Finite Group Databasebeta

← #6484997

Subgroups — #6484997

Order 512.

1701 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35247C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#33100C212yesmaximal, complemented
4#33096C212yesmaximal
5#29909C212yesmaximal, complemented
6#29906C212yesmaximal
7#26330C214yesmaximal, complemented
8#26288C214yesmaximal, complemented
9#25062C212yesmaximal, complemented
10#25056C212yesmaximal, complemented
11#24994C212yesmaximal, complemented
12#24992C212yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21867C211yesmaximal, characteristic
15#21820C211yesmaximal, characteristic, complemented
16#21388C211yesmaximal, characteristic
17#21365C211yesmaximal, characteristic, complemented
18C2×#529C2211yescharacteristic
19C22×C4×Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2855C2212yes—
24#2854C2212yes—
25#2637—22no—
26#2592C2212yes—
27#2584C2212yes—
28#2582C2211yescharacteristic
29#2582C2211yescharacteristic
30#2538C2211yescharacteristic
31#2538C2211yescharacteristic
32#2486—21no—
33#2413—22no—
34#2409—21no—
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4—21no—
37Q8×C4⋊C4—22no—
38Q8×C4⋊C4C2212yes—
39Q8×C4⋊C4C2212yes—
40Q8×C4⋊C4C2214yes—
41#2323—21no—
42#2323—21no—
43C2×#389—22no—
44C2×#389—21no—
45#2285C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#384C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2100C2214yes—
51#2099C2214yes—
52#2095C2214yes—
53#2093C2214yes—
54#2083C2214yes—
55#2083C2214yes—
56#2072C2214yes—
57#2071C2214yes—
58#1992C2212yes—
59#1989C2212yes—
60#1985C2214yes—
61#1984C2212yes—
62#1984C2212yes—
63#1983C2212yes—
64#1959C2212yes—
65#1948C2212yes—
66#1947C2212yes—
67#1947C2214yes—
68#1928C2214yes—
69#1927C2214yes—
70#1926C2214yes—
71#1925C2214yes—
72#1921C2214yes—
73#1920C2214yes—
74#1919C2214yes—
75#1918C2214yes—
76#1874C2211yescharacteristic
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1872C2211yescharacteristic
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1853C2212yes—
83#1852C2212yes—
84#1852C2212yes—
85#1852C2212yes—
86#1845C2212yes—
87#1834C2212yes—
88#1817C2212yes—
89#1817C2212yes—
90#1803C2212yes—
91#1802C2212yes—
92#1776C2214yes—
93C4×#103C2212yes—
94C4×#102C2212yes—
95C23×Q8D411yescharacteristic
96#529—24no—
97#529—24no—
98C2×#126—21no—
99C2×#125C2311yescharacteristic
100C2×#124C2311yescharacteristic
101C2×#123—21no—
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×#121C2311yescharacteristic
107C2×C4×Q8—24no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8—48no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D412yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C2×#117—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4D412yes—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C42—22no—
134C22×C42—21no—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#432—24no—
138#431—24no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2312yes—
154C2×#97C2312yes—
155C2×C4×C8C2312yes—
156#398—22no—
157#398—22no—
158#398—22no—
159#397—22no—
160#396—22no—
161#396—22no—
162#395—22no—
163#393—22no—
164#391—21no—
165#391—21no—
166#391C2312yes—
167#391C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—24no—
177#389—24no—
178#389—24no—
179#389—24no—
180#389—24no—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—22no—
190#387—22no—
191#386—22no—
192#385—22no—
193#384—22no—
194#384—22no—
195#384—21no—
196#384—21no—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2314yes—
201#384C2314yes—
202#384C2312yes—
203#382—21no—
204#381—22no—
205#380—21no—
206#380—21no—
207#380—21no—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C2×#95—22no—
220C2×#95—21no—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8C2×D412yes—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2411yescharacteristic
242C22×D4—21no—
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256#126—44no—
257#125—22no—
258#125—22no—
259#124—22no—
260#124—22no—
261#123—44no—
262#123—84no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267#121—22no—
268#121—22no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—28no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—24no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—416no—
281C4×Q8—48no—
282C4×Q8—48no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—44no—
286#117—44no—
287#117—82no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—48no—
353C2×C42—44no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42#4114yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—44no—
365#107—44no—
366#107—44no—
367#106—28no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370#103—24no—
371#103—42no—
372#103—44no—
373#102—24no—
374#102—42no—
375#102—44no—
376C22⋊C8—22no—
377C22⋊C8—22no—
378C22⋊C8—24no—
379#97—24no—
380C4×C8—24no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395C24—21no—
396C24C22×D411yescharacteristic
397C2×Q8—232no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8AGL(1,ℤ/8)14yes—
410C2×Q8#13714yes—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4#12314yes—
451C22×C4#12314yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2511yescharacteristic
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C42−1+414yes—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—44no—
532C42—48no—
533C42—42no—
534C42—48no—
535C42—44no—
536C42—82no—
537C42#12314yes—
538C42#12314yes—
539C42C2×#4112yes—
540C42C2×#4112yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C2×#12312yes—
555C23#57612yes—
556C23#57812yes—
557C23C23×D411yescharacteristic
558Q8—216no—
559Q8—28no—
560Q8—28no—
561Q8—42no—
562Q8—42no—
563D4—42no—
564D4—42no—
565D4—84no—
566C2×C4—216no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4C2×#12312yes—
619C2×C4C2×#12312yes—
620C2×C4C2×#12312yes—
621C2×C4C2×#12312yes—
622C2×C4C2×#12312yes—
623C2×C4C2×AGL(1,ℤ/8)14yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×#13714yes—
626C2×C4C2×#13712yes—
627C2×C4C22×#4111yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×2−1+412yes—
630C2×C4#58512yes—
631C8—44no—
632C8—42no—
633C8—42no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22#309814yes—
648C22C22×#12311yescharacteristic
649C22#348611yescharacteristic
650C22#348711yescharacteristic
651C22C22×AGL(1,ℤ/8)11yescharacteristic
652C22C22×#13711yescharacteristic
653C22C2×#57611yescharacteristic
654C22C2×#57811yescharacteristic
655C4—22no—
656C4—24no—
657C4—24no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—84no—
671C4#309214yes—
672C4#346112yes—
673C4#350712yes—
674C2—24no—
675C2—161no—
676C2#3518211yescharacteristic
677C2C2×#309211yescharacteristic
678C2C2×#309811yescharacteristic
679C2#6183711yescharacteristic
680C2#6184411yescharacteristic
681C2#6190011yescharacteristic
682C2#6190711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
3. #33100C2semidirect
5. #29909C2semidirect
7. #26330C2semidirect
8. #26288C2semidirect
9. #25062C2semidirect
10. #25056C2semidirect
11. #24994C2semidirect
12. #24992C2semidirect
13. #21982C2semidirect
15. #21820C2semidirect
17. #21365C2semidirect

Maximal subgroups

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

subgroupindex
#352472
C2×Q8×C4⋊C42
#331002
#330962
#299092
#299062
#263302
#262882
#250622
#250562
#249942
#249922
#219822
#218672
#218202
#213882
#213652