Finite Group Databasebeta

← #6660694

Subgroups — #6660694

Order 512.

2218 conjugacy classes of subgroups.

The subgroup lattice has 769 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic, complemented
3#32722C214yesmaximal, complemented
4#30240C212yesmaximal, complemented
5#30187C212yesmaximal, complemented
6#29951C212yesmaximal, complemented
7#29922C212yesmaximal, complemented
8#29713C212yesmaximal, complemented
9#29663C212yesmaximal, complemented
10#27678C212yesmaximal, complemented
11#24751C214yesmaximal, complemented
12#24381C214yesmaximal, complemented
13#24209C212yesmaximal, complemented
14C2×#1797C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#525—21no—
18C22×C22⋊Q8C2212yes—
19C23×C42C2211yescharacteristic
20C2×#465C2218yes—
21C2×#461C2214yescomplemented
22C2×#448C2214yes—
23C22×C4⋊C8C2212yes—
24C22×C22⋊C8C2212yes—
25#2857—21no—
26#2857—21no—
27#2857—21no—
28#2857—21no—
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2212yes—
32#2857C2212yes—
33#2857C2212yes—
34#2857C2212yes—
35#2856C2211yescharacteristic
36#2856C2211yescharacteristic
37#2854—21no—
38#2854C2212yes—
39#2850—21no—
40#2850—21no—
41#2847—21no—
42#2847C2212yes—
43C4×#121—21no—
44C4×#121—21no—
45C4×#121C2212yes—
46C4×#121C2212yes—
47#2297—21no—
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#389—21no—
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×#380—21no—
54C2×#380—21no—
55#2100C2214yescomplemented
56#2096C2214yescomplemented
57#2094C2214yescomplemented
58#2072C2214yescomplemented
59#2066C2218yescomplemented
60#2055C2218yescomplemented
61#2043C2214yescomplemented
62#2035C2218yes—
63#2024C2214yescomplemented
64#2014C2214yescomplemented
65#2013C2214yescomplemented
66#2000C2218yes—
67#1981C2214yescomplemented
68#1948C2214yescomplemented
69#1892C2218yes—
70#1874C2214yes—
71#1853C2214yes—
72#1802C2214yescomplemented
73#1797C2212yes—
74#1797C2212yes—
75#1797C2212yes—
76#1797C2212yes—
77#1787C2214yes—
78C2×#336C2212yes—
79C22×#41—21no—
80C23×Q8C2312yes—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C22×SD16C2318yes—
85C23×C8C2314yes—
86#532—44no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—44no—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#124—21no—
95#525—48no—
96C2×#123—22no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—48no—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2314yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#121C2312yes—
112C2×#120—22no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4C2312yes—
117C2×#117—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C22⋊C4—22no—
125C22×C42—44no—
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#465—416no—
139#461—48no—
140#448—48no—
141C2×#106C2318yes—
142C2×C4⋊C8—44no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2318yes—
146C2×#103C2318yes—
147C2×#103C2318yes—
148C2×#102C2318yes—
149C2×C22⋊C8—44no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152#391—21no—
153#391—21no—
154#391—22no—
155#391—22no—
156#391C2314yes—
157#391C2314yes—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#382—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C2×#95—22no—
212C2×#95—22no—
213#340C2318yes—
214#340C2318yes—
215#336C2314yes—
216#336C2314yes—
217C25C2×D411yescharacteristic
218C2×#41—44no—
219C2×#41—44no—
220C2×#41—44no—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8—24no—
224C22×Q8—24no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8C2414yes—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C2412yes—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4SD1618yescomplemented
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×SD16—48no—
266C2×SD16—48no—
267C2×SD16—48no—
268C22×C8—28no—
269C22×C8—44no—
270C22×C8—44no—
271C22×C8C2418yes—
272C4⋊Q8—28no—
273C4⋊Q8—42no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277#127—42no—
278#124—44no—
279#123—48no—
280C22⋊Q8—216no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286#121—24no—
287#121—28no—
288#121—24no—
289#121—44no—
290#121—44no—
291#121—44no—
292#121—44no—
293#121—44no—
294#121—44no—
295#120—48no—
296C4×D4—28no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—48no—
300#117—44no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
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—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
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—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—48no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#106—48no—
373C4⋊C8—28no—
374C4⋊C8—44no—
375#103—48no—
376#103—48no—
377#103—48no—
378#102—48no—
379C22⋊C8—28no—
380C22⋊C8—44no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—41no—
397C24—41no—
398C24C2×SD1614yes—
399C24C22×D411yescharacteristic
400C24C22×D411yescharacteristic
401C24C22×D411yescharacteristic
402#41—816no—
403C2×Q8—24no—
404C2×Q8—28no—
405C2×Q8—28no—
406C2×Q8—24no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—88no—
414C2×Q8—88no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—416no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—416no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4C2×SD1614yes—
500C22×C4C2×SD1614yes—
501C22×C4C2×SD1614yes—
502C22×C4C2×SD1614yes—
503C22×C4C2×SD1614yes—
504C22×C4C2×SD1614yes—
505C22×C4#13714yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4H(5,𝔽2)14yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C42−1+412yes—
520C22×C4C2511yescharacteristic
521SD16—816no—
522C2×C8—28no—
523C2×C8—28no—
524C2×C8—28no—
525C2×C8—88no—
526C4⋊C4—28no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—48no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—44no—
578C42H(5,𝔽2)14yes—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—28no—
591C23—28no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23#53512yes—
608C23#53712yes—
609C23C22×SD1612yes—
610C23C22×SD1612yes—
611C23C22×SD1612yes—
612C23C2×#13712yes—
613C23#57512yes—
614C23C23×D411yescharacteristic
615Q8—48no—
616Q8—48no—
617Q8—88no—
618Q8—84no—
619D4—44no—
620D4—44no—
621D4—84no—
622D4—84no—
623D4—88no—
624D4—88no—
625D4—88no—
626C2×C4—28no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—88no—
685C2×C4—88no—
686C2×C4—88no—
687C2×C4#53412yes—
688C2×C4#53412yes—
689C2×C4#53512yes—
690C2×C4#53712yes—
691C2×C4C22×SD1612yes—
692C2×C4C22×SD1612yes—
693C2×C4C22×SD1612yes—
694C2×C4C22×SD1612yes—
695C2×C4C2×#13712yes—
696C2×C4C2×#13712yes—
697C2×C4#57512yes—
698C2×C4C2×H(5,𝔽2)12yes—
699C2×C4C2×H(5,𝔽2)12yes—
700C2×C4C2×H(5,𝔽2)11yescharacteristic
701C2×C4C2×2−1+411yescharacteristic
702C8—48no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—22no—
716C22—22no—
717C22—22no—
718C22—416no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22#320614yes—
731C22#324614yes—
732C22C2×#53411yescharacteristic
733C22C2×#53511yescharacteristic
734C22C2×#53711yescharacteristic
735C22C23×SD1611yescharacteristic
736C22C22×#13711yescharacteristic
737C22C2×#57511yescharacteristic
738C22C2×#57511yescharacteristic
739C4—22no—
740C4—22no—
741C4—22no—
742C4—22no—
743C4—22no—
744C4—22no—
745C4—22no—
746C4—22no—
747C4—41no—
748C4—41no—
749C4—82no—
750C4—84no—
751C4—82no—
752C4—84no—
753C4—82no—
754C4#318714yes—
755C4#353612yes—
756C4#353712yes—
757C2—24no—
758C2—24no—
759C2—42no—
760C2—162no—
761C2#3743911yescharacteristic
762C2C2×#318711yescharacteristic
763C2#6075111yescharacteristic
764C2C2×#320611yescharacteristic
765C2#6079011yescharacteristic
766C2C2×#324611yescharacteristic
767C2#6091811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2857C2semidirect
3. #32722C2semidirect
4. #30240C2semidirect
5. #30187C2semidirect
6. #29951C2semidirect
7. #29922C2semidirect
8. #29713C2semidirect
9. #29663C2semidirect
10. #27678C2semidirect
11. #24751C2semidirect
12. #24381C2semidirect
13. #24209C2semidirect
14. C2×#1797C2semidirect
21. C2×#461C22semidirect
55. #2100C22semidirect
56. #2096C22semidirect
57. #2094C22semidirect
58. #2072C22semidirect
59. #2066C22semidirect
60. #2055C22semidirect
61. #2043C22semidirect
63. #2024C22semidirect
64. #2014C22semidirect
65. #2013C22semidirect
67. #1981C22semidirect
68. #1948C22semidirect
72. #1802C22semidirect
246. C23×C4SD16semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#28572
#327222
#302402
#301872
#299512
#299222
#297132
#296632
#276782
#247512
#243812
#242092
C2×#17972