Finite Group Databasebeta

← #6564377

Subgroups — #6564377

Order 512.

1951 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3303C212yesmaximal, complemented
2#36187C211yesmaximal, characteristic, complemented
3#34316C211yesmaximal, characteristic, complemented
4#29868C211yesmaximal, characteristic, complemented
5#29716C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28574C212yesmaximal, complemented
8#28157C211yesmaximal, characteristic, complemented
9#27652C212yesmaximal, complemented
10#25319C212yesmaximal, complemented
11#25203C214yesmaximal, complemented
12#24677C212yesmaximal, complemented
13#24535C214yesmaximal, complemented
14#23676C211yesmaximal, characteristic, complemented
15#23627C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22489C212yesmaximal, complemented
18C2×#555C2212yes—
19C2×#547C2211yescharacteristic
20#3303—216no—
21C2×#487C2212yes—
22C2×#485C2214yes—
23C2×#476C2212yes—
24C2×#465C2214yes—
25C2×#461C2212yes—
26C2×#449C2214yes—
27C2×C4×D8C2212yes—
28C2×C8×D4C2212yes—
29C2×#416C2214yes—
30#2868C2211yescharacteristic
31#2856C2212yes—
32#2563C2212yes—
33#2537—21no—
34#2537—21no—
35#2500C2214yes—
36#2491—22no—
37#2491—22no—
38#2374—22no—
39#2365C2211yescharacteristic
40#2363—21no—
41#2362C2212yes—
42#2354—21no—
43#2354—21no—
44#2354C2211yescharacteristic
45#2354C2211yescharacteristic
46#2345—21no—
47#2341C2212yes—
48#2335—22no—
49#2324—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2297C2212yes—
54#2293C2211yescharacteristic
55#2285C2212yes—
56#2274—21no—
57C42×D4C2211yescharacteristic
58#2084C2212yes—
59#2084C2212yes—
60#2083C2211yescharacteristic
61#2074C2214yes—
62#2073C2214yes—
63#2072C2212yes—
64#2072C2212yes—
65#2072C2212yes—
66#2071C2212yes—
67#2071C2211yescharacteristic
68#2071C2212yes—
69#2067C2212yes—
70#2055C2214yes—
71#2054C2214yes—
72#2054C2212yes—
73#2041C2211yescharacteristic
74#2014C2212yes—
75#2011C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1949C2214yes—
79#1939C2212yes—
80#1926C2214yes—
81#1893C2214yes—
82#1880C2214yes—
83#1880C2214yes—
84#1875C2214yes—
85#1850C2214yes—
86#1831C2211yescharacteristic
87#1825C2214yes—
88#1805C2214yes—
89#1803C2214yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1735C2214yes—
95C2×#336C2212yes—
96C22×#41C2312yes—
97C22×SD16C2314yes—
98C22×D8C2312yes—
99C23×C8C2314yes—
100#555—28no—
101#555—28no—
102#547—24no—
103#547D418yescomplemented
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#124—24no—
110C2×#123—22no—
111C2×C22⋊Q8C2312yes—
112C2×#121—21no—
113C2×#121—22no—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C2×#117—21no—
128C2×#117—21no—
129C2×#117C2314yes—
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C42C2312yes—
133#487—28no—
134#485—216no—
135#476—28no—
136#465—216no—
137#461—28no—
138#449—216no—
139C4×D8—28no—
140C8×D4—28no—
141C2×#107C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2312yes—
146#416—216no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2312yes—
158#394—24no—
159#394—24no—
160#393—24no—
161#391C2312yes—
162#391C2312yes—
163#390—21no—
164#390—22no—
165#390—22no—
166#390—21no—
167#390—21no—
168#390—22no—
169#390—22no—
170#390C2312yes—
171#390C2314yes—
172#390C2311yescharacteristic
173#390C2312yes—
174#390C2311yescharacteristic
175#389—21no—
176#389—21no—
177#389—21no—
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#388—22no—
181#388—22no—
182#388—22no—
183#388—21no—
184#388—21no—
185#388—22no—
186#388—24no—
187#388—24no—
188#387—22no—
189#387—22no—
190#387—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#382C2314yes—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380—22no—
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2314yes—
242C2×#41—24no—
243C2×#41—44no—
244C2×#41—44no—
245C2×#41C2×D418yes—
246C22×Q8—22no—
247C22×Q8C2412yes—
248C22×D4—22no—
249C22×D4—24no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C2×D411yescharacteristic
253C22×D4C2412yes—
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C2×SD16—28no—
262C2×SD16—44no—
263C2×SD16—44no—
264C2×D8—24no—
265C2×D8—42no—
266C2×D8—42no—
267C22×C8—28no—
268C22×C8—44no—
269C22×C8—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—22no—
275C4⋊Q8—24no—
276C4⋊Q8—82no—
277C4⋊Q8—82no—
278C4⋊Q8C2×D414yes—
279#127—42no—
280#124—416no—
281#123—216no—
282C22⋊Q8—24no—
283C22⋊Q8C2×D418yes—
284#121—28no—
285#121—28no—
286#121—28no—
287#121—48no—
288#121—84no—
289#121—84no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—24no—
296C4×D4—42no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4C2×D414yes—
300C4×D4C2×D414yes—
301C4×D4#4114yes—
302#117—28no—
303#117—28no—
304#117—44no—
305#117—48no—
306#117—44no—
307#117—44no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—24no—
372#106—28no—
373C4⋊C8—24no—
374C4⋊C8—44no—
375C4⋊C8—42no—
376#103—28no—
377#103—28no—
378#103—28no—
379#102—24no—
380#102—28no—
381#102—24no—
382#102—42no—
383#102—44no—
384#102—42no—
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—21no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—21no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—21no—
402C24—22no—
403C24C22×D412yes—
404#41—416no—
405#41—88no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—44no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8C22×D414yes—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—48no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4#12114yes—
437C2×D4C22×D412yes—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D414yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4114yes—
473C22×C4C2511yescharacteristic
474SD16—88no—
475D8—48no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—88no—
486C4⋊C4—24no—
487C4⋊C4—28no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4#12114yes—
513C4⋊C4C22×D412yes—
514C22⋊C4—28no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—82no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4#12118yes—
534C42—22no—
535C42—22no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—42no—
540C42—42no—
541C42—41no—
542C42—44no—
543C42—48no—
544C42—42no—
545C42—82no—
546C42—82no—
547C42—82no—
548C42#13514yes—
549C42C22×D412yes—
550C42C2×#4112yes—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23C2×#12114yes—
572C23C23×D411yescharacteristic
573Q8—28no—
574Q8—44no—
575Q8—164no—
576D4—24no—
577D4—28no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—84no—
584D4—84no—
585D4—88no—
586D4—168no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—48no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4—88no—
634C2×C4C2×#12114yes—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#13512yes—
638C2×C4C2×#13512yes—
639C2×C4C2×#13512yes—
640C2×C4#57512yes—
641C2×C4#57712yes—
642C2×C4C23×D411yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4#58512yes—
645C2×C4#58512yes—
646C8—42no—
647C8—44no—
648C8—42no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22C22×#12111yescharacteristic
671C22#348511yescharacteristic
672C22#350011yescharacteristic
673C22C22×#13511yescharacteristic
674C22C2×#57511yescharacteristic
675C22C2×#57711yescharacteristic
676C22#358511yescharacteristic
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—24no—
682C4—42no—
683C4—42no—
684C4—41no—
685C4—44no—
686C4—42no—
687C4—81no—
688C4—84no—
689C4—81no—
690C4—82no—
691C4—82no—
692C4#303514yes—
693C4#348512yes—
694C4#352112yes—
695C2—44no—
696C2—82no—
697C2—162no—
698C2#3600911yescharacteristic
699C2C2×#303511yescharacteristic
700C2#6043111yescharacteristic
701C2#6170011yescharacteristic
702C2#6173111yescharacteristic
703C2#6253211yescharacteristic
704C2#6258611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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
1. C2×#3303C2semidirect
2. #36187C2semidirect
3. #34316C2semidirect
4. #29868C2semidirect
5. #29716C2semidirect
6. #28717C2semidirect
7. #28574C2semidirect
8. #28157C2semidirect
9. #27652C2semidirect
10. #25319C2semidirect
11. #25203C2semidirect
12. #24677C2semidirect
13. #24535C2semidirect
14. #23676C2semidirect
15. #23627C2semidirect
16. #22696C2semidirect
17. #22489C2semidirect
103. #547D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33032
#361872
#343162
#298682
#297162
#287172
#285742
#281572
#276522
#253192
#252032
#246772
#245352
#236762
#236272
#226962
#224892