Finite Group Databasebeta

← #6623895

Subgroups — #6623895

Order 512.

1809 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2C4×#550C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#33032C212yesmaximal, complemented
5#28508C211yesmaximal, characteristic, complemented
6#28382C212yesmaximal, complemented
7#28252C211yesmaximal, characteristic, complemented
8#28201C212yesmaximal, complemented
9#27509C214yesmaximal, complemented
10#27081C214yesmaximal, complemented
11#25062C212yesmaximal, complemented
12#24188C212yesmaximal
13#23101C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22811C212yesmaximal, complemented
16#21613C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#550C2211yescharacteristic
19C2×C4×#41—21no—
20C2×#474C2212yes—
21C2×#474C2212yes—
22C2×#474C2212yes—
23C2×#460C2212yes—
24C2×#461C2211yescharacteristic
25C2×#461C2212yes—
26C2×C4×SD16C2212yes—
27#2868C2211yescharacteristic
28#2862C2212yes—
29#2582C2212yes—
30#2571—21no—
31#2570—22no—
32#2568—21no—
33#2511C2214yes—
34#2446—22no—
35#2366C2211yescharacteristic
36#2366C2212yes—
37#2365—21no—
38#2363—22no—
39#2363—21no—
40#2355—21no—
41#2355C2212yes—
42#2354C2211yescharacteristic
43#2354C2212yes—
44Q8×C4⋊C4C2211yescharacteristic
45#2341—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2212yes—
49C4×#127C2211yescharacteristic
50C4×#121—21no—
51C4×#121—22no—
52C42×Q8C2211yescharacteristic
53C42×D4C2212yes—
54C42×D4C2211yescharacteristic
55#2095C2214yes—
56#2093C2214yes—
57#2083C2214yes—
58#2055C2214yes—
59#2041C2214yes—
60#2014C2214yes—
61#1988C2212yes—
62#1984C2212yes—
63#1981C2211yescharacteristic
64#1973C2214yes—
65#1973C2214yes—
66#1957C2212yes—
67#1948C2211yescharacteristic
68#1947C2212yes—
69#1947C2212yes—
70#1928C2214yes—
71#1928C2214yes—
72#1927C2212yes—
73#1925C2214yes—
74#1925C2214yes—
75#1921C2214yes—
76#1917C2214yes—
77#1877C2214yes—
78#1877C2214yes—
79#1877C2212yes—
80#1864C2214yes—
81#1845C2212yes—
82#1842C2211yescharacteristic
83#1831C2212yes—
84#1803C2212yes—
85#1802C2212yes—
86#1802C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#103C2212yes—
95C22×#41—21no—
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98#550—24no—
99#550—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2312yes—
105C2×#127C2311yescharacteristic
106C2×#121—21no—
107C2×#121—22no—
108C4×#41—48no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—21no—
128C2×#117—22no—
129C22×C42—22no—
130C22×C42—21no—
131#474—28no—
132#474—44no—
133#474—44no—
134#461—24no—
135#461—44no—
136#460—28no—
137C4×SD16—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#400—24no—
159#400—24no—
160#398—24no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391C2312yes—
166#391C2314yes—
167#391C2314yes—
168#390—21no—
169#390—22no—
170#390—22no—
171#390—21no—
172#390—22no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#389—22no—
178#389—21no—
179#389—22no—
180#389—21no—
181#389—22no—
182#389—21no—
183#389C2314yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—24no—
188#387—24no—
189#387—24no—
190#386—22no—
191#386—22no—
192#386—22no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#382—22no—
201#382—22no—
202#382—22no—
203#380—21no—
204#380—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—21no—
227C43C2312yes—
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237#336C2314yes—
238C2×#41—28no—
239C2×#41—44no—
240C2×#41—44no—
241C22×Q8—22no—
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2412yes—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—21no—
248C22×D4C2412yes—
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—21no—
256C2×SD16—24no—
257C2×SD16—28no—
258C2×SD16—24no—
259C2×SD16—42no—
260C2×SD16—44no—
261C2×SD16—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—24no—
268C4⋊Q8—22no—
269C4⋊Q8—44no—
270C4⋊Q8—82no—
271#127—24no—
272#127—24no—
273#127—22no—
274#127#4114yes—
275#121—28no—
276#121—216no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×Q8—84no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—42no—
295C4×D4—41no—
296C4×D4—48no—
297C4×D4—48no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—84no—
301C4×D4—82no—
302C4×D4#4114yes—
303#117—44no—
304#117—48no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
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—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—24no—
351C2×C42—48no—
352C2×C42—44no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—28no—
367#107—44no—
368#107—44no—
369#106—24no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—22no—
373C4⋊C8—42no—
374C4⋊C8#4114yescomplemented
375#103—24no—
376#103—24no—
377#103—24no—
378#103—44no—
379#102—24no—
380#102—28no—
381#102—28no—
382#102—44no—
383#102—44no—
384#97—44no—
385C4×C8—24no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—21no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—22no—
403C24—21no—
404#41—432no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8#13714yes—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—81no—
436C2×D4—82no—
437C2×D4—81no—
438C2×D4—82no—
439C2×D4C2×#4114yes—
440C2×D4C2×#4112yes—
441C22×C4—28no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—81no—
465C22×C4—82no—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2511yescharacteristic
477SD16—48no—
478SD16—88no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8C2×#4114yes—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4C2×#4112yes—
520C4⋊C42−1+412yes—
521C22⋊C4—28no—
522C22⋊C4—24no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C42—24no—
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—44no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—84no—
553C42AGL(1,ℤ/8)14yes—
554C42C2×#4112yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42H(5,𝔽2)12yes—
558C23—24no—
559C23—22no—
560C23—42no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—41no—
570C23—42no—
571C23—41no—
572C23C23×D411yescharacteristic
573Q8—28no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—88no—
579D4—22no—
580D4—24no—
581D4—22no—
582D4—24no—
583D4—416no—
584D4—48no—
585D4—42no—
586D4—41no—
587D4—42no—
588D4—41no—
589D4—84no—
590D4—88no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—416no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—84no—
635C2×C4—88no—
636C2×C4—84no—
637C2×C4—88no—
638C2×C4#52512yes—
639C2×C4#52512yes—
640C2×C4#55012yes—
641C2×C4#55012yes—
642C2×C4C2×AGL(1,ℤ/8)12yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C2×#13714yes—
646C2×C4C2×#13712yes—
647C2×C4#57612yes—
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C2×C4#58511yescharacteristic
654C8—24no—
655C8—42no—
656C8—44no—
657C22—44no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—81no—
664C22—82no—
665C22—82no—
666C22—81no—
667C22—82no—
668C22—81no—
669C22—82no—
670C22—81no—
671C22C2×#52511yescharacteristic
672C22C2×#55011yescharacteristic
673C22#348911yescharacteristic
674C22C22×AGL(1,ℤ/8)11yescharacteristic
675C22C22×#13711yescharacteristic
676C22C2×#57611yescharacteristic
677C22#358611yescharacteristic
678C4—24no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—41no—
688C4—82no—
689C4—82no—
690C4—81no—
691C4—82no—
692C4—84no—
693C4#311014yes—
694C4#339914yes—
695C4#347511yescharacteristic
696C4#348211yescharacteristic
697C4#351111yescharacteristic
698C4#351411yescharacteristic
699C2—84no—
700C2—82no—
701C2#3650411yescharacteristic
702C2C2×#311011yescharacteristic
703C2#6059611yescharacteristic
704C2C2×#339911yescharacteristic
705C2#6140411yescharacteristic
706C2#6210911yescharacteristic
707C2#6214911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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. #36598C2semidirect
3. #33195C2semidirect
4. #33032C2semidirect
5. #28508C2semidirect
6. #28382C2semidirect
7. #28252C2semidirect
8. #28201C2semidirect
9. #27509C2semidirect
10. #27081C2semidirect
11. #25062C2semidirect
13. #23101C2semidirect
14. #22825C2semidirect
15. #22811C2semidirect
16. #21613C2semidirect
17. C4×#461C2semidirect
374. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×#5502
#331952
#330322
#285082
#283822
#282522
#282012
#275092
#270812
#250622
#241882
#231012
#228252
#228112
#216132
C4×#4612