Finite Group Databasebeta

← #6539651

Subgroups — #6539651

Order 512.

2105 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3299C212yesmaximal
2D4×C4⋊Q8C211yesmaximal, characteristic
3#34332C211yesmaximal, characteristic
4#29799C211yesmaximal, characteristic, complemented
5#29757C212yesmaximal, complemented
6#28748C211yesmaximal, characteristic, complemented
7#28575C212yesmaximal, complemented
8#28206C211yesmaximal, characteristic, complemented
9#25278C212yesmaximal, complemented
10#25082C214yesmaximal, complemented
11#24732C212yesmaximal, complemented
12#24482C214yesmaximal, complemented
13#24212C212yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15C2×#2043C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic
17D4×#103C212yesmaximal
18C2×#554C2211yescharacteristic
19C2×D4×Q8C2212yes—
20C22×C4⋊Q8C2212yes—
21#3299—216no—
22C2×#487C2212yes—
23C2×#484C2214yes—
24C2×#477C2212yes—
25C2×#467C2214yes—
26C2×#462C2212yes—
27C2×#462C2214yes—
28C2×#451C2214yes—
29C2×C4×Q16C2212yes—
30C2×C8×D4C2212yes—
31C22×#103C2214yes—
32C22×#103C2214yes—
33#2868C2211yescharacteristic
34#2857C2212yes—
35#2551—21no—
36#2551—21no—
37#2546C2212yes—
38#2440—22no—
39#2440—22no—
40#2434C2214yes—
41#2370—21no—
42#2369C2211yescharacteristic
43#2360—22no—
44#2360C2212yes—
45Q8×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4—21no—
47D4×C4⋊C4—21no—
48D4×C4⋊C4—21no—
49D4×C4⋊C4C2211yescharacteristic
50#2345C2212yes—
51#2330—21no—
52#2329—22no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C22⋊Q8C2212yes—
56#2295C2211yescharacteristic
57C2×#389C2212yes—
58#2275—21no—
59C42×D4C2211yescharacteristic
60#2083C2212yes—
61#2083C2211yescharacteristic
62#2072C2211yescharacteristic
63#2068C2214yes—
64#2068C2214yes—
65#2068C2212yes—
66#2056C2212yes—
67#2043C2212yes—
68#2043C2212yes—
69#2043C2212yes—
70#2043C2212yes—
71#2043C2212yes—
72#2043C2211yescharacteristic
73#2043C2214yes—
74#2043C2214yes—
75#2016C2212yes—
76#2016C2212yes—
77#2016C2211yescharacteristic
78#1981C2214yes—
79#1981C2212yes—
80#1981C2214yes—
81#1956C2212yes—
82#1939C2212yes—
83#1928C2214yes—
84#1928C2214yes—
85#1892C2214yes—
86#1878C2214yes—
87#1872C2214yes—
88#1850C2214yes—
89#1831C2211yescharacteristic
90#1803C2212yes—
91#1797C2212yes—
92#1787C2214yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C2×#336C2212yes—
96C23×Q8C2314yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C23×C8C2314yes—
100#554—24no—
101#554D418yes—
102D4×Q8—28no—
103D4×Q8—28no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—48no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2311yescharacteristic
114C2×#125—21no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8C2314yes—
118C2×C22⋊Q8C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C42C2312yes—
135#487—28no—
136#484—216no—
137#477—28no—
138#467—216no—
139#462—28no—
140#462—48no—
141#451—216no—
142C4×Q16—28no—
143C8×D4—28no—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2312yes—
149C2×#103—216no—
150C2×#103—48no—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×C22⋊C8C2314yes—
165C2×C4×C8C2312yes—
166#398—24no—
167#395—24no—
168#395—24no—
169#391—21no—
170#391—22no—
171#391C2312yes—
172#391C2311yescharacteristic
173#391C2314yes—
174#391C2311yescharacteristic
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389C2312yes—
189#389C2311yescharacteristic
190#389C2312yes—
191#389C2312yes—
192#389C2311yescharacteristic
193#387—21no—
194#387—22no—
195#387—21no—
196#387—22no—
197#387—22no—
198#387—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—24no—
203#385—24no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#380—22no—
212#380—22no—
213#380—22no—
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4C2312yes—
234#376—21no—
235C2×#95—22no—
236C43C2311yescharacteristic
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2312yes—
244#336C2314yes—
245C22×Q8—21no—
246C22×Q8—24no—
247C22×Q8—28no—
248C22×Q8—28no—
249C22×Q8—28no—
250C22×Q8—44no—
251C22×Q8—44no—
252C22×Q8—44no—
253C22×Q8C2414yes—
254C22×Q8C2411yescharacteristic
255C22×Q8C2412yes—
256C22×D4C2×D411yescharacteristic
257C23×C4—22no—
258C23×C4—24no—
259C23×C4—22no—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C2×Q16—24no—
264C2×Q16—24no—
265C2×Q16—28no—
266C2×Q16—42no—
267C2×Q16—44no—
268C2×Q16—44no—
269C22×C8—28no—
270C22×C8—44no—
271C22×C8—44no—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—28no—
277C4⋊Q8—24no—
278C4⋊Q8—22no—
279C4⋊Q8—24no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283C4⋊Q8—48no—
284C4⋊Q8—48no—
285C4⋊Q8—48no—
286C4⋊Q8C2×D414yes—
287#125—28no—
288C22⋊Q8—24no—
289C22⋊Q8—216no—
290C22⋊Q8—28no—
291C22⋊Q8—28no—
292C22⋊Q8—416no—
293C22⋊Q8C2×D418yes—
294C4×Q8—22no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—44no—
298C4×Q8—82no—
299C4×Q8C2×D414yes—
300C4×D4—28no—
301C4×D4—24no—
302C4×D4—48no—
303C4×D4—44no—
304C4×D4Q1618yescomplemented
305C4×D4C2×D414yes—
306C4×D4#4114yes—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—48no—
337C2×C4⋊C4—48no—
338C2×C4⋊C4—84no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D418yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—44no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#107—24no—
373#107—28no—
374C4⋊C8—24no—
375C4⋊C8—42no—
376C4⋊C8—44no—
377#103—24no—
378#103—24no—
379#103—28no—
380#103—24no—
381#103—28no—
382#103—28no—
383#103—28no—
384#103—28no—
385#103—44no—
386#103—44no—
387#103—42no—
388#103—44no—
389#103—44no—
390C22⋊C8—28no—
391C4×C8—24no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—21no—
407#95—24no—
408#95—24no—
409#95C2×D412yes—
410#95C2×D412yes—
411#95C2×D412yes—
412C24C22×D412yes—
413C2×Q8—24no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—48no—
429C2×Q8—416no—
430C2×Q8—416no—
431C2×Q8—48no—
432C2×Q8—81no—
433C2×Q8—81no—
434C2×Q8—84no—
435C2×Q8—88no—
436C2×Q8—88no—
437C2×Q8C22×D412yes—
438C2×Q8H(5,𝔽2)14yes—
439C2×D4#12114yes—
440C2×D4C2×Q1614yes—
441C2×D4#13714yes—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—84no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4C2×Q1618yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D414yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4114yes—
479C22×C4C2511yescharacteristic
480Q16—48no—
481Q16—88no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—88no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—28no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—81no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4#12114yes—
528C4⋊C4C2×Q1614yes—
529C4⋊C4C22×D412yes—
530C4⋊C4C22×D414yes—
531C22⋊C4—28no—
532C22⋊C4—24no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4#12118yes—
536C22⋊C4C2×Q1618yes—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—42no—
544C42—42no—
545C42—41no—
546C42—44no—
547C42—42no—
548C42—82no—
549C42—82no—
550C42—82no—
551C42C2×Q1614yes—
552C42C22×D412yes—
553C42C2×#4112yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23C2×#12114yes—
559C23C22×Q1614yes—
560C23C2×#13714yes—
561C23C23×D411yescharacteristic
562Q8—24no—
563Q8—24no—
564Q8—24no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—44no—
569Q8—44no—
570Q8—44no—
571Q8—44no—
572Q8—84no—
573Q8—816no—
574D4#462116yescomplemented
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—88no—
625C2×C4—816no—
626C2×C4—164no—
627C2×C4C2×#12112yes—
628C2×C4C2×#12114yes—
629C2×C4C2×#12112yes—
630C2×C4C22×Q1612yes—
631C2×C4C22×Q1612yes—
632C2×C4C22×Q1614yes—
633C2×C4C22×Q1612yes—
634C2×C4C2×#13712yes—
635C2×C4#57712yes—
636C2×C4C23×D411yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)12yes—
639C2×C4#58512yes—
640C8—42no—
641C8—44no—
642C8—42no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22C2×#46218yes—
650C22C22×#12111yescharacteristic
651C22#348411yescharacteristic
652C22#350011yescharacteristic
653C22C23×Q1611yescharacteristic
654C22C22×#13711yescharacteristic
655C22C2×#57711yescharacteristic
656C22#358611yescharacteristic
657C4—22no—
658C4—22no—
659C4—22no—
660C4—24no—
661C4—24no—
662C4—22no—
663C4—42no—
664C4—44no—
665C4—41no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—81no—
671C4—84no—
672C4—82no—
673C4—81no—
674C4C2×#46214yes—
675C4#348812yes—
676C4#351312yes—
677C2—28no—
678C2#3572811yescharacteristic
679C2C22×#46211yescharacteristic
680C2#6042911yescharacteristic
681C2#6156311yescharacteristic
682C2#6159311yescharacteristic
683C2#6253011yescharacteristic
684C2#6257711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
4. #29799C2semidirect
5. #29757C2semidirect
6. #28748C2semidirect
7. #28575C2semidirect
8. #28206C2semidirect
9. #25278C2semidirect
10. #25082C2semidirect
11. #24732C2semidirect
12. #24482C2semidirect
13. #24212C2semidirect
14. #23679C2semidirect
15. C2×#2043C2semidirect
304. C4×D4Q16semidirect
574. D4#462semidirect

Maximal subgroups

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

subgroupindex
C2×#32992
D4×C4⋊Q82
#343322
#297992
#297572
#287482
#285752
#282062
#252782
#250822
#247322
#244822
#242122
#236792
C2×#20432
#226962
D4×#1032