Finite Group Databasebeta

← #6484457

Subgroups — #6484457

Order 512.

1798 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2C4×#535C211yesmaximal, characteristic
3#32955C212yesmaximal, complemented
4#32953C212yesmaximal, complemented
5#29964C212yesmaximal, complemented
6#29962C212yesmaximal, complemented
7#29268C212yesmaximal, complemented
8#29265C212yesmaximal, complemented
9#29204C212yesmaximal, complemented
10#29200C212yesmaximal, complemented
11#29149C211yesmaximal, characteristic, complemented
12#29140C211yesmaximal, characteristic, complemented
13#28520C211yesmaximal, characteristic, complemented
14#28508C211yesmaximal, characteristic, complemented
15#28281C212yesmaximal, complemented
16#28279C212yesmaximal, complemented
17#21991C212yesmaximal, complemented
18#21813C212yesmaximal
19#20812C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#475C2212yescomplemented
24C2×#474C2212yes—
25C2×#474C2212yes—
26C2×#474C2212yescomplemented
27C2×#433C2212yes—
28#2868C2211yescharacteristic, complemented
29#2859C2211yescharacteristic, complemented
30#2857C2212yes—
31#2647C2212yescomplemented
32#2629C2212yes—
33#2507—22no—
34#2497—22no—
35#2446—22no—
36#2429—22no—
37#2330C2212yescomplemented
38#2328C2211yescharacteristic
39#2328C2211yescharacteristic
40#2326C2211yescharacteristic
41#2326C2211yescharacteristic
42#2324—22no—
43#2321—22no—
44#2317C2211yescharacteristic
45#2314C2212yes—
46#2311C2211yescharacteristic
47#2309C2211yescharacteristic
48C4×C4⋊Q8C2212yescomplemented
49C4×#127C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C4×#124C2211yescharacteristic
52C4×#124C2211yescharacteristic
53C4×#121—22no—
54C4×#120—22no—
55C4×#117C2211yescharacteristic
56#2095C2212yescomplemented
57#2094C2212yes—
58#2093C2212yes—
59#2093C2212yescomplemented
60#2068C2214yescomplemented
61#2067C2214yes—
62#2067C2214yescomplemented
63#2065C2214yes—
64#2026C2212yescomplemented
65#2025C2212yes—
66#2025C2212yes—
67#2025C2212yescomplemented
68#1983C2214yes—
69#1982C2214yes—
70#1961C2212yes—
71#1959C2212yes—
72#1958C2212yes—
73#1957C2212yes—
74#1949C2214yescomplemented
75#1947C2214yescomplemented
76#1864C2214yescomplemented
77#1863C2214yescomplemented
78#1862C2214yes—
79#1862C2214yes—
80#1854C2214yes—
81#1852C2214yes—
82#1847C2212yescomplemented
83#1845C2212yescomplemented
84#1842C2212yescomplemented
85#1836C2212yescomplemented
86#1835C2212yescomplemented
87#1832C2212yescomplemented
88#1803C2212yescomplemented
89#1802C2212yes—
90#1802C2212yes—
91#1802C2212yescomplemented
92#1798C2212yes—
93#1797C2212yes—
94#1788C2214yes—
95#1786C2214yes—
96#1768C2212yescomplemented
97#1767C2211yescharacteristic, complemented
98C4×C4⋊C8C2211yescharacteristic, complemented
99C22×#41C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C24×C4—21no—
102#535—24no—
103#535Q818yescomplemented
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#125C2312yes—
109C2×#124C2311yescharacteristic
110C2×#124C2311yescharacteristic
111C2×C22⋊Q8—22no—
112C2×#121—22no—
113C2×#120—22no—
114C4×#41—44no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—44no—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—21no—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#475—44no—
140#474—44no—
141#474—44no—
142#474—44no—
143#433—28no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2314yes—
161C2×#97C2312yes—
162C2×C4×C8C2312yes—
163#400—24no—
164#398—24no—
165#398—24no—
166#396—24no—
167#391—22no—
168#391—22no—
169#391C2311yescharacteristic
170#391C2311yescharacteristic
171#390—22no—
172#390—22no—
173#390—22no—
174#390C2311yescharacteristic
175#390C2311yescharacteristic
176#390C2311yescharacteristic
177#390C2311yescharacteristic
178#390C2311yescharacteristic
179#390C2311yescharacteristic
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—24no—
186#389C2314yes—
187#389C2314yes—
188#388—24no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#387—24no—
194#387—24no—
195#386—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#386—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384—22no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#384C2312yes—
209#382C2312yes—
210#382C2311yescharacteristic
211#382C2311yescharacteristic
212#380—22no—
213#380—22no—
214#380C2314yes—
215#379—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224#376C2311yescharacteristic
225#376C2311yescharacteristic
226C43C2311yescharacteristic
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235C25—21no—
236C2×#41—22no—
237C2×#41—42no—
238C2×#41—42no—
239C2×#41C2×Q814yes—
240C22×Q8—22no—
241C22×Q8C2411yescharacteristic
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—42no—
248C22×D4—42no—
249C22×D4C2×Q814yes—
250C22×D4C2411yescharacteristic
251C22×D4C2411yescharacteristic
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—44no—
262C23×C4—84no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2411yescharacteristic
266C22×C8C2414yes—
267C22×C8C2414yes—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—84no—
271#127—22no—
272#127—22no—
273#127C2×Q814yes—
274#127C2×Q814yes—
275#125—44no—
276#124—22no—
277#124—22no—
278#124C2×Q814yes—
279#124C2×Q814yes—
280C22⋊Q8—84no—
281#121—216no—
282#120—216no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×Q8—84no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—42no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—84no—
297C4×D4—84no—
298#117—42no—
299#117—44no—
300#117—42no—
301#117C2×Q814yes—
302#117#4114yes—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2414yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—24no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—44no—
363#107—44no—
364#107—44no—
365#106—44no—
366C4⋊C8—28no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—44no—
370#103—44no—
371#103—44no—
372#102—44no—
373#102—44no—
374#102—44no—
375#102—44no—
376#102—44no—
377#102—44no—
378C22⋊C8—28no—
379#97—24no—
380C4×C8—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—21no—
395C24—21no—
396C24—21no—
397C24—24no—
398C24—42no—
399C24—42no—
400C24—82no—
401C24—82no—
402C24C22×D411yescharacteristic
403#41—48no—
404#41—48no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8C22×Q812yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—48no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—48no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4C22×Q812yes—
436C2×D4C22×Q812yes—
437C2×D4C22×Q812yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—816no—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×Q812yes—
479C22×C4C2×#4112yes—
480C22×C4C2511yescharacteristic
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4C22⋊Q818yes—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4C22⋊Q818yes—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—84no—
541C42C22×Q812yes—
542C42C22×Q812yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C42H(5,𝔽2)14yes—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—48no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—88no—
567C23—84no—
568C23—88no—
569C23—84no—
570C23C2×C22⋊Q812yes—
571C23C23×D411yescharacteristic
572Q8—24no—
573Q8—42no—
574Q8—88no—
575D4—24no—
576D4—24no—
577D4—24no—
578D4—48no—
579D4—48no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—88no—
584D4—88no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—48no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—82no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4C2×C22⋊Q814yes—
632C2×C4C2×C22⋊Q814yes—
633C2×C4C2×C22⋊Q812yes—
634C2×C4#53512yes—
635C2×C4#53512yes—
636C2×C4#57614yes—
637C2×C4C23×Q811yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4#58512yes—
641C8—44no—
642C8—44no—
643C22—24no—
644C22—22no—
645C22—42no—
646C22—41no—
647C22—41no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—816no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22C22×C22⋊Q811yescharacteristic
665C22C2×#53511yescharacteristic
666C22#344911yescharacteristic
667C22C2×#57612yes—
668C22#358612yes—
669C4—24no—
670C4—24no—
671C4—22no—
672C4—22no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—41no—
677C4—41no—
678C4—84no—
679C4—81no—
680C4—81no—
681C4—81no—
682C4—84no—
683C4—81no—
684C4#322014yes—
685C4#350112yes—
686C4#350612yes—
687C2—42no—
688C2—82no—
689C2—82no—
690C2—82no—
691C2#3517211yescharacteristic
692C2#6049412yes—
693C2C2×#322011yescharacteristic
694C2#6080911yescharacteristic
695C2#6098812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×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
1. #35284C2semidirect
3. #32955C2semidirect
4. #32953C2semidirect
5. #29964C2semidirect
6. #29962C2semidirect
7. #29268C2semidirect
8. #29265C2semidirect
9. #29204C2semidirect
10. #29200C2semidirect
11. #29149C2semidirect
12. #29140C2semidirect
13. #28520C2semidirect
14. #28508C2semidirect
15. #28281C2semidirect
16. #28279C2semidirect
17. #21991C2semidirect
19. #20812C2semidirect
23. C2×#475C22semidirect
26. C2×#474C22semidirect
28. #2868C22semidirect
29. #2859C22semidirect
31. #2647C22semidirect
37. #2330C22semidirect
48. C4×C4⋊Q8C22semidirect
56. #2095C22semidirect
59. #2093C22semidirect
60. #2068C22semidirect
62. #2067C22semidirect
64. #2026C22semidirect
67. #2025C22semidirect
74. #1949C22semidirect
75. #1947C22semidirect
76. #1864C22semidirect
77. #1863C22semidirect
82. #1847C22semidirect
83. #1845C22semidirect
84. #1842C22semidirect
85. #1836C22semidirect
86. #1835C22semidirect
87. #1832C22semidirect
88. #1803C22semidirect
91. #1802C22semidirect
96. #1768C22semidirect
97. #1767C22semidirect
98. C4×C4⋊C8C22semidirect
103. #535Q8semidirect

Maximal subgroups

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

subgroupindex
#352842
C4×#5352
#329552
#329532
#299642
#299622
#292682
#292652
#292042
#292002
#291492
#291402
#285202
#285082
#282812
#282792
#219912
#218132
#208122