Finite Group Databasebeta

← #6852435

Subgroups — #6852435

Order 512.

1831 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
1#36545C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic
3#32953C211yesmaximal, characteristic
4#32654C214yesmaximal, complemented
5#32572C214yesmaximal, complemented
6#29956C212yesmaximal
7#29298C212yesmaximal, complemented
8#29244C212yesmaximal, complemented
9#28264C212yesmaximal, complemented
10#27948C211yesmaximal, characteristic, complemented
11#27681C211yesmaximal, characteristic, complemented
12#25469C212yesmaximal, complemented
13#25059C212yesmaximal, complemented
14#24328C212yesmaximal, complemented
15Q8×C22⋊C8C211yesmaximal, characteristic, complemented
16#22043C212yesmaximal
17C4×#450C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic, complemented
19C22×C22⋊Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#474C2212yes—
23C2×#450C2211yescharacteristic
24C2×C8×Q8C2212yescomplemented
25C2×C4×SD16C2212yes—
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2584C2212yescomplemented
29#2582C2211yescharacteristic, complemented
30#2581C2212yescomplemented
31#2546C2211yescharacteristic, complemented
32#2538—21no—
33#2538—21no—
34#2538—22no—
35#2538C2212yes—
36#2538C2214yes—
37#2538C2212yes—
38#2473—22no—
39#2462—22no—
40#2446—21no—
41#2434—21no—
42C2×#396—22no—
43#2365—21no—
44#2360—21no—
45#2355C2211yescharacteristic, complemented
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#390C2212yes—
49C2×#389C2212yes—
50C2×#384C2211yescharacteristic
51C2×#380—21no—
52#2100C2212yescomplemented
53#2095C2212yescomplemented
54#2092C2214yes—
55#2086C2214yescomplemented
56#2085C2214yescomplemented
57#2084C2214yescomplemented
58#2067C2214yes—
59#2066C2214yescomplemented
60#2065C2214yes—
61#2059C2214yescomplemented
62#2025C2214yes—
63#1990C2212yescomplemented
64#1984C2212yescomplemented
65#1984C2214yescomplemented
66#1968C2214yes—
67#1947C2212yescomplemented
68#1942C2214yescomplemented
69#1941C2214yescomplemented
70#1934C2214yes—
71#1926C2214yes—
72#1920C2214yescomplemented
73#1880C2214yes—
74#1877C2212yes—
75#1872C2212yes—
76#1872C2211yescharacteristic
77#1864C2212yes—
78#1863C2214yescomplemented
79#1862C2214yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1852C2212yes—
84#1852C2212yes—
85#1844C2212yescomplemented
86#1803C2212yescomplemented
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1786C2211yescharacteristic
91#1786C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99#557—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—22no—
104C2×#125C2312yes—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—48no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C4×Q8—21no—
112C2×C4×Q8—44no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D412yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—44no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D412yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#474—44no—
139#450Q818yescomplemented
140C8×Q8—44no—
141C4×SD16—44no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2312yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2314yes—
161#400—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#396—24no—
166#396—24no—
167#396—24no—
168#396—24no—
169#395—22no—
170#395—22no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391C2312yes—
176#391C2311yescharacteristic
177#390—22no—
178#390—22no—
179#390C2314yes—
180#390C2314yes—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—22no—
186#389—22no—
187#389C2314yes—
188#389C2312yes—
189#389C2314yes—
190#389C2312yes—
191#389C2312yes—
192#389C2311yescharacteristic
193#389C2312yes—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—21no—
201#385—22no—
202#385—22no—
203#385—22no—
204#385—21no—
205#384—21no—
206#384—21no—
207#384C2311yescharacteristic
208#384C2312yes—
209#384C2312yes—
210#382C2312yes—
211#380—21no—
212#380—22no—
213#380—22no—
214#380—24no—
215#380C2311yescharacteristic
216#380C2312yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C2×#95—22no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236C25—21no—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—21no—
240C22×Q8—42no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2×D412yes—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—42no—
248C22×D4—42no—
249C22×D4C2×Q814yes—
250C22×D4C2412yes—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C2×SD16—42no—
264C2×SD16—42no—
265C2×SD16C2×Q818yescomplemented
266C22×C8C2412yes—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272#125—44no—
273#125—48no—
274C22⋊Q8—22no—
275C22⋊Q8—48no—
276C22⋊Q8—416no—
277C22⋊Q8—48no—
278C22⋊Q8—44no—
279C22⋊Q8C2×Q814yes—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—42no—
286C4×Q8—42no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—44no—
294C4×D4—44no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4—84no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363#107—44no—
364#107—44no—
365#106—44no—
366#106—44no—
367#106—42no—
368C4⋊C8—24no—
369C4⋊C8—44no—
370#103—44no—
371#103—42no—
372#102—44no—
373#102—42no—
374#102—42no—
375#102—44no—
376#102C2×Q818yescomplemented
377C22⋊C8—24no—
378C22⋊C8—22no—
379C22⋊C8C2×Q814yes—
380C4×C8—24no—
381C4×C8—44no—
382#95—24no—
383#95—24no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399C24—24no—
400C24—22no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24C22×D411yescharacteristic
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—42no—
419C2×Q8—416no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8AGL(1,ℤ/8)14yes—
424C2×Q8C22×Q812yes—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—48no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—48no—
435C2×D4C22×Q814yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—816no—
470C22×C4#13514yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×Q812yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483SD16—48no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8C22×Q814yes—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—81no—
526C4⋊C4—81no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4C22×Q812yes—
531C22⋊C4—24no—
532C22⋊C4—44no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C42—22no—
541C42—22no—
542C42—24no—
543C42—22no—
544C42—24no—
545C42—42no—
546C42—42no—
547C42—44no—
548C42—42no—
549C42—42no—
550C42—82no—
551C42H(5,𝔽2)14yes—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—24no—
557C23—48no—
558C23—48no—
559C23—42no—
560C23—48no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23D4×Q812yes—
571C23C2×#13512yes—
572C23#57712yes—
573C23C23×D411yescharacteristic
574Q8—28no—
575Q8—24no—
576Q8—48no—
577Q8—44no—
578Q8—42no—
579Q8—84no—
580D4—24no—
581D4—24no—
582D4—48no—
583D4—42no—
584D4—42no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—44no—
634C2×C4—88no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4—88no—
639C2×C4#53412yes—
640C2×C4#53412yes—
641C2×C4D4×Q812yes—
642C2×C4C2×#13512yes—
643C2×C4C2×#13512yes—
644C2×C4C2×AGL(1,ℤ/8)14yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4C23×Q811yescharacteristic
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4C2×2−1+411yescharacteristic
649C2×C4#58512yes—
650C8—24no—
651C8—42no—
652C8—44no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—816no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22#338114yes—
668C22C2×#53411yescharacteristic
669C22C2×D4×Q811yescharacteristic
670C22#348711yescharacteristic
671C22C22×#13511yescharacteristic
672C22C22×AGL(1,ℤ/8)11yescharacteristic
673C22C2×#57711yescharacteristic
674C22#358611yescharacteristic
675C4—22no—
676C4—24no—
677C4—22no—
678C4—22no—
679C4—42no—
680C4—41no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—82no—
686C4—82no—
687C4—81no—
688C4—82no—
689C4—81no—
690C4—84no—
691C4#318814yes—
692C4#350112yes—
693C4#354112yes—
694C2—24no—
695C2—84no—
696C2#3959511yescharacteristic
697C2C2×#318811yescharacteristic
698C2#6076311yescharacteristic
699C2C2×#338111yescharacteristic
700C2Q8×AGL(1,ℤ/8)11yescharacteristic
701C2#6188211yescharacteristic
702C2#6190111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. #36545C2semidirect
4. #32654C2semidirect
5. #32572C2semidirect
7. #29298C2semidirect
8. #29244C2semidirect
9. #28264C2semidirect
10. #27948C2semidirect
11. #27681C2semidirect
12. #25469C2semidirect
13. #25059C2semidirect
14. #24328C2semidirect
15. Q8×C22⋊C8C2semidirect
18. C2×#557C22semidirect
24. C2×C8×Q8C22semidirect
28. #2584C22semidirect
29. #2582C22semidirect
30. #2581C22semidirect
31. #2546C22semidirect
45. #2355C22semidirect
52. #2100C22semidirect
53. #2095C22semidirect
55. #2086C22semidirect
56. #2085C22semidirect
57. #2084C22semidirect
59. #2066C22semidirect
61. #2059C22semidirect
63. #1990C22semidirect
64. #1984C22semidirect
65. #1984C22semidirect
67. #1947C22semidirect
68. #1942C22semidirect
69. #1941C22semidirect
72. #1920C22semidirect
78. #1863C22semidirect
85. #1844C22semidirect
86. #1803C22semidirect
139. #450Q8semidirect
265. C2×SD16C2×Q8semidirect
376. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365452
C2×#25382
#329532
#326542
#325722
#299562
#292982
#292442
#282642
#279482
#276812
#254692
#250592
#243282
Q8×C22⋊C82
#220432
C4×#4502