C2×C4#54712yes—
634C2×C4C2×#13512yes—
635C2×C4C2×#13512yes—
636C2×C4C2×#13512yes—
637C2×C4#57512yes—
638C2×C4#57712yes—
639C2×C4C23×D411yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C2×C4#58511yescharacteristic
645C8—24no—
646C8—44no—
647C8—42no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22C22×#12111yescharacteristic
663C22C2×#54711yescharacteristic
664C22#348911yescharacteristic
665C22C22×#13511yescharacteristic
666C22C2×#57511yescharacteristic
667C22C2×#57711yescharacteristic
668C22#358611yescharacteristic
669C4—22no—
670C4—22no—
671C4—22no—
672C4—24no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—41no—
678C4—42no—
679C4—82no—
680C4—81no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4#303514yes—
685C4#334014yescomplemented
686C4D4×#4111yescharacteristic
687C4#348211yescharacteristic
688C4#349911yescharacteristic
689C4#351511yescharacteristic
690C2—44no—
691C2—82no—
692C2#3625911yescharacteristic
693C2C2×#303511yescharacteristic
694C2#6042911yescharacteristic
695C2C2×#334011yescharacteristic
696C2#6125711yescharacteristic
697C2#6209711yescharacteristic
698C2#6213511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36188C2semidirect
4. #33195C2semidirect
5. #33114C2semidirect
6. #28687C2semidirect
7. #28284C2semidirect
8. #27646C2semidirect
9. #25454C2semidirect
10. #25205C2semidirect
11. #24751C2semidirect
12. #24681C2semidirect
13. #24586C2semidirect
14. D4×C4⋊C8C2semidirect
15. #22490C2semidirect
16. C4×#461C2semidirect
17. #21169C2semidirect
685. C4#3340semidirect

Maximal subgroups

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

subgroupindex
C2×#33402
#361882
C4×#5552
#331952
#331142
#286872
#282842
#276462
#254542
#252052
#247512
#246812
#245862
D4×C4⋊C82
#224902
C4×#4612
#211692