Finite Group Databasebeta

← #6395706

Subgroups — #6395706

Order 512.

2364 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3217C211yesmaximal, characteristic
2#34210C211yesmaximal, characteristic, complemented
3#34151C211yesmaximal, characteristic, complemented
4#23231C214yesmaximal, complemented
5#23195C214yesmaximal, complemented
6#22880C214yesmaximal, complemented
7#22864C214yesmaximal, complemented
8#22151C212yesmaximal, complemented
9#22147C212yesmaximal, complemented
10#21890C211yesmaximal, characteristic, complemented
11#21505C212yesmaximal, complemented
12C4×#459C212yesmaximal, complemented
13#21364C211yesmaximal, characteristic, complemented
14#21260C211yesmaximal, characteristic, complemented
15#20549C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C2×#529C2211yescharacteristic
18C2×#518—21no—
19#3217C4116yescomplemented
20C2×#496C2212yes—
21C2×#490C2212yes—
22C2×#460C2212yes—
23C2×#459C2212yes—
24C2×#450C2212yes—
25C2×#447C2212yes—
26C2×#440C2214yescomplemented
27C2×C4×D8C2214yescomplemented
28C2×#432C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#418C2214yes—
31#2374—22no—
32#2373—22no—
33#2370C2212yescomplemented
34#2367C2212yescomplemented
35#2366—22no—
36#2366C2214yes—
37#2363C2212yes—
38#2362C2212yes—
39D4×C22⋊C4—22no—
40D4×C22⋊C4C2214yes—
41#2328—21no—
42#2328—21no—
43#2309—21no—
44C4×#127C2212yes—
45C4×#127C2212yes—
46#2298—21no—
47#2295C2211yescharacteristic, complemented
48#2293—21no—
49#2293C2211yescharacteristic, complemented
50#2290—21no—
51#2285C2211yescharacteristic
52C42×Q8C2211yescharacteristic, complemented
53C42×D4C2211yescharacteristic, complemented
54C4×#117—21no—
55C4×#117C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#1989C2212yescomplemented
58#1983C2212yescomplemented
59#1982C2212yes—
60#1982C2212yes—
61#1982C2214yes—
62#1972C2214yescomplemented
63#1968C2214yescomplemented
64#1948C2212yescomplemented
65#1947C2212yescomplemented
66#1936C2214yescomplemented
67#1934C2214yescomplemented
68#1925C2214yescomplemented
69#1925C2214yescomplemented
70#1895C2212yes—
71#1894C2212yes—
72#1880C2214yescomplemented
73#1879C2214yescomplemented
74#1879C2214yescomplemented
75#1877C2214yescomplemented
76#1864C2214yescomplemented
77#1864C2214yescomplemented
78#1854C2214yes—
79#1803C2212yescomplemented
80#1802C2212yescomplemented
81#1776C2214yescomplemented
82#1771C2214yescomplemented
83#1767C2212yescomplemented
84C4×C4⋊C8C2212yescomplemented
85#1762C2214yescomplemented
86C4×#102C2214yescomplemented
87#1753C2214yes—
88C23×D4—21no—
89C23×D4—21no—
90C23×D4C2312yes—
91C22×SD16C2314yes—
92C22×D8C2314yes—
93C22×M16C2312yes—
94C2×#127—21no—
95C2×#127—21no—
96C2×#127—24no—
97C2×#127C2×C418yescomplemented
98C2×#127C2312yes—
99C2×#127C2312yes—
100#529—24no—
101#529C2×C418yescomplemented
102C2×#126—21no—
103C2×#125C2311yescharacteristic
104C2×#124C2311yescharacteristic
105C2×#123—21no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×#121C2311yescharacteristic
108#518—48no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—21no—
112C2×C4×D4—24no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2314yes—
120C2×C4×D4C2314yes—
121C2×#117—21no—
122C2×#117—21no—
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C2×#117D411yescharacteristic
126C2×#117C2312yes—
127C22×C22⋊C4—24no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130#496C2×C4116yescomplemented
131#490C2×C4116yescomplemented
132#460C2×C4116yescomplemented
133#459C2×C4116yescomplemented
134#450C2×C4116yescomplemented
135#447C2×C4116yescomplemented
136#440—48no—
137C4×D8—48no—
138#432C2×C418yescomplemented
139C2×#107C2314yes—
140C2×#106C2314yes—
141#423—24no—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145#418—48no—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×#97C2314yes—
156C2×#97C2312yes—
157C2×C4×C8C2314yes—
158#391C2312yes—
159#390—24no—
160#390—22no—
161#390—22no—
162#390—22no—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2314yes—
167#390C2312yes—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#386—22no—
182#386—24no—
183#386—24no—
184#386—24no—
185#386—24no—
186#385—22no—
187#384C2312yes—
188#382—21no—
189#380—21no—
190#380C2312yes—
191#379—24no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376—21no—
216#376—21no—
217#376D411yescharacteristic
218#376D411yescharacteristic
219C43—21no—
220C43—21no—
221C43C2311yescharacteristic
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2314yes—
228#336C2314yes—
229C25—22no—
230C22×Q8C2411yescharacteristic
231C22×D4—21no—
232C22×D4—28no—
233C22×D4—28no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—24no—
240C22×D4—44no—
241C22×D4—44no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4—84no—
245C22×D4C22×C418yes—
246C22×D4C2411yescharacteristic
247C22×D4C2414yes—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16C22×C4116yes—
258C2×D8—44no—
259C2×D8—44no—
260C2×D8C22×C4116yes—
261C2×M16—24no—
262C2×M16—44no—
263C22×C8C2414yes—
264C22×C8C2414yes—
265#127—28no—
266#127—28no—
267#127—24no—
268#127—24no—
269#127C22×C418yes—
270#127C22×C418yes—
271#126—28no—
272#125—22no—
273#125C22×C414yes—
274#124—22no—
275#124C22×C414yes—
276#123—28no—
277C22⋊Q8—22no—
278C22⋊Q8C22×C414yes—
279#121—22no—
280#121C22×C414yes—
281C4×Q8—22no—
282C4×Q8—24no—
283C4×Q8—42no—
284C4×Q8C22×C414yes—
285C4×D4—28no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—42no—
289C4×D4—48no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—416no—
295C4×D4—48no—
296C4×D4—84no—
297C4×D4—84no—
298C4×D4C22×C414yes—
299#117—28no—
300#117—24no—
301#117—24no—
302#117—42no—
303#117—44no—
304#117—42no—
305#117—44no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—416no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42C22×C414yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42#4114yes—
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—28no—
366#106—44no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8C22×C418yes—
370#103—44no—
371#102—28no—
372#102—28no—
373#102—44no—
374#102—44no—
375#102—44no—
376#102C22×C4116yes—
377#102C22×C4116yes—
378C22⋊C8C22×C418yes—
379#97—44no—
380#97C22×C418yes—
381C4×C8—44no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—21no—
398C24—24no—
399C24—28no—
400C24—44no—
401C24—44no—
402C24—84no—
403C24—84no—
404C24C22×D411yescharacteristic
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8C23×C412yes—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—21no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—416no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—432no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—816no—
436C2×D4—82no—
437C2×D4C23×C418yes—
438C2×D4C23×C412yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—82no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4C4×D414yes—
463C22×C4C4×D414yes—
464C22×C4C23×C412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2511yescharacteristic
476SD16—416no—
477D8—416no—
478M16—84no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8C23×C418yes—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—82no—
503C4⋊C4C23×C412yes—
504C4⋊C4C23×C412yes—
505C4⋊C4H(5,𝔽2)14yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—44no—
536C42—44no—
537C42—44no—
538C42—42no—
539C42—42no—
540C42—82no—
541C42—82no—
542C42C4×D414yes—
543C42C4×D414yes—
544C42AGL(1,ℤ/8)14yes—
545C42C23×C412yes—
546C42C23×C412yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C23—22no—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—28no—
555C23—416no—
556C23—44no—
557C23—44no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—41no—
564C23—41no—
565C23—44no—
566C23—88no—
567C23—88no—
568C23—168no—
569C23—168no—
570C23C2×C4×D412yes—
571C23C23×D411yescharacteristic
572Q8—24no—
573Q8—41no—
574Q8—41no—
575D4—24no—
576D4—28no—
577D4—28no—
578D4—416no—
579D4—416no—
580D4—416no—
581D4—41no—
582D4—44no—
583D4—41no—
584D4—44no—
585D4—88no—
586D4—88no—
587D4—84no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
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—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—88no—
627C2×C4—168no—
628C2×C4—168no—
629C2×C4C2×C4×D412yes—
630C2×C4C2×C4×D412yes—
631C2×C4C2×C4×D412yes—
632C2×C4C2×C4×D412yes—
633C2×C4C2×C4×D412yes—
634C2×C4C2×AGL(1,ℤ/8)14yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4#57512yes—
637C2×C4C24×C411yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4#58512yes—
641C8—28no—
642C8—44no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—42no—
647C22—42no—
648C22—48no—
649C22—42no—
650C22—48no—
651C22—42no—
652C22—42no—
653C22—48no—
654C22—82no—
655C22—84no—
656C22—84no—
657C22—82no—
658C22—84no—
659C22—84no—
660C22—1616no—
661C22C22×C4×D411yescharacteristic
662C22#348812yes—
663C22C22×AGL(1,ℤ/8)11yescharacteristic
664C22C2×#57511yescharacteristic
665C22#358512yes—
666C4—24no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—42no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—84no—
679C4—82no—
680C4C4×AGL(1,ℤ/8)14yes—
681C4C4×H(5,𝔽2)12yes—
682C4#350811yescharacteristic
683C4#351011yescharacteristic
684C2—42no—
685C2—82no—
686C2—88no—
687C2#3426911yescharacteristic
688C2C2×C4×AGL(1,ℤ/8)11yescharacteristic
689C2#6016811yescharacteristic
690C2#6195912yes—
691C2#6197012yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3217C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34210C2semidirect
3. #34151C2semidirect
4. #23231C2semidirect
5. #23195C2semidirect
6. #22880C2semidirect
7. #22864C2semidirect
8. #22151C2semidirect
9. #22147C2semidirect
10. #21890C2semidirect
11. #21505C2semidirect
12. C4×#459C2semidirect
13. #21364C2semidirect
14. #21260C2semidirect
15. #20549C2semidirect
19. #3217C4semidirect
26. C2×#440C22semidirect
27. C2×C4×D8C22semidirect
33. #2370C22semidirect
34. #2367C22semidirect
47. #2295C22semidirect
49. #2293C22semidirect
52. C42×Q8C22semidirect
53. C42×D4C22semidirect
57. #1989C22semidirect
58. #1983C22semidirect
62. #1972C22semidirect
63. #1968C22semidirect
64. #1948C22semidirect
65. #1947C22semidirect
66. #1936C22semidirect
67. #1934C22semidirect
68. #1925C22semidirect
69. #1925C22semidirect
72. #1880C22semidirect
73. #1879C22semidirect
74. #1879C22semidirect
75. #1877C22semidirect
76. #1864C22semidirect
77. #1864C22semidirect
79. #1803C22semidirect
80. #1802C22semidirect
81. #1776C22semidirect
82. #1771C22semidirect
83. #1767C22semidirect
84. C4×C4⋊C8C22semidirect
85. #1762C22semidirect
86. C4×#102C22semidirect
97. C2×#127C2×C4semidirect
101. #529C2×C4semidirect
130. #496C2×C4semidirect
131. #490C2×C4semidirect
132. #460C2×C4semidirect
133. #459C2×C4semidirect
134. #450C2×C4semidirect
135. #447C2×C4semidirect
138. #432C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32172
#342102
#341512
#232312
#231952
#228802
#228642
#221512
#221472
#218902
#215052
C4×#4592
#213642
#212602
#205492