Finite Group Databasebeta

← #6848833

Subgroups — #6848833

Order 512.

1769 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#34929C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#32112C214yesmaximal, complemented
5#30909C214yesmaximal, complemented
6#29661C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#28252C211yesmaximal, characteristic, complemented
9#28141C212yesmaximal, complemented
10#27628C212yesmaximal, complemented
11#26024C212yesmaximal, complemented
12#25385C212yesmaximal, complemented
13#24753C212yesmaximal, complemented
14#24630C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21885C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#555—21no—
19C2×#533C2212yes—
20C2×#475C2212yes—
21C2×#465C2214yes—
22C2×#460C2211yescharacteristic
23C2×#460C2212yes—
24C2×#461C2212yes—
25C2×C4×SD16C2212yes—
26C2×C8×D4C2212yes—
27C22×C4⋊C8C2212yes—
28#2868C2211yescharacteristic
29#2856C2212yes—
30#2563—21no—
31#2556—22no—
32#2551C2211yescharacteristic
33#2546C2211yescharacteristic
34#2542C2211yescharacteristic
35#2538C2211yescharacteristic
36#2537—21no—
37#2537C2212yes—
38#2506—21no—
39#2500—22no—
40#2492C2212yes—
41#2491—22no—
42#2491C2214yes—
43#2452—21no—
44#2429C2212yes—
45#2396—22no—
46#2354—21no—
47#2354C2212yes—
48C4×C4⋊Q8—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52#2293—21no—
53#2285C2212yes—
54C42×Q8C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2077C2214yes—
57#2065C2214yes—
58#2055C2214yes—
59#2055C2214yes—
60#2035C2214yes—
61#2024C2214yes—
62#2023C2214yes—
63#2013C2214yes—
64#2000C2214yes—
65#2000C2214yes—
66#1988C2212yes—
67#1987C2214yes—
68#1982C2211yescharacteristic
69#1981C2214yes—
70#1956C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2211yescharacteristic
74#1939C2212yes—
75#1927C2214yes—
76#1926C2212yes—
77#1920C2214yes—
78#1893C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1880C2212yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×SD16C2312yes—
96C22×SD16C2314yes—
97C23×C8C2314yes—
98#555—48no—
99#533—28no—
100#533—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#125—22no—
110C2×#124—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×#121—22no—
116C2×#121—22no—
117C2×#121—22no—
118C2×#121C2314yes—
119C2×C4×Q8—22no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2311yescharacteristic
130C2×#117—22no—
131C2×#117—22no—
132C2×#117—22no—
133C22×C42C2312yes—
134#475—44no—
135#465—48no—
136#461—44no—
137#460—24no—
138#460—44no—
139C4×SD16—44no—
140C8×D4—44no—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8—44no—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161#400—22no—
162#399—22no—
163#398—22no—
164#398—22no—
165#396—22no—
166#396—22no—
167#395—22no—
168#394—24no—
169#394—24no—
170#393—22no—
171#393—24no—
172#391—22no—
173#391C2312yes—
174#391C2312yes—
175#390—21no—
176#390—21no—
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—21no—
188#389—21no—
189#389—22no—
190#389C2312yes—
191#388—22no—
192#388—22no—
193#388—24no—
194#388—24no—
195#387—22no—
196#385—22no—
197#384—21no—
198#384—21no—
199#384—21no—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#382—22no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380C2312yes—
209#380C2314yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2314yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—22no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4C2312yes—
228C43—21no—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8C2412yes—
242C22×Q8C2411yescharacteristic
243C22×D4—24no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C2×SD16—24no—
253C2×SD16—42no—
254C2×SD16—42no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C22×C8—44no—
259C22×C8—44no—
260C22×C8—44no—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—42no—
269C4⋊Q8—42no—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—82no—
275#127—22no—
276#127—22no—
277#125—48no—
278#124—416no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—42no—
283C22⋊Q8—42no—
284C22⋊Q8—42no—
285C22⋊Q8—48no—
286#121—48no—
287#121—48no—
288#121—44no—
289#121—44no—
290#121—84no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×Q8—48no—
298C4×Q8—41no—
299C4×Q8#4114yes—
300C4×D4—22no—
301C4×D4—24no—
302C4×D4—22no—
303C4×D4—24no—
304C4×D4—42no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4SD1618yescomplemented
308#117—48no—
309#117—48no—
310#117—48no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—21no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#106—28no—
377#106—42no—
378C4⋊C8—22no—
379C4⋊C8—22no—
380C4⋊C8—24no—
381C4⋊C8—24no—
382C4⋊C8—44no—
383C4⋊C8—42no—
384#103—44no—
385#103—42no—
386#103—44no—
387#103—44no—
388#103—44no—
389#102—24no—
390#102—24no—
391#102—42no—
392#102—44no—
393C22⋊C8—44no—
394C4×C8—24no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—24no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—22no—
410C24C22×D412yes—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8C2×#4112yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—48no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—84no—
445C2×D4—84no—
446C2×D4C2×SD1614yes—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4C2×SD1618yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2511yescharacteristic
480SD16—48no—
481SD16—88no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—88no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—81no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4C2×SD1614yes—
527C4⋊C4C2×#4112yes—
528C4⋊C4H(5,𝔽2)12yes—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C22⋊C4—44no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—82no—
542C22⋊C4C2×SD1618yes—
543C42—24no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—21no—
551C42—21no—
552C42—41no—
553C42—42no—
554C42—42no—
555C42—44no—
556C42—48no—
557C42—42no—
558C42—42no—
559C42—81no—
560C42C2×SD1614yes—
561C42C2×#4112yes—
562C42H(5,𝔽2)12yes—
563C42H(5,𝔽2)12yes—
564C422−1+412yes—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23C22×SD1614yes—
579C23C23×D411yescharacteristic
580Q8—22no—
581Q8—22no—
582Q8—41no—
583Q8—41no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—88no—
589Q8—164no—
590D4—28no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—84no—
596D4—84no—
597D4—168no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—216no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—84no—
645C2×C4—88no—
646C2×C4#53712yes—
647C2×C4#53712yes—
648C2×C4#54612yes—
649C2×C4#54612yes—
650C2×C4C22×SD1614yes—
651C2×C4C22×SD1612yes—
652C2×C4C22×SD1612yes—
653C2×C4C22×SD1612yes—
654C2×C4#57512yes—
655C2×C4#57812yes—
656C2×C4C22×#4111yescharacteristic
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C2×C4#58511yescharacteristic
662C8—42no—
663C8—42no—
664C8—44no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22C2×#53711yescharacteristic
680C22C2×#54611yescharacteristic
681C22#348811yescharacteristic
682C22C23×SD1611yescharacteristic
683C22C2×#57511yescharacteristic
684C22C2×#57811yescharacteristic
685C22#358511yescharacteristic
686C4—22no—
687C4—24no—
688C4—22no—
689C4—22no—
690C4—41no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—81no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—84no—
701C4#324614yes—
702C4#329814yes—
703C4#350611yescharacteristic
704C4#351111yescharacteristic
705C4#354711yescharacteristic
706C4#354911yescharacteristic
707C2—44no—
708C2—162no—
709C2#3946911yescharacteristic
710C2C2×#324611yescharacteristic
711C2#6093011yescharacteristic
712C2C2×#329811yescharacteristic
713C2#6116611yescharacteristic
714C2#6195111yescharacteristic
715C2#6198511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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
2. #34929C2semidirect
3. #33196C2semidirect
4. #32112C2semidirect
5. #30909C2semidirect
6. #29661C2semidirect
7. #29141C2semidirect
8. #28252C2semidirect
9. #28141C2semidirect
10. #27628C2semidirect
11. #26024C2semidirect
12. #25385C2semidirect
13. #24753C2semidirect
14. #24630C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21885C2semidirect
17. C4×#460C2semidirect
307. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361872
#349292
#331962
#321122
#309092
#296612
#291412
#282522
#281412
#276282
#260242
#253852
#247532
#246302
D4×C4⋊C82
#218852
C4×#4602