Finite Group Databasebeta

← #6564378

Subgroups — #6564378

Order 512.

2205 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×#3298C212yesmaximal, complemented
2#36179C211yesmaximal, characteristic
3#34334C211yesmaximal, characteristic, complemented
4#29866C211yesmaximal, characteristic, complemented
5#29713C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28574C212yesmaximal, complemented
8#28157C211yesmaximal, characteristic, complemented
9#27647C212yesmaximal, complemented
10#25320C212yesmaximal, complemented
11#25204C214yesmaximal, complemented
12#24678C212yesmaximal, complemented
13#24528C214yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15C2×#2072C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22485C212yesmaximal, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#547—21no—
20C2×#547—21no—
21C2×#547C2212yes—
22#3298—216no—
23C2×#487C2212yes—
24C2×#483C2214yes—
25C2×#475C2212yes—
26C2×#465C2214yes—
27C2×#461C2212yes—
28C2×#448C2214yes—
29C2×C4×SD16C2212yes—
30C2×C8×D4C2212yes—
31C22×#103C2214yes—
32#2868C2211yescharacteristic
33#2856C2212yes—
34#2542—21no—
35#2542—21no—
36#2542C2212yes—
37#2501—22no—
38#2501—22no—
39#2501C2214yes—
40#2369—21no—
41#2368—22no—
42#2365C2211yescharacteristic
43#2360C2212yes—
44#2355C2211yescharacteristic
45#2349—21no—
46D4×C4⋊C4C2211yescharacteristic
47Q8×C22⋊C4C2212yes—
48#2331—21no—
49#2329—22no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2295C2211yescharacteristic
54C2×#391C2212yes—
55C2×#384C2212yes—
56#2275—21no—
57C42×D4C2211yescharacteristic
58#2084C2212yes—
59#2084C2211yescharacteristic
60#2084C2212yes—
61#2072C2212yes—
62#2072C2212yes—
63#2072C2212yes—
64#2072C2211yescharacteristic
65#2072C2214yes—
66#2072C2214yes—
67#2072C2212yes—
68#2072C2212yes—
69#2065C2212yes—
70#2055C2214yes—
71#2055C2214yes—
72#2055C2212yes—
73#2041C2211yescharacteristic
74#2014C2211yescharacteristic
75#2014C2212yes—
76#1990C2212yes—
77#1988C2212yes—
78#1948C2214yes—
79#1939C2212yes—
80#1926C2214yes—
81#1893C2214yes—
82#1880C2214yes—
83#1880C2214yes—
84#1874C2214yes—
85#1850C2214yes—
86#1831C2211yescharacteristic
87#1823C2214yes—
88#1803C2214yes—
89#1803C2214yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94C2×#340C2214yes—
95C2×#336C2212yes—
96C22×#41—22no—
97C23×Q8C2312yes—
98C22×SD16C2312yes—
99C22×SD16C2314yes—
100C23×C8C2314yes—
101D4×Q8—24no—
102D4×Q8D418yescomplemented
103#547—28no—
104#547—28no—
105#547—48no—
106#547—48no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×#127C2311yescharacteristic
113C2×#123—24no—
114C2×#123—24no—
115C2×#123—24no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8C2312yes—
121C2×#121—22no—
122C2×#121—22no—
123C2×#121C2314yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4D411yescharacteristic
132C2×C4×D4D411yescharacteristic
133C2×C4×D4C2312yes—
134C2×C4×D4C2311yescharacteristic
135C22×C4⋊C4—22no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4C2314yes—
138C22×C4⋊C4C2312yes—
139C22×C42C2312yes—
140#487—28no—
141#483—216no—
142#475—28no—
143#465—216no—
144#461—28no—
145#448—216no—
146C4×SD16—28no—
147C8×D4—28no—
148C2×#106C2314yes—
149C2×#106C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2312yes—
153C2×#103—216no—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×C22⋊C8C2314yes—
164C2×C4×C8C2312yes—
165#399—24no—
166#399—24no—
167#399—24no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391C2312yes—
173#391C2311yescharacteristic
174#391C2312yes—
175#391C2312yes—
176#391C2311yescharacteristic
177#390—21no—
178#390—21no—
179#390—22no—
180#390—22no—
181#390C2312yes—
182#390C2314yes—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389C2311yescharacteristic
189#389C2311yescharacteristic
190#387—21no—
191#387—22no—
192#387—21no—
193#387—22no—
194#387—22no—
195#387—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#384—21no—
200#384—21no—
201#384—22no—
202#384—22no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2314yes—
207#384C2312yes—
208#382—21no—
209#380—22no—
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4C2312yes—
228#376—21no—
229C2×#95—22no—
230C2×#95—22no—
231C43C2311yescharacteristic
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2311yescharacteristic
242#336C2314yes—
243#336C2312yes—
244C2×#41—48no—
245C2×#41—48no—
246C2×#41—48no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—24no—
250C22×Q8—44no—
251C22×Q8—44no—
252C22×Q8C2×D418yes—
253C22×Q8C2412yes—
254C22×Q8C2411yescharacteristic
255C22×D4—24no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4C2×D411yescharacteristic
259C22×D4C2412yes—
260C23×C4—22no—
261C23×C4—24no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2412yes—
267C2×SD16—24no—
268C2×SD16—28no—
269C2×SD16—42no—
270C2×SD16—42no—
271C2×SD16—44no—
272C2×SD16—44no—
273C22×C8—28no—
274C22×C8—44no—
275C22×C8—44no—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C22×C8C2412yes—
279C4⋊Q8—28no—
280C4⋊Q8—24no—
281C4⋊Q8—22no—
282C4⋊Q8—24no—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285C4⋊Q8C2×D414yes—
286#127—42no—
287#123—416no—
288#123—416no—
289#123—416no—
290C22⋊Q8—24no—
291C22⋊Q8—216no—
292C22⋊Q8—48no—
293C22⋊Q8—48no—
294C22⋊Q8—48no—
295C22⋊Q8C2×D418yes—
296#121—28no—
297#121—28no—
298#121—48no—
299#121—48no—
300C4×Q8—22no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8C2×D414yes—
304C4×D4—28no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—44no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4SD1618yescomplemented
312C4×D4C2×D414yes—
313C4×D4#4114yes—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—48no—
339C2×C4⋊C4—48no—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2414yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—44no—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#106—24no—
377#106—28no—
378C4⋊C8—24no—
379C4⋊C8—44no—
380C4⋊C8—42no—
381#103—24no—
382#103—28no—
383#103—28no—
384#103—28no—
385#103—28no—
386#102—24no—
387#102—42no—
388#102—44no—
389#102—42no—
390C22⋊C8—28no—
391C4×C8—24no—
392#95—21no—
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—21no—
402#95—21no—
403#95—21no—
404#95—22no—
405#95—22no—
406#95—21no—
407#95C2×D412yes—
408#95C2×D412yes—
409#95C2×D412yes—
410C24—22no—
411C24C22×D412yes—
412#41—832no—
413C2×Q8—28no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—416no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×Q8—88no—
427C2×Q8C22×D412yes—
428C2×Q8C22×D414yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—48no—
440C2×D4—48no—
441C2×D4—48no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4#12114yes—
445C2×D4C2×SD1614yes—
446C2×D4AGL(1,ℤ/8)14yes—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4C2×SD1618yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4114yes—
486C22×C4C2511yescharacteristic
487SD16—48no—
488SD16—88no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—88no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—81no—
525C4⋊C4—81no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4#12114yes—
530C4⋊C4C2×SD1614yes—
531C4⋊C4C22×D412yes—
532C22⋊C4—28no—
533C22⋊C4—24no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4#12118yes—
542C22⋊C4C2×SD1618yes—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—42no—
550C42—42no—
551C42—41no—
552C42—42no—
553C42—82no—
554C42—82no—
555C42—82no—
556C42C2×SD1614yes—
557C42C22×D412yes—
558C42C2×#4112yes—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23C2×#12114yes—
571C23C22×SD1614yes—
572C23C2×AGL(1,ℤ/8)14yes—
573C23C23×D411yescharacteristic
574Q8—24no—
575Q8—28no—
576Q8—44no—
577Q8—42no—
578Q8—88no—
579Q8—88no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—84no—
585D4—84no—
586D4—816no—
587D4#460116yes—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—88no—
635C2×C4—816no—
636C2×C4—88no—
637C2×C4—164no—
638C2×C4C2×#12112yes—
639C2×C4C2×#12114yes—
640C2×C4C2×#12112yes—
641C2×C4C22×SD1612yes—
642C2×C4C22×SD1612yes—
643C2×C4C22×SD1614yes—
644C2×C4C22×SD1612yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4#57812yes—
647C2×C4C23×D411yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4#58512yes—
650C2×C4#58512yes—
651C8—42no—
652C8—44no—
653C8—42no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22C2×#46018yes—
668C22C22×#12111yescharacteristic
669C22#348511yescharacteristic
670C22#350011yescharacteristic
671C22C23×SD1611yescharacteristic
672C22C22×AGL(1,ℤ/8)11yescharacteristic
673C22C2×#57811yescharacteristic
674C22#358611yescharacteristic
675C4—22no—
676C4—22no—
677C4—22no—
678C4—24no—
679C4—22no—
680C4—24no—
681C4—42no—
682C4—42no—
683C4—41no—
684C4—42no—
685C4—44no—
686C4—81no—
687C4—82no—
688C4—81no—
689C4—82no—
690C4—84no—
691C4C2×#46014yes—
692C4#348512yes—
693C4#352112yes—
694C2—28no—
695C2—162no—
696C2#3600911yescharacteristic
697C2C22×#46011yescharacteristic
698C2#6043211yescharacteristic
699C2#6169511yescharacteristic
700C2#6173711yescharacteristic
701C2#6252611yescharacteristic
702C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
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×#3298C2semidirect
3. #34334C2semidirect
4. #29866C2semidirect
5. #29713C2semidirect
6. #28717C2semidirect
7. #28574C2semidirect
8. #28157C2semidirect
9. #27647C2semidirect
10. #25320C2semidirect
11. #25204C2semidirect
12. #24678C2semidirect
13. #24528C2semidirect
14. #23678C2semidirect
15. C2×#2072C2semidirect
16. #22696C2semidirect
17. #22485C2semidirect
102. D4×Q8D4semidirect
311. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32982
#361792
#343342
#298662
#297132
#287172
#285742
#281572
#276472
#253202
#252042
#246782
#245282
#236782
C2×#20722
#226962
#224852