Finite Group Databasebeta

← #6621049

Subgroups — #6621049

Order 512.

1919 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3360C212yesmaximal, complemented
2#36559C211yesmaximal, characteristic, complemented
3C4×#556C211yesmaximal, characteristic, complemented
4#33200C212yesmaximal, complemented
5#33197C211yesmaximal, characteristic, complemented
6#28721C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28253C212yesmaximal
9#28252C211yesmaximal, characteristic, complemented
10#25327C214yesmaximal, complemented
11#25319C212yesmaximal, complemented
12#24685C214yesmaximal, complemented
13#24677C212yesmaximal, complemented
14Q8×C4⋊C8C211yesmaximal, characteristic, complemented
15#22519C212yesmaximal, complemented
16C4×#460C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C2×#554C2212yes—
20#3360—216no—
21C2×#488C2214yes—
22C2×#487C2212yes—
23C2×#479C2214yes—
24C2×#476C2212yes—
25C2×#460C2212yes—
26C2×#463C2214yes—
27C2×#461C2212yes—
28C2×#459C2211yescharacteristic
29C2×C8×Q8C2212yes—
30C2×C4×SD16C2214yes—
31C2×C4×D8C2212yes—
32#2868C2211yescharacteristic
33#2859C2212yes—
34#2566C2214yes—
35#2563C2212yes—
36#2538—21no—
37#2538—21no—
38#2452—22no—
39#2452—22no—
40#2373—22no—
41#2367C2212yes—
42#2365C2211yescharacteristic
43#2362—21no—
44#2356C2212yes—
45#2355C2211yescharacteristic
46#2354C2211yescharacteristic
47D4×C4⋊C4—21no—
48D4×C4⋊C4—21no—
49#2347—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×#127C2212yes—
53C4×#126—22no—
54C4×#124C2212yes—
55C4×C22⋊Q8—21no—
56C42×Q8C2212yes—
57C42×Q8C2211yescharacteristic
58C42×D4C2211yescharacteristic
59C4×#117—21no—
60#2074C2214yes—
61#2073C2214yes—
62#2072C2214yes—
63#2055C2214yes—
64#2055C2214yes—
65#2054C2214yes—
66#1991C2214yes—
67#1990C2212yes—
68#1989C2214yes—
69#1987C2212yes—
70#1982C2211yescharacteristic
71#1949C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1947C2211yescharacteristic
75#1939C2212yes—
76#1939C2214yes—
77#1926C2214yes—
78#1925C2212yes—
79#1880C2214yes—
80#1879C2212yes—
81#1842C2211yescharacteristic
82#1842C2212yes—
83#1805C2214yes—
84#1804C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1802C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C4×#102C2214yes—
96C22×SD16C2314yes—
97C22×D8C2312yes—
98#556—24no—
99#556D418yescomplemented
100#554—28no—
101#554—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#126—22no—
108C2×#125C2314yes—
109C2×#124C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—24no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—21no—
130C22×C4⋊C4—22no—
131C22×C42—21no—
132#488—216no—
133#487—28no—
134#479—216no—
135#476—28no—
136#463—216no—
137#461—28no—
138#460—28no—
139#459—24no—
140C8×Q8—28no—
141C4×SD16—216no—
142C4×D8—28no—
143C2×#107C2312yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2314yes—
163#396—24no—
164#396—24no—
165#393—24no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—22no—
172#390—22no—
173#390C2314yes—
174#390C2312yes—
175#390C2312yes—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#385—22no—
193#385—24no—
194#385—24no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2312yes—
201#383—22no—
202#382C2314yes—
203#382C2312yes—
204#380—22no—
205#380—22no—
206#380—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—22no—
231#376—22no—
232C43—22no—
233C43C2311yescharacteristic
234C43C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244C22×Q8—22no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2412yes—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—24no—
256C2×SD16—28no—
257C2×SD16—28no—
258C2×SD16—44no—
259C2×D8—24no—
260C2×D8—24no—
261C2×D8—42no—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—24no—
268C4⋊Q8—82no—
269C4⋊Q8—82no—
270C4⋊Q8C2×D414yes—
271#127—22no—
272#127#4114yes—
273#126—216no—
274#125—28no—
275#125—28no—
276#124—24no—
277#124C2×D418yes—
278C22⋊Q8—28no—
279C22⋊Q8—48no—
280C22⋊Q8—416no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—42no—
287C4×Q8—42no—
288C4×Q8C2×D418yes—
289C4×Q8C2×D414yes—
290C4×Q8#4114yes—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—28no—
294C4×D4—22no—
295C4×D4—24no—
296C4×D4—28no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—41no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—84no—
303C4×D4—84no—
304C4×D4C2×D414yes—
305#117—28no—
306#117—44no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—44no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371#107—24no—
372#106—28no—
373C4⋊C8—24no—
374C4⋊C8—28no—
375C4⋊C8—22no—
376C4⋊C8—24no—
377C4⋊C8—22no—
378C4⋊C8—42no—
379C4⋊C8—44no—
380#103—28no—
381#103—28no—
382#103—28no—
383#102—24no—
384#102—28no—
385#102—24no—
386#102—28no—
387#102—24no—
388#102—24no—
389C4×C8—24no—
390C4×C8—28no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407C24—21no—
408C24—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—44no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8#12114yes—
420C2×Q8C22×D414yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C22×D412yes—
438C2×D4C2×#4114yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—48no—
458C22×C4—81no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2511yescharacteristic
472SD16—416no—
473D8—48no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C4⋊C4—24no—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4#12114yes—
514C4⋊C4#12118yes—
515C4⋊C4C22×D412yes—
516C4⋊C42−1+412yes—
517C22⋊C4—28no—
518C22⋊C4—24no—
519C22⋊C4—28no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—82no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C42—28no—
531C42—24no—
532C42—22no—
533C42—24no—
534C42—22no—
535C42—21no—
536C42—22no—
537C42—21no—
538C42—22no—
539C42—42no—
540C42—42no—
541C42—41no—
542C42—44no—
543C42—44no—
544C42—44no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—82no—
549C42#13514yes—
550C42C22×D412yes—
551C42C22×D414yes—
552C42C2×#4112yes—
553C42C2×#4114yes—
554C42C2×#4112yes—
555C422−1+412yes—
556C23—22no—
557C23—24no—
558C23—42no—
559C23—41no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—41no—
570C23C23×D411yescharacteristic
571Q8—28no—
572Q8—24no—
573Q8—24no—
574Q8—42no—
575Q8—42no—
576Q8—164no—
577D4—22no—
578D4—24no—
579D4—24no—
580D4—22no—
581D4—48no—
582D4—48no—
583D4—41no—
584D4—42no—
585D4—42no—
586D4—41no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—88no—
631C2×C4—168no—
632C2×C4C2×#12114yes—
633C2×C4C2×#12114yes—
634C2×C4C2×#12112yes—
635C2×C4C2×#12112yes—
636C2×C4#54812yes—
637C2×C4#54812yes—
638C2×C4C2×#13512yes—
639C2×C4C2×#13512yes—
640C2×C4C2×#13512yes—
641C2×C4#57512yes—
642C2×C4#57712yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×2−1+411yescharacteristic
647C2×C4C2×2−1+411yescharacteristic
648C2×C4#58511yescharacteristic
649C8—42no—
650C8—44no—
651C8—42no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—82no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—82no—
663C22—81no—
664C22—82no—
665C22—82no—
666C22C22×#12111yescharacteristic
667C22C2×#54811yescharacteristic
668C22#348911yescharacteristic
669C22C22×#13511yescharacteristic
670C22C2×#57511yescharacteristic
671C22C2×#57711yescharacteristic
672C22#358511yescharacteristic
673C4—24no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—24no—
678C4—44no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—41no—
683C4—42no—
684C4—84no—
685C4—81no—
686C4—82no—
687C4—82no—
688C4#303514yes—
689C4#336014yescomplemented
690C4D4×#4111yescharacteristic
691C4#348211yescharacteristic
692C4#349911yescharacteristic
693C4#351811yescharacteristic
694C2—82no—
695C2—84no—
696C2#3644411yescharacteristic
697C2C2×#303511yescharacteristic
698C2#6043111yescharacteristic
699C2C2×#336011yescharacteristic
700C2#6131811yescharacteristic
701C2#6209811yescharacteristic
702C2#6214511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
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×#3360C2semidirect
2. #36559C2semidirect
3. C4×#556C2semidirect
4. #33200C2semidirect
5. #33197C2semidirect
6. #28721C2semidirect
7. #28717C2semidirect
9. #28252C2semidirect
10. #25327C2semidirect
11. #25319C2semidirect
12. #24685C2semidirect
13. #24677C2semidirect
14. Q8×C4⋊C8C2semidirect
15. #22519C2semidirect
16. C4×#460C2semidirect
99. #556D4semidirect
689. C4#3360semidirect

Maximal subgroups

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

subgroupindex
C2×#33602
#365592
C4×#5562
#332002
#331972
#287212
#287172
#282532
#282522
#253272
#253192
#246852
#246772
Q8×C4⋊C82
#225192
C4×#4602
C4×#4592