Finite Group Databasebeta

← #6856742

Subgroups — #6856742

Order 512.

1693 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic, complemented
3#33162C211yesmaximal, characteristic, complemented
4#32108C214yesmaximal, complemented
5#30902C214yesmaximal, complemented
6#29795C212yesmaximal
7#29259C212yesmaximal, complemented
8#28754C212yesmaximal, complemented
9#28748C211yesmaximal, characteristic, complemented
10#28414C212yesmaximal, complemented
11#28279C211yesmaximal, characteristic, complemented
12#27776C212yesmaximal, complemented
13#25650C212yesmaximal, complemented
14#24185C212yesmaximal, complemented
15#22825C211yesmaximal, characteristic, complemented
16#22152C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556C2212yes—
19C2×#533—21no—
20C2×#489C2214yes—
21C2×#487C2212yes—
22C2×#474C2212yes—
23C2×#461C2212yes—
24C2×#461C2212yes—
25C2×#461C2212yes—
26C2×#459C2211yescharacteristic
27C2×C4×D8C2212yes—
28#2868C2211yescharacteristic
29#2858C2212yes—
30#2556—22no—
31#2551C2211yescharacteristic
32#2547C2212yes—
33#2546C2211yescharacteristic
34#2545C2214yes—
35#2542C2211yescharacteristic
36#2538—21no—
37#2538—21no—
38#2538C2211yescharacteristic
39#2512—22no—
40#2512—22no—
41#2506C2212yes—
42#2492—21no—
43#2452C2212yes—
44#2429—21no—
45#2396—22no—
46#2354—21no—
47#2354—21no—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51C4×#127C2211yescharacteristic
52C4×C22⋊Q8—21no—
53#2293C2212yes—
54C42×Q8C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2095C2214yes—
57#2067C2214yes—
58#2043C2214yes—
59#2042C2214yes—
60#2042C2214yes—
61#2042C2214yes—
62#2042C2214yes—
63#2024C2214yes—
64#2012C2214yes—
65#2012C2214yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1982C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1973C2214yes—
75#1959C2214yes—
76#1956C2212yes—
77#1947C2211yescharacteristic
78#1947C2212yes—
79#1935C2214yes—
80#1925C2212yes—
81#1917C2214yes—
82#1880C2214yes—
83#1879C2212yes—
84#1863C2214yes—
85#1863C2214yes—
86#1863C2214yes—
87#1842C2211yescharacteristic
88#1831C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×SD16C2314yes—
96C22×D8C2312yes—
97#556—44no—
98#556—44no—
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#127C2311yescharacteristic
108C2×#126—24no—
109C2×#125—22no—
110C2×#124C2314yes—
111C2×#124C2314yes—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117—22no—
130C2×#117—22no—
131C22×C42—21no—
132#489—48no—
133#487—44no—
134#474—44no—
135#461—28no—
136#461—44no—
137#461—44no—
138#459—24no—
139C4×D8—44no—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×#97C2314yes—
161C2×C4×C8C2312yes—
162#400—22no—
163#399—22no—
164#398—22no—
165#398—22no—
166#397—24no—
167#396—22no—
168#396—22no—
169#396—24no—
170#396—24no—
171#395—22no—
172#393—22no—
173#391—21no—
174#391—21no—
175#391—22no—
176#391C2314yes—
177#390—21no—
178#390—21no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390—22no—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#389—21no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—21no—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#388—22no—
197#388—22no—
198#388—24no—
199#388—24no—
200#387—22no—
201#385—22no—
202#384—21no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2312yes—
209#380—22no—
210#380—22no—
211#380—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D412yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C43C2311yescharacteristic
231C43C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2414yes—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2412yes—
246C22×D4C2411yescharacteristic
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—21no—
251C2×SD16—28no—
252C2×SD16—44no—
253C2×SD16—44no—
254C2×D8—24no—
255C2×D8—42no—
256C2×D8—42no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—42no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272#127—22no—
273#127—22no—
274#126—416no—
275#125—48no—
276#124—44no—
277#124—44no—
278#124—44no—
279#124—44no—
280C22⋊Q8—44no—
281C22⋊Q8—48no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—48no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—42no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×Q8—48no—
292C4×Q8—44no—
293C4×D4—22no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—41no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4#4114yes—
303#117—48no—
304#117—48no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—44no—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2412yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—42no—
371#107—44no—
372C4⋊C8—22no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375C4⋊C8—22no—
376C4⋊C8—42no—
377#103—28no—
378#103—28no—
379#103—44no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384#102—24no—
385#102—24no—
386#102—42no—
387#102—42no—
388#102—44no—
389#102—44no—
390#97—44no—
391C4×C8—24no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—21no—
407C24—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—82no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C2×#4112yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—81no—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2511yescharacteristic
471SD16—88no—
472D8—48no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4C2×#4112yes—
519C4⋊C4H(5,𝔽2)12yes—
520C22⋊C4—24no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—82no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—24no—
538C42—22no—
539C42—21no—
540C42—22no—
541C42—22no—
542C42—21no—
543C42—22no—
544C42—41no—
545C42—42no—
546C42—42no—
547C42—48no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—44no—
552C42AGL(1,ℤ/8)14yes—
553C42C2×#4112yes—
554C42H(5,𝔽2)12yes—
555C42H(5,𝔽2)12yes—
556C422−1+412yes—
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—
571C23C23×D411yescharacteristic
572Q8—44no—
573Q8—44no—
574Q8—44no—
575Q8—44no—
576Q8—44no—
577Q8—88no—
578Q8—84no—
579D4—22no—
580D4—22no—
581D4—42no—
582D4—42no—
583D4—41no—
584D4—42no—
585D4—41no—
586D4—42no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4#53512yes—
636C2×C4#53512yes—
637C2×C4#54812yes—
638C2×C4#54812yes—
639C2×C4C2×AGL(1,ℤ/8)12yes—
640C2×C4C2×AGL(1,ℤ/8)12yes—
641C2×C4C2×AGL(1,ℤ/8)12yes—
642C2×C4#57512yes—
643C2×C4#57812yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×2−1+411yescharacteristic
649C2×C4#58511yescharacteristic
650C8—42no—
651C8—44no—
652C8—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22C2×#53511yescharacteristic
668C22C2×#54811yescharacteristic
669C22#348811yescharacteristic
670C22C22×AGL(1,ℤ/8)11yescharacteristic
671C22C2×#57511yescharacteristic
672C22C2×#57811yescharacteristic
673C22#358611yescharacteristic
674C4—22no—
675C4—22no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—41no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—82no—
685C4—84no—
686C4—82no—
687C4—81no—
688C4—82no—
689C4#321414yes—
690C4#336714yes—
691C4#350611yescharacteristic
692C4#350811yescharacteristic
693C4#354811yescharacteristic
694C4#355411yescharacteristic
695C2—82no—
696C2—162no—
697C2#3972511yescharacteristic
698C2C2×#321411yescharacteristic
699C2#6083711yescharacteristic
700C2C2×#336711yescharacteristic
701C2#6132311yescharacteristic
702C2#6196211yescharacteristic
703C2#6199811yescharacteristic
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. #36594C2semidirect
2. #34929C2semidirect
3. #33162C2semidirect
4. #32108C2semidirect
5. #30902C2semidirect
7. #29259C2semidirect
8. #28754C2semidirect
9. #28748C2semidirect
10. #28414C2semidirect
11. #28279C2semidirect
12. #27776C2semidirect
13. #25650C2semidirect
14. #24185C2semidirect
15. #22825C2semidirect
16. #22152C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#349292
#331622
#321082
#309022
#297952
#292592
#287542
#287482
#284142
#282792
#277762
#256502
#241852
#228252
#221522
C4×#4592