Finite Group Databasebeta

← #6567457

Subgroups — #6567457

Order 512.

1941 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3393C212yesmaximal, complemented
2#36594C211yesmaximal, characteristic, complemented
3#34322C211yesmaximal, characteristic, complemented
4#29801C212yesmaximal, complemented
5#29798C211yesmaximal, characteristic, complemented
6#28442C212yesmaximal
7#28440C211yesmaximal, characteristic, complemented
8#28164C212yesmaximal, complemented
9#28157C211yesmaximal, characteristic, complemented
10#25374C214yesmaximal, complemented
11#25365C212yesmaximal, complemented
12#24736C214yesmaximal, complemented
13#24727C212yesmaximal, complemented
14#23679C212yesmaximal, complemented
15#23676C211yesmaximal, characteristic
16#22734C211yesmaximal, characteristic, complemented
17#22541C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#556C2211yescharacteristic
20#3393—216no—
21C2×#495C2214yes—
22C2×#494C2212yes—
23C2×#464C2214yes—
24C2×#464C2214yes—
25C2×#464C2214yes—
26C2×#462C2214yes—
27C2×#463C2214yes—
28C2×#461C2212yes—
29C2×#461C2212yes—
30C2×C8×Q8C2212yes—
31C2×C4×Q16C2214yes—
32C2×C4×D8C2212yes—
33#2868C2211yescharacteristic
34#2858C2212yes—
35#2547C2212yes—
36#2545C2214yes—
37#2538—21no—
38#2538—21no—
39#2512—22no—
40#2512—22no—
41#2373—22no—
42#2367C2212yes—
43#2365C2211yescharacteristic
44#2360—21no—
45#2355C2211yescharacteristic
46#2354—21no—
47#2354—21no—
48#2351C2212yes—
49D4×C4⋊C4C2211yescharacteristic
50#2346—21no—
51#2334—22no—
52#2323—21no—
53C4×C4⋊Q8C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C4×#124C2212yes—
57#2295C2212yes—
58#2293C2211yescharacteristic
59#2274—21no—
60C42×Q8C2211yescharacteristic
61#2083C2211yescharacteristic
62#2072C2212yes—
63#2071C2211yescharacteristic
64#2068C2214yes—
65#2067C2212yes—
66#2056C2214yes—
67#2054C2212yes—
68#2043C2212yes—
69#2043C2212yes—
70#2042C2214yes—
71#2041C2211yescharacteristic
72#2016C2212yes—
73#2016C2212yes—
74#2015C2214yes—
75#2014C2212yes—
76#2013C2212yes—
77#2013C2212yes—
78#2011C2211yescharacteristic
79#1983C2214yes—
80#1981C2212yes—
81#1958C2214yes—
82#1956C2212yes—
83#1939C2212yes—
84#1939C2214yes—
85#1880C2214yes—
86#1878C2214yes—
87#1877C2214yes—
88#1831C2212yes—
89#1831C2211yescharacteristic
90#1805C2214yes—
91#1802C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2214yes—
95C4×#102C2212yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100C22×D8C2312yes—
101#556—24no—
102#556—28no—
103#556—28no—
104#556D418yescomplemented
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#127C2311yescharacteristic
110C2×#126—22no—
111C2×#126—24no—
112C2×#124C2314yes—
113C2×#124C2312yes—
114C2×#124C2314yes—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—22no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C2×#117—22no—
131C2×#117—21no—
132C22×C4⋊C4—21no—
133#495—216no—
134#494—28no—
135#464—216no—
136#464—216no—
137#464—48no—
138#463—48no—
139#462—48no—
140#461—28no—
141#461—28no—
142C8×Q8—28no—
143C4×Q16—216no—
144C4×D8—28no—
145C2×#107C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2314yes—
163#397—24no—
164#396—24no—
165#396—24no—
166#391—21no—
167#391C2314yes—
168#391C2312yes—
169#391C2312yes—
170#390—21no—
171#390—21no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390C2312yes—
177#390C2311yescharacteristic
178#390C2311yescharacteristic
179#389—21no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389C2311yescharacteristic
187#389C2312yes—
188#389C2311yescharacteristic
189#388—22no—
190#388—22no—
191#388—21no—
192#388—22no—
193#388—21no—
194#388—22no—
195#388—24no—
196#388—24no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#385—22no—
202#385—22no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#382—22no—
210#382C2312yes—
211#380—21no—
212#380—22no—
213#380—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4D412yes—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4D412yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234#376—22no—
235C43C2311yescharacteristic
236C43C2312yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2314yes—
244#336C2312yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2414yes—
247C22×Q8C2412yes—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2412yes—
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—22no—
255C2×Q16—28no—
256C2×Q16—28no—
257C2×Q16—28no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×SD16—28no—
262C2×SD16—28no—
263C2×SD16—44no—
264C2×SD16—44no—
265C2×SD16—44no—
266C2×SD16—44no—
267C2×D8—24no—
268C2×D8—24no—
269C2×D8—42no—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—22no—
275C4⋊Q8—24no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8C2×D414yes—
279#127—42no—
280#126—216no—
281#126—416no—
282#124—28no—
283#124—24no—
284#124—28no—
285#124—44no—
286#124—44no—
287#124C2×D418yes—
288C22⋊Q8—28no—
289C22⋊Q8—48no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—42no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×Q8C2×D418yes—
298C4×Q8C2×D414yes—
299C4×Q8#4114yes—
300C4×D4—24no—
301C4×D4—22no—
302C4×D4—24no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4C2×D414yes—
307#117—28no—
308#117—48no—
309#117—44no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—21no—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2412yes—
375#107—24no—
376C4⋊C8—24no—
377C4⋊C8—28no—
378C4⋊C8—24no—
379C4⋊C8—28no—
380#103—28no—
381#103—28no—
382#103—28no—
383#103—44no—
384#103—44no—
385#103—44no—
386#102—24no—
387#102—24no—
388#102—24no—
389#102—42no—
390#102—44no—
391C4×C8—24no—
392C4×C8—28no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—21no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—22no—
405#95—21no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—21no—
410C24—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8#12114yes—
423C2×Q8C22×D414yes—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C22×D412yes—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2511yescharacteristic
467Q16—416no—
468Q16—88no—
469SD16—88no—
470SD16—88no—
471D8—48no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8H(5,𝔽2)14yes—
481C4⋊C4—24no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—44no—
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—42no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4#12114yes—
511C4⋊C4#12118yes—
512C4⋊C4C22×D412yes—
513C22⋊C4—28no—
514C22⋊C4—24no—
515C22⋊C4—28no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—28no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—42no—
537C42—41no—
538C42—42no—
539C42—48no—
540C42—42no—
541C42—82no—
542C42—82no—
543C42—82no—
544C42#13514yes—
545C42C22×D412yes—
546C42C22×D414yes—
547C42C2×#4112yes—
548C42C2×#4114yes—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C23×D411yescharacteristic
564Q8—28no—
565Q8—28no—
566Q8—44no—
567Q8—44no—
568Q8—44no—
569Q8—44no—
570Q8—44no—
571D4—24no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—84no—
578D4—84no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—48no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—88no—
623C2×C4—164no—
624C2×C4C2×#12114yes—
625C2×C4C2×#12112yes—
626C2×C4C2×#12114yes—
627C2×C4C2×#12112yes—
628C2×C4C2×#13512yes—
629C2×C4C2×#13512yes—
630C2×C4C2×#13512yes—
631C2×C4#57512yes—
632C2×C4#57812yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)12yes—
636C2×C4#58512yes—
637C8—24no—
638C8—44no—
639C8—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22C22×#12111yescharacteristic
655C22#348711yescharacteristic
656C22#349811yescharacteristic
657C22C22×#13511yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57811yescharacteristic
660C22#358611yescharacteristic
661C4—24no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—24no—
667C4—44no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—42no—
672C4—81no—
673C4—84no—
674C4—81no—
675C4—82no—
676C4—82no—
677C4#303514yes—
678C4#348812yes—
679C4#351712yes—
680C2—82no—
681C2—162no—
682C2#3602911yescharacteristic
683C2C2×#303511yescharacteristic
684C2#6043211yescharacteristic
685C2#6188211yescharacteristic
686C2#6190511yescharacteristic
687C2#6241811yescharacteristic
688C2#6245611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2011C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3393C2semidirect
2. #36594C2semidirect
3. #34322C2semidirect
4. #29801C2semidirect
5. #29798C2semidirect
7. #28440C2semidirect
8. #28164C2semidirect
9. #28157C2semidirect
10. #25374C2semidirect
11. #25365C2semidirect
12. #24736C2semidirect
13. #24727C2semidirect
14. #23679C2semidirect
16. #22734C2semidirect
17. #22541C2semidirect
104. #556D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33932
#365942
#343222
#298012
#297982
#284422
#284402
#281642
#281572
#253742
#253652
#247362
#247272
#236792
#236762
#227342
#225412