Finite Group Databasebeta

← #6800681

Subgroups — #6800681

Order 512.

1711 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34934C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#32430C214yesmaximal, complemented
5#31638C214yesmaximal, complemented
6#29981C212yesmaximal, complemented
7#29139C212yesmaximal, complemented
8#28716C211yesmaximal, characteristic, complemented
9#28404C212yesmaximal
10#28252C211yesmaximal, characteristic, complemented
11#28157C212yesmaximal, complemented
12#27729C212yesmaximal, complemented
13#25932C212yesmaximal, complemented
14#24329C212yesmaximal, complemented
15#22821C211yesmaximal, characteristic, complemented
16#22146C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×Q82C2212yes—
19C2×#529—21no—
20C2×#486C2212yes—
21C2×#476C2212yes—
22C2×#460C2212yes—
23C2×#461C2212yes—
24C2×#461C2211yescharacteristic
25C2×C4×SD16C2212yes—
26#2868C2212yes—
27#2868C2211yescharacteristic
28#2567C2212yes—
29#2567C2214yes—
30#2554—22no—
31#2552C2211yescharacteristic
32#2551—22no—
33#2551—21no—
34#2551—22no—
35#2551—21no—
36#2548C2211yescharacteristic
37#2541C2211yescharacteristic
38#2538C2211yescharacteristic
39#2512C2212yes—
40#2490—21no—
41#2445C2212yes—
42#2426—21no—
43#2392—22no—
44Q8×C4⋊C4—21no—
45Q8×C4⋊C4—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#127C2211yescharacteristic
49C4×#127C2212yes—
50C4×#121—21no—
51#2293C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2100C2214yes—
55#2094C2214yes—
56#2072C2214yes—
57#2071C2214yes—
58#2058C2214yes—
59#2055C2214yes—
60#2023C2214yes—
61#2023C2214yes—
62#2023C2214yes—
63#2014C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1984C2212yes—
67#1981C2211yescharacteristic
68#1968C2214yes—
69#1957C2214yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1933C2214yes—
74#1927C2212yes—
75#1916C2214yes—
76#1880C2214yes—
77#1880C2214yes—
78#1880C2214yes—
79#1877C2212yes—
80#1865C2214yes—
81#1842C2211yescharacteristic
82#1834C2214yes—
83#1831C2212yes—
84#1803C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C22×SD16C2312yes—
96C22×SD16C2314yes—
97Q82—44no—
98Q82—44no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×#127C2311yescharacteristic
109C2×#127C2312yes—
110#529—48no—
111C2×#126—22no—
112C2×#125—21no—
113C2×#124—21no—
114C2×#123—22no—
115C2×C22⋊Q8—21no—
116C2×#121—21no—
117C2×#121—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2314yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—22no—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117—22no—
133C22×C42—21no—
134#486—44no—
135#476—44no—
136#461—24no—
137#461—28no—
138#460—44no—
139C4×SD16—44no—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2314yes—
158#398—22no—
159#397—22no—
160#396—22no—
161#396—22no—
162#395—24no—
163#395—24no—
164#395—24no—
165#394—22no—
166#394—22no—
167#394—22no—
168#392—22no—
169#391C2312yes—
170#391C2314yes—
171#390—21no—
172#390—21no—
173#390—22no—
174#390—22no—
175#390—21no—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#390C2312yes—
181#389—24no—
182#389—22no—
183#389—24no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—21no—
189#389—22no—
190#389C2312yes—
191#389C2312yes—
192#388—22no—
193#388—22no—
194#388—22no—
195#386—22no—
196#384—21no—
197#384—22no—
198#384—22no—
199#384—21no—
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2312yes—
205#384C2314yes—
206#380—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C43C2312yes—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C22×Q8—24no—
238C22×Q8—22no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4C2412yes—
246C22×D4C2411yescharacteristic
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—28no—
254C2×SD16—42no—
255C2×SD16—44no—
256C2×SD16—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—416no—
266C4⋊Q8—48no—
267C4⋊Q8—84no—
268C4⋊Q8—82no—
269C4⋊Q8—82no—
270C4⋊Q8—84no—
271C4⋊Q8#4114yes—
272#127—24no—
273#127—22no—
274#127—22no—
275#127—42no—
276#126—48no—
277#125—44no—
278#124—44no—
279#123—48no—
280C22⋊Q8—44no—
281#121—44no—
282#121—44no—
283C4×Q8—24no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—48no—
291C4×Q8SD1618yescomplemented
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—24no—
299C4×D4—41no—
300C4×D4—44no—
301C4×D4—48no—
302C4×D4—82no—
303C4×D4—84no—
304#117—48no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C42—21no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2412yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#106—42no—
371#106—44no—
372C4⋊C8—22no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375C4⋊C8—22no—
376C4⋊C8—42no—
377#103—28no—
378#103—24no—
379#103—42no—
380#103—42no—
381#102—42no—
382#102—44no—
383#102—44no—
384#102—44no—
385#102—44no—
386C4×C8—24no—
387C4×C8—44no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—21no—
401C24—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—48no—
410C2×Q8—44no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—84no—
414C2×Q8—84no—
415C2×Q8C2×SD1614yes—
416C2×Q8C2×#4114yes—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—82no—
437C2×D4—81no—
438C2×D4—82no—
439C2×D4—81no—
440C2×D42−1+412yes—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—81no—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2511yescharacteristic
471SD16—48no—
472SD16—88no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—82no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—82no—
517C4⋊C4C2×SD1614yes—
518C4⋊C4C2×SD1618yes—
519C4⋊C4#13714yes—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C42—24no—
535C42—22no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—22no—
542C42—22no—
543C42—21no—
544C42—41no—
545C42—42no—
546C42—48no—
547C42—44no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42C2×SD1614yes—
552C42C2×SD1618yes—
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—28no—
573Q8—24no—
574Q8—24no—
575Q8—42no—
576Q8—42no—
577Q8—164no—
578Q8—168no—
579D4—22no—
580D4—22no—
581D4—41no—
582D4—42no—
583D4—42no—
584D4—41no—
585D4—42no—
586D4—42no—
587D4—88no—
588D4—88no—
589D4—84no—
590D4—84no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4—84no—
634C2×C4#53512yes—
635C2×C4#53512yes—
636C2×C4#54812yes—
637C2×C4#54812yes—
638C2×C4C22×SD1612yes—
639C2×C4C22×SD1614yes—
640C2×C4C22×SD1614yes—
641C2×C4C22×SD1612yes—
642C2×C4C22×SD1612yes—
643C2×C4C2×#13714yes—
644C2×C4C2×#13712yes—
645C2×C4#57612yes—
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C2×C4#58511yescharacteristic
652C8—44no—
653C8—42no—
654C8—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22C2×#53511yescharacteristic
670C22C2×#54811yescharacteristic
671C22#348911yescharacteristic
672C22C23×SD1611yescharacteristic
673C22C22×#13711yescharacteristic
674C22C2×#57611yescharacteristic
675C22#358511yescharacteristic
676C4—24no—
677C4—22no—
678C4—22no—
679C4—24no—
680C4—22no—
681C4—24no—
682C4—42no—
683C4—41no—
684C4—44no—
685C4—81no—
686C4—81no—
687C4—82no—
688C4—81no—
689C4—84no—
690C4—82no—
691C4#320614yes—
692C4#336214yes—
693C4#351411yescharacteristic
694C4#352011yescharacteristic
695C4#354711yescharacteristic
696C4#355211yescharacteristic
697C2—82no—
698C2—162no—
699C2#3912411yescharacteristic
700C2C2×#320611yescharacteristic
701C2#6083211yescharacteristic
702C2C2×#336211yescharacteristic
703C2#6131911yescharacteristic
704C2#6209611yescharacteristic
705C2#6212711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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. Q8×C4⋊Q8C2semidirect
3. #33197C2semidirect
4. #32430C2semidirect
5. #31638C2semidirect
6. #29981C2semidirect
7. #29139C2semidirect
8. #28716C2semidirect
10. #28252C2semidirect
11. #28157C2semidirect
12. #27729C2semidirect
13. #25932C2semidirect
14. #24329C2semidirect
15. #22821C2semidirect
16. #22146C2semidirect
17. C4×#461C2semidirect
291. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349342
#331972
#324302
#316382
#299812
#291392
#287162
#284042
#282522
#281572
#277292
#259322
#243292
#228212
#221462
C4×#4612