Finite Group Databasebeta

← #6433140

Subgroups — #6433140

Order 512.

3123 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36614C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic, complemented
3#29897C214yesmaximal, complemented
4#29896C214yesmaximal, complemented
5#29895C211yesmaximal, characteristic, complemented
6#29894C212yesmaximal, complemented
7#29893C211yesmaximal, characteristic, complemented
8#29892C211yesmaximal, characteristic, complemented
9#29891C212yesmaximal, complemented
10#29890C211yesmaximal, characteristic, complemented
11#23638C214yesmaximal, complemented
12#23479C214yesmaximal, complemented
13#20908C212yesmaximal, complemented
14#20843C212yesmaximal, complemented
15C2×#1779C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#525C2212yes—
19C22×#117C2212yes—
20C2×#427C2214yescomplemented
21C2×#416C2214yescomplemented
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2214yes—
24#2849—21no—
25#2849—22no—
26#2848C2211yescharacteristic
27#2848C2211yescharacteristic
28#2848C2212yes—
29#2847C2211yescharacteristic
30#2847C2211yescharacteristic
31#2847C2212yes—
32#2846—21no—
33C4×#120—21no—
34C4×#120—21no—
35C4×#120—22no—
36#2280—22no—
37#2280—22no—
38#2280C2212yes—
39#2280C2211yescharacteristic
40#2280C2214yes—
41#2280C2214yes—
42C2×#381—21no—
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×#379—21no—
47C2×C4×C22⋊C4—21no—
48C2×C4×C22⋊C4—21no—
49#2077C2214yescomplemented
50#2077C2214yescomplemented
51#2077C2214yescomplemented
52#2077C2214yescomplemented
53#2076C2214yescomplemented
54#2076C2214yescomplemented
55#2075C2214yescomplemented
56#2075C2214yescomplemented
57#2036C2214yescomplemented
58#2036C2214yescomplemented
59#2035C2214yescomplemented
60#2035C2214yescomplemented
61#2034C2214yescomplemented
62#2034C2214yescomplemented
63#2033C2214yescomplemented
64#2033C2214yescomplemented
65#1875C2214yescomplemented
66#1874C2212yescomplemented
67#1874C2212yescomplemented
68#1873C2214yescomplemented
69#1872C2212yescomplemented
70#1872C2212yescomplemented
71#1854C2214yescomplemented
72#1854C2214yescomplemented
73#1853C2212yescomplemented
74#1853C2212yescomplemented
75#1852C2212yescomplemented
76#1852C2212yescomplemented
77#1779C2211yescharacteristic
78#1779C2212yes—
79#1779C2211yescharacteristic
80#1779C2214yes—
81#1735C2214yescomplemented
82#1735C2214yescomplemented
83C26D411yescharacteristic
84C22×#41C2311yescharacteristic
85C24×C4—28no—
86C24×C4—48no—
87C24×C4D411yescharacteristic
88C24×C4D412yes—
89C24×C4D411yescharacteristic
90C24×C4D411yescharacteristic
91C24×C4D412yes—
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C23×C8C2314yes—
98#525—28no—
99#525—48no—
100C2×#123—22no—
101C2×#123—21no—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C22⋊Q8C2311yescharacteristic
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×#121C2311yescharacteristic
107C2×#121C2311yescharacteristic
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×#120—21no—
111C2×#120—22no—
112C2×#120—21no—
113C2×#120—22no—
114C2×#120—21no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×#117—22no—
120C2×#117—22no—
121C2×#117—48no—
122C2×#117C2314yes—
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C2×#117C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—22no—
129C22×C22⋊C4—22no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C42—22no—
133#427—48no—
134C2×#107C2314yescomplemented
135C2×#107C2314yescomplemented
136C2×#106C2314yescomplemented
137C2×#106C2314yescomplemented
138#416—48no—
139C2×#103C2314yescomplemented
140C2×#103C2314yescomplemented
141C2×#102C2314yescomplemented
142C2×#102C2314yescomplemented
143C2×C22⋊C8—28no—
144C2×C22⋊C8—48no—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152#386—21no—
153#386—21no—
154#386—22no—
155#386—22no—
156#386—22no—
157#386—22no—
158#386—22no—
159#385—21no—
160#385—21no—
161#385—22no—
162#385—22no—
163#385—22no—
164#385—22no—
165#385—22no—
166#381—22no—
167#381—21no—
168#381—24no—
169#381—24no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—22no—
174#380C2312yes—
175#380C2312yes—
176#380C2314yes—
177#380C2314yes—
178#380C2312yes—
179#380C2312yes—
180#380C2314yes—
181#380C2314yes—
182#380C2311yescharacteristic
183#380C2311yescharacteristic
184#380C2312yes—
185#379—21no—
186#379—22no—
187#379—21no—
188#379—22no—
189#379—21no—
190#379—24no—
191#379—24no—
192C4×C22⋊C4—24no—
193C4×C22⋊C4—24no—
194C4×C22⋊C4—24no—
195C4×C22⋊C4—24no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—21no—
199C4×C22⋊C4—21no—
200C2×#95—21no—
201C2×#95—22no—
202C2×#95—21no—
203#340C2314yescomplemented
204#340C2314yescomplemented
205#340C2314yescomplemented
206#340C2314yescomplemented
207#340C2314yescomplemented
208#340C2314yescomplemented
209#340C2314yescomplemented
210#340C2314yescomplemented
211C25—24no—
212C25—24no—
213C25—44no—
214C25—42no—
215C25—44no—
216C25C2×D411yescharacteristic
217C25C2×D412yes—
218C25C2×D411yescharacteristic
219C25C2×D412yes—
220C25C2×D411yescharacteristic
221C2×#41—24no—
222C2×#41—28no—
223C22×Q8C2411yescharacteristic
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2411yescharacteristic
230C23×C4—28no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—28no—
235C23×C4—28no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—28no—
239C23×C4—28no—
240C23×C4—28no—
241C23×C4—28no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—464no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4C2×D412yes—
267C23×C4C2×D412yes—
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4#4118yes—
271C23×C4C2412yes—
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C23×C4C2411yescharacteristic
275C23×C4C2411yescharacteristic
276C22×C8—28no—
277C22×C8—28no—
278C22×C8—28no—
279C22×C8C2414yes—
280C22×C8C2414yes—
281#123—48no—
282#123—44no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287#121—24no—
288#121—24no—
289#121—44no—
290#121—44no—
291#120—48no—
292#120—44no—
293#120—48no—
294#120—44no—
295#120—44no—
296C4×D4—48no—
297C4×D4—48no—
298C4×D4—44no—
299C4×D4—44no—
300#117—24no—
301#117—28no—
302#117—28no—
303#117—28no—
304#117—44no—
305#117—44no—
306#117—48no—
307#117—48no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
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—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4—44no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—48no—
354#107—44no—
355#107—44no—
356#106—44no—
357#106—44no—
358#103—44no—
359#103—44no—
360#102—44no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364C22⋊C8—28no—
365C22⋊C8—44no—
366C22⋊C8—44no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—22no—
386C24—24no—
387C24—22no—
388C24—24no—
389C24—22no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—24no—
394C24—22no—
395C24—24no—
396C24—22no—
397C24—21no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—416no—
409C24—416no—
410C24—416no—
411C24—432no—
412C24—432no—
413C24#12314yes—
414C24#13514yes—
415C24C22×D411yescharacteristic
416C24C22×D412yes—
417C24C22×D412yes—
418C24C22×D411yescharacteristic
419C24C22×D411yescharacteristic
420#41—416no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—42no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—232no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—432no—
486C22×C4—416no—
487C22×C4—416no—
488C22×C4—432no—
489C22×C4—416no—
490C22×C4—416no—
491C22×C4—432no—
492C22×C4—416no—
493C22×C4—432no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—42no—
501C22×C4—44no—
502C22×C4#12314yes—
503C22×C4#12314yes—
504C22×C4#12314yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C2×#4114yes—
511C22×C4C2×#4114yes—
512C22×C4C2×#4114yes—
513C22×C4H(5,𝔽2)18yes—
514C22×C4C2511yescharacteristic
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—416no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—42no—
554C22⋊C4—48no—
555C22⋊C4—44no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C42—48no—
560C42—42no—
561C42—48no—
562C42—44no—
563C23—22no—
564C23—21no—
565C23—22no—
566C23—22no—
567C23—21no—
568C23—216no—
569C23—28no—
570C23—28no—
571C23—28no—
572C23—28no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—21no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—21no—
589C23—44no—
590C23—48no—
591C23—48no—
592C23—48no—
593C23—48no—
594C23—48no—
595C23—48no—
596C23—48no—
597C23—48no—
598C23—48no—
599C23—48no—
600C23—48no—
601C23—48no—
602C23—44no—
603C23—44no—
604C23—48no—
605C23—48no—
606C23—48no—
607C23—44no—
608C23—48no—
609C23—48no—
610C23—4128no—
611C23—48no—
612C23—44no—
613C23—48no—
614C23—41no—
615C23—42no—
616C23—41no—
617C23—42no—
618C23—41no—
619C23C2×#12312yes—
620C23C2×#12312yes—
621C23C2×#12312yes—
622C23#53414yes—
623C23C2×#13514yes—
624C23C2×#13512yes—
625C23#57514yes—
626C23#57512yes—
627C23C23×D411yescharacteristic
628Q8—44no—
629D4—44no—
630D4—88no—
631D4—88no—
632D4—84no—
633D4—84no—
634D4—84no—
635C2×C4—28no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—48no—
649C2×C4—48no—
650C2×C4—48no—
651C2×C4—48no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—48no—
657C2×C4—48no—
658C2×C4—48no—
659C2×C4—48no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—88no—
680C2×C4—88no—
681C2×C4—84no—
682C2×C4C2×#12312yes—
683C2×C4C2×#12312yes—
684C2×C4C2×#12312yes—
685C2×C4#53414yes—
686C2×C4#538116yes—
687C2×C4C22×#4112yes—
688C2×C4C2×H(5,𝔽2)14yes—
689C8—44no—
690C8—44no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—24no—
703C22—22no—
704C22—24no—
705C22—22no—
706C22—24no—
707C22—22no—
708C22—416no—
709C22—416no—
710C22—416no—
711C22—432no—
712C22—432no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—42no—
723C22—82no—
724C22—82no—
725C22—81no—
726C22—81no—
727C22—81no—
728C22#262214yes—
729C22#309114yes—
730C22C22×#12311yescharacteristic
731C22C2×#53412yes—
732C22C22×#13511yescharacteristic
733C22C2×#57512yes—
734C22C2×#57511yescharacteristic
735C4—22no—
736C4—22no—
737C4—42no—
738C4—42no—
739C4—84no—
740C4—82no—
741C4—84no—
742C4—82no—
743C4—81no—
744C4—81no—
745C4—81no—
746C4#262214yes—
747C4C2×#53814yes—
748C2—24no—
749C2—24no—
750C2—44no—
751C2—44no—
752C2—42no—
753C2—161no—
754C2C2×#262211yescharacteristic
755C2C2×#309111yescharacteristic
756C2#6051111yescharacteristic
757C2#6073412yes—
758C2#6074112yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radical#36614C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2280C2semidirect
3. #29897C2semidirect
4. #29896C2semidirect
5. #29895C2semidirect
6. #29894C2semidirect
7. #29893C2semidirect
8. #29892C2semidirect
9. #29891C2semidirect
10. #29890C2semidirect
11. #23638C2semidirect
12. #23479C2semidirect
13. #20908C2semidirect
14. #20843C2semidirect
15. C2×#1779C2semidirect
20. C2×#427C22semidirect
21. C2×#416C22semidirect
49. #2077C22semidirect
50. #2077C22semidirect
51. #2077C22semidirect
52. #2077C22semidirect
53. #2076C22semidirect
54. #2076C22semidirect
55. #2075C22semidirect
56. #2075C22semidirect
57. #2036C22semidirect
58. #2036C22semidirect
59. #2035C22semidirect
60. #2035C22semidirect
61. #2034C22semidirect
62. #2034C22semidirect
63. #2033C22semidirect
64. #2033C22semidirect
65. #1875C22semidirect
66. #1874C22semidirect
67. #1874C22semidirect
68. #1873C22semidirect
69. #1872C22semidirect
70. #1872C22semidirect
71. #1854C22semidirect
72. #1854C22semidirect
73. #1853C22semidirect
74. #1853C22semidirect
75. #1852C22semidirect
76. #1852C22semidirect
81. #1735C22semidirect
82. #1735C22semidirect
134. C2×#107C23semidirect
135. C2×#107C23semidirect
136. C2×#106C23semidirect
137. C2×#106C23semidirect
139. C2×#103C23semidirect
140. C2×#103C23semidirect
141. C2×#102C23semidirect
142. C2×#102C23semidirect
203. #340C23semidirect
204. #340C23semidirect
205. #340C23semidirect
206. #340C23semidirect
207. #340C23semidirect
208. #340C23semidirect
209. #340C23semidirect
210. #340C23semidirect

Maximal subgroups

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

subgroupindex
#366142
C2×#22802
#298972
#298962
#298952
#298942
#298932
#298922
#298912
#298902
#236382
#234792
#209082
#208432
C2×#17792