Finite Group Databasebeta

← #6623447

Subgroups — #6623447

Order 512.

1827 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2C4×#550C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#33072C212yesmaximal, complemented
5#29139C211yesmaximal, characteristic, complemented
6#29130C212yesmaximal, complemented
7#28252C211yesmaximal, characteristic, complemented
8#28157C212yesmaximal, complemented
9#27445C214yesmaximal, complemented
10#26912C214yesmaximal, complemented
11#24996C212yesmaximal, complemented
12#24188C212yesmaximal
13#23003C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22759C212yesmaximal, complemented
16#21504C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#550C2211yescharacteristic
19C2×C4×#41—21no—
20C2×#474C2212yes—
21C2×#460C2212yes—
22C2×#461C2212yes—
23C2×#461C2211yescharacteristic
24C2×#440C2214yes—
25C2×C4×SD16C2212yes—
26#2868C2211yescharacteristic
27#2859C2212yes—
28#2582C2212yes—
29#2567—21no—
30#2552—21no—
31#2550—22no—
32#2507—22no—
33#2451C2214yes—
34#2369—21no—
35#2366C2212yes—
36#2366C2211yescharacteristic
37#2363—22no—
38#2363—21no—
39#2355—21no—
40#2354C2212yes—
41#2354C2211yescharacteristic
42Q8×C4⋊C4C2211yescharacteristic
43Q8×C4⋊C4C2212yes—
44#2341—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#127C2212yes—
49C4×#121—21no—
50C4×#121—22no—
51C42×Q8C2211yescharacteristic
52C42×D4C2212yes—
53C42×D4C2211yescharacteristic
54#2100C2214yes—
55#2072C2214yes—
56#2055C2214yes—
57#2041C2214yes—
58#2023C2214yes—
59#2014C2214yes—
60#1988C2212yes—
61#1984C2212yes—
62#1981C2211yescharacteristic
63#1957C2212yes—
64#1948C2211yescharacteristic
65#1947C2212yes—
66#1947C2212yes—
67#1936C2214yes—
68#1936C2214yes—
69#1928C2214yes—
70#1927C2212yes—
71#1921C2214yes—
72#1919C2214yes—
73#1917C2214yes—
74#1880C2214yes—
75#1880C2214yes—
76#1877C2214yes—
77#1877C2212yes—
78#1864C2214yes—
79#1862C2214yes—
80#1842C2211yescharacteristic
81#1834C2212yes—
82#1831C2212yes—
83#1803C2212yes—
84#1802C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2211yescharacteristic
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95C22×#41—21no—
96C22×SD16C2314yes—
97C22×SD16C2312yes—
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×#125—22no—
107C2×#121—21no—
108C2×#121—22no—
109C4×#41—48no—
110C2×C4×Q8—21no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—22no—
130C22×C42—22no—
131C22×C42—21no—
132#474—44no—
133#461—28no—
134#461—24no—
135#460—28no—
136#440—48no—
137C4×SD16—28no—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#398—24no—
157#398—24no—
158#395—24no—
159#391—22no—
160#391—24no—
161#391C2312yes—
162#391C2314yes—
163#390—21no—
164#390—22no—
165#390—22no—
166#390—21no—
167#390—22no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—22no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—24no—
182#389C2314yes—
183#389C2314yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—24no—
188#386—22no—
189#386—22no—
190#386—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382—22no—
200#380—21no—
201#380—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C43C2312yes—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—28no—
238C2×#41—44no—
239C2×#41—44no—
240C22×Q8—22no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4C2411yescharacteristic
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C2×SD16—28no—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—28no—
259C2×SD16—42no—
260C2×SD16—44no—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C4⋊Q8—22no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8—44no—
268C4⋊Q8—82no—
269C4⋊Q8#4114yescomplemented
270#127—24no—
271#127—24no—
272#127—22no—
273#127#4114yes—
274#125—84no—
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—44no—
285C4×Q8—44no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—24no—
295C4×D4—42no—
296C4×D4—41no—
297C4×D4—48no—
298C4×D4—48no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—82no—
302C4×D4—84no—
303C4×D4#4114yes—
304#117—44no—
305#117—48no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—24no—
354C2×C42—48no—
355C2×C42—44no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2412yes—
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107—28no—
370#106—24no—
371C4⋊C8—24no—
372C4⋊C8—22no—
373C4⋊C8—24no—
374C4⋊C8—22no—
375C4⋊C8—42no—
376#103—28no—
377#103—24no—
378#103—24no—
379#103—24no—
380#102—24no—
381#102—28no—
382#102—44no—
383#102—44no—
384#102—44no—
385#97—44no—
386C4×C8—24no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—21no—
392#95—24no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—24no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—22no—
402C24—21no—
403#41—432no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—44no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8#13714yes—
415C2×Q8C2×#4114yes—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—44no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—81no—
435C2×D4—82no—
436C2×D4—81no—
437C2×D4—82no—
438C2×D4C2×#4114yes—
439C2×D4C2×#4112yes—
440C22×C4—28no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—81no—
464C22×C4—82no—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476SD16—48no—
477SD16—416no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4C2×#4112yes—
521C4⋊C4H(5,𝔽2)12yes—
522C22⋊C4—28no—
523C22⋊C4—24no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C42—24no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—21no—
543C42—22no—
544C42—21no—
545C42—42no—
546C42—41no—
547C42—44no—
548C42—44no—
549C42—44no—
550C42—42no—
551C42—42no—
552C42—42no—
553C42—84no—
554C42AGL(1,ℤ/8)14yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C422−1+412yes—
559C23—24no—
560C23—22no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—41no—
566C23—42no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—42no—
572C23—41no—
573C23C23×D411yescharacteristic
574Q8—28no—
575Q8—24no—
576Q8—24no—
577Q8—42no—
578Q8—42no—
579Q8—164no—
580D4—22no—
581D4—24no—
582D4—22no—
583D4—24no—
584D4—416no—
585D4—48no—
586D4—42no—
587D4—41no—
588D4—42no—
589D4—41no—
590D4—84no—
591D4—88no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—416no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4—84no—
639C2×C4#52512yes—
640C2×C4#52512yes—
641C2×C4#54812yes—
642C2×C4#54812yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4C2×#13714yes—
647C2×C4C2×#13712yes—
648C2×C4#57712yes—
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×2−1+411yescharacteristic
654C2×C4#58511yescharacteristic
655C8—42no—
656C8—44no—
657C8—42no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—44no—
663C22—42no—
664C22—81no—
665C22—82no—
666C22—81no—
667C22—82no—
668C22—81no—
669C22—82no—
670C22—82no—
671C22—81no—
672C22C2×#52511yescharacteristic
673C22C2×#54811yescharacteristic
674C22#348811yescharacteristic
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22C22×#13711yescharacteristic
677C22C2×#57711yescharacteristic
678C22#358511yescharacteristic
679C4—22no—
680C4—24no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—41no—
687C4—42no—
688C4—44no—
689C4—42no—
690C4—84no—
691C4—82no—
692C4—81no—
693C4—82no—
694C4#311014yes—
695C4#336814yes—
696C4#347511yescharacteristic
697C4#348111yescharacteristic
698C4#351111yescharacteristic
699C4#351511yescharacteristic
700C2—84no—
701C2—82no—
702C2#3649211yescharacteristic
703C2C2×#311011yescharacteristic
704C2#6059711yescharacteristic
705C2C2×#336811yescharacteristic
706C2#6131911yescharacteristic
707C2#6196111yescharacteristic
708C2#6200211yescharacteristic
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. #36563C2semidirect
3. #33195C2semidirect
4. #33072C2semidirect
5. #29139C2semidirect
6. #29130C2semidirect
7. #28252C2semidirect
8. #28157C2semidirect
9. #27445C2semidirect
10. #26912C2semidirect
11. #24996C2semidirect
13. #23003C2semidirect
14. #22825C2semidirect
15. #22759C2semidirect
16. #21504C2semidirect
17. C4×#461C2semidirect
269. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
#365632
C4×#5502
#331952
#330722
#291392
#291302
#282522
#281572
#274452
#269122
#249962
#241882
#230032
#228252
#227592
#215042
C4×#4612