Finite Group Databasebeta

← #6564151

Subgroups — #6564151

Order 512.

1859 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36493C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29927C212yesmaximal, complemented
4#29920C211yesmaximal, characteristic, complemented
5#27913C212yesmaximal, complemented
6#27909C211yesmaximal, characteristic, complemented
7#27680C211yesmaximal, characteristic, complemented
8#27677C212yesmaximal, complemented
9#26244C214yesmaximal, complemented
10#26232C212yesmaximal, complemented
11#25022C214yesmaximal, complemented
12#25016C212yesmaximal, complemented
13#23761C212yesmaximal, complemented
14#23756C211yesmaximal, characteristic, complemented
15#22728C212yesmaximal, complemented
16#22546C212yesmaximal, complemented
17#22532C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#484C2214yes—
22C2×#481C2214yes—
23C2×#480C2214yes—
24C2×#446C2212yes—
25#2856C2211yescharacteristic
26#2850C2212yes—
27#2541C2211yescharacteristic
28#2540C2212yes—
29#2538C2211yescharacteristic
30#2535C2212yes—
31#2471—22no—
32#2456—22no—
33#2448—21no—
34#2431—21no—
35#2366—21no—
36#2363—21no—
37#2355C2211yescharacteristic
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2212yes—
40D4×C4⋊C4—21no—
41D4×C4⋊C4—22no—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2214yes—
44#2317—21no—
45#2317—22no—
46C4×C22⋊Q8C2211yescharacteristic
47C4×#121C2212yes—
48#2285C2211yescharacteristic
49#2285C2212yes—
50C2×#384C2211yescharacteristic
51C2×#384C2212yes—
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2096C2211yescharacteristic
55#2096C2212yes—
56#2093C2212yes—
57#2093C2212yes—
58#2093C2212yes—
59#2084C2211yescharacteristic
60#2077C2212yes—
61#2076C2214yes—
62#2072C2211yescharacteristic
63#2071C2212yes—
64#2036C2214yes—
65#2035C2212yes—
66#2034C2214yes—
67#2034C2214yes—
68#2033C2214yes—
69#1983C2214yes—
70#1982C2212yes—
71#1959C2214yes—
72#1957C2212yes—
73#1955C2214yes—
74#1954C2212yes—
75#1949C2214yes—
76#1948C2212yes—
77#1947C2214yes—
78#1947C2212yes—
79#1874C2214yes—
80#1872C2212yes—
81#1853C2212yes—
82#1852C2214yes—
83#1847C2214yes—
84#1845C2212yes—
85#1803C2212yes—
86#1802C2214yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1776C2214yes—
90C4×#106C2212yes—
91#1762C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41D411yescharacteristic
96#559—28no—
97C2×#127—21no—
98C2×#127C2311yescharacteristic
99C2×#124—21no—
100C2×#124C2312yes—
101C2×#123—22no—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#121C2312yes—
109#520—48no—
110C4×#41—28no—
111C2×C4×Q8—22no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—24no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D412yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C2×#117—21no—
126C2×#117D411yescharacteristic
127C2×#117D412yes—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2314yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#484—48no—
137#481—48no—
138#480—48no—
139#446—28no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2312yes—
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161#399—22no—
162#397—22no—
163#396—22no—
164#392—22no—
165#392—22no—
166#392—22no—
167#391—21no—
168#391C2311yescharacteristic
169#390—21no—
170#390—21no—
171#390—22no—
172#390—22no—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#389—22no—
177#389—24no—
178#389—22no—
179#389—24no—
180#389—21no—
181#389C2311yescharacteristic
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—21no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#385—21no—
193#385—24no—
194#385—24no—
195#385—24no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384—22no—
200#384C2314yes—
201#384C2312yes—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2311yescharacteristic
206#382—22no—
207#382—21no—
208#382C2312yes—
209#380—22no—
210#380—21no—
211#380C2312yes—
212#380C2314yes—
213#380C2311yescharacteristic
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2314yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—24no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2×D412yes—
246C22×D4C2×D411yescharacteristic
247C22×D4C2412yes—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2412yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260#127—24no—
261#127—44no—
262#124—28no—
263#124—44no—
264#123—48no—
265C22⋊Q8—22no—
266C22⋊Q8—41no—
267C22⋊Q8—44no—
268#121—24no—
269#121—42no—
270#121—44no—
271#121—44no—
272#121—44no—
273#121—82no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—48no—
277C4×Q8#4118yes—
278C4×D4—24no—
279C4×D4—28no—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—416no—
284C4×D4—48no—
285C4×D4—44no—
286C4×D4—84no—
287#117—24no—
288#117—28no—
289#117—48no—
290#117—44no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
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—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—84no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—28no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358#107—28no—
359#107—44no—
360#107—44no—
361#106—24no—
362#106—24no—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8—28no—
366#103—24no—
367#103—42no—
368#103—44no—
369#102—28no—
370#102—44no—
371#102—44no—
372#102—44no—
373C22⋊C8—24no—
374C22⋊C8—28no—
375C22⋊C8—42no—
376C22⋊C8—44no—
377#97—28no—
378C4×C8—24no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—24no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—24no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392C24—22no—
393C24C22×D411yescharacteristic
394C24C22×D412yes—
395#41—232no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8#12314yes—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4#13714yes—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—84no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4AGL(1,ℤ/8)14yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2511yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8H(5,𝔽2)14yes—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4#12314yes—
514C4⋊C4#12318yes—
515C4⋊C42−1+412yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—81no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C42—24no—
534C42—22no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—48no—
539C42—42no—
540C42—44no—
541C42—42no—
542C42—82no—
543C42C2×#4114yes—
544C42C2×#4118yes—
545C23—21no—
546C23—22no—
547C23—21no—
548C23—22no—
549C23—21no—
550C23—22no—
551C23—22no—
552C23—21no—
553C23—22no—
554C23—21no—
555C23—22no—
556C23—21no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23C2×AGL(1,ℤ/8)12yes—
565C23C2×#13714yes—
566C23C23×D411yescharacteristic
567Q8—28no—
568Q8—42no—
569Q8—42no—
570D4—216no—
571D4—28no—
572D4—44no—
573D4—44no—
574D4—88no—
575D4—84no—
576D4—162no—
577D4—164no—
578D4—164no—
579C2×C4—216no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—164no—
628C2×C4—164no—
629C2×C4—168no—
630C2×C4C2×#12314yes—
631C2×C4C2×#12312yes—
632C2×C4C2×#12314yes—
633C2×C4C2×#12312yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4C2×#13712yes—
637C2×C4#57612yes—
638C2×C4C22×#4112yes—
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4C2×2−1+411yescharacteristic
641C2×C4#58511yescharacteristic
642C8—24no—
643C8—42no—
644C8—44no—
645C22—22no—
646C22—24no—
647C22—22no—
648C22—24no—
649C22—22no—
650C22—24no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22—41no—
655C22—42no—
656C22—41no—
657C22—42no—
658C22—41no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22C22×#12311yescharacteristic
667C22#348911yescharacteristic
668C22#349811yescharacteristic
669C22C22×AGL(1,ℤ/8)11yescharacteristic
670C22C22×#13711yescharacteristic
671C22C2×#57611yescharacteristic
672C22#358611yescharacteristic
673C4—22no—
674C4—24no—
675C4—22no—
676C4—24no—
677C4—22no—
678C4—42no—
679C4—41no—
680C4—42no—
681C4—81no—
682C4—82no—
683C4—81no—
684C4—84no—
685C4—82no—
686C4—82no—
687C4—81no—
688C4#309214yes—
689C4#351712yes—
690C4#352212yes—
691C2—42no—
692C2—44no—
693C2—162no—
694C2#3600511yescharacteristic
695C2C2×#309211yescharacteristic
696C2#6052111yescharacteristic
697C2#6211111yescharacteristic
698C2#6212711yescharacteristic
699C2#6242311yescharacteristic
700C2#6244811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36493C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34183C2semidirect
3. #29927C2semidirect
4. #29920C2semidirect
5. #27913C2semidirect
6. #27909C2semidirect
7. #27680C2semidirect
8. #27677C2semidirect
9. #26244C2semidirect
10. #26232C2semidirect
11. #25022C2semidirect
12. #25016C2semidirect
13. #23761C2semidirect
14. #23756C2semidirect
15. #22728C2semidirect
16. #22546C2semidirect
17. #22532C2semidirect

Maximal subgroups

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

subgroupindex
#364932
#341832
#299272
#299202
#279132
#279092
#276802
#276772
#262442
#262322
#250222
#250162
#237612
#237562
#227282
#225462
#225322