Finite Group Databasebeta

← #6660709

Subgroups — #6660709

Order 512.

2192 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#32723C214yesmaximal, complemented
4#30248C212yesmaximal, complemented
5#30199C212yesmaximal, complemented
6#29953C212yesmaximal, complemented
7#29923C212yesmaximal, complemented
8#29713C212yesmaximal, complemented
9#29663C212yesmaximal, complemented
10#27677C212yesmaximal, complemented
11#24759C214yesmaximal, complemented
12#24387C214yesmaximal, complemented
13#24209C212yesmaximal, complemented
14#20626C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#525—21no—
18C22×C22⋊Q8C2212yes—
19C23×C42C2211yescharacteristic
20C2×#468C2218yes—
21C2×#461C2214yescomplemented
22C2×#448C2214yes—
23C2×#423C2212yes—
24C2×#407C2212yes—
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#2099C2214yescomplemented
56#2095C2214yescomplemented
57#2093C2214yescomplemented
58#2072C2214yescomplemented
59#2068C2218yescomplemented
60#2055C2218yescomplemented
61#2043C2214yescomplemented
62#2036C2218yes—
63#2024C2214yescomplemented
64#2014C2214yescomplemented
65#2013C2214yescomplemented
66#2000C2218yes—
67#1981C2214yescomplemented
68#1947C2214yescomplemented
69#1894C2218yes—
70#1874C2214yes—
71#1852C2214yes—
72#1802C2214yescomplemented
73#1797C2212yes—
74#1797C2212yes—
75#1797C2212yes—
76#1797C2212yes—
77#1787C2214yes—
78#1728C2212yes—
79C22×#41—21no—
80C23×Q8C2312yes—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C22×SD16C2318yes—
85C22×M16C2314yes—
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#468—416no—
139#461—48no—
140#448—48no—
141C2×#107C2318yes—
142#423—44no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2318yes—
146C2×#103C2318yes—
147C2×#103C2318yes—
148C2×#102C2318yes—
149#407—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—24no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—42no—
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—
268C2×M16—28no—
269C2×M16—44no—
270C2×M16—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—48no—
322C2×C4⋊C4—44no—
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#107—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—22no—
397C24—41no—
398C24—41no—
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—42no—
410C2×Q8—44no—
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—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
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—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
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—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—416no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—416no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4AGL(1,ℤ/8)14yes—
504C22×C4#13714yes—
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—
522M16—88no—
523C2×C8—28no—
524C2×C8—28no—
525C2×C8—28no—
526C4⋊C4—28no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—48no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—48no—
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—28no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23#53512yes—
610C23#53712yes—
611C23C2×AGL(1,ℤ/8)12yes—
612C23C2×#13712yes—
613C23C2×#13712yes—
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—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—44no—
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—44no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—88no—
685C2×C4—88no—
686C2×C4—88no—
687C2×C4#53412yes—
688C2×C4#53412yes—
689C2×C4#53512yes—
690C2×C4#53712yes—
691C2×C4C2×AGL(1,ℤ/8)12yes—
692C2×C4C2×AGL(1,ℤ/8)12yes—
693C2×C4C2×#13712yes—
694C2×C4C2×#13712yes—
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—416no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—41no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22#325414yes—
732C22C2×#53411yescharacteristic
733C22C2×#53511yescharacteristic
734C22C2×#53711yescharacteristic
735C22C22×AGL(1,ℤ/8)11yescharacteristic
736C22C22×#13711yescharacteristic
737C22C22×#13711yescharacteristic
738C22C2×#57511yescharacteristic
739C4—22no—
740C4—22no—
741C4—22no—
742C4—22no—
743C4—22no—
744C4—22no—
745C4—22no—
746C4—22no—
747C4—41no—
748C4—41no—
749C4—84no—
750C4—82no—
751C4—84no—
752C4—82no—
753C4—82no—
754C4#319314yes—
755C4#353612yes—
756C4#353712yes—
757C2—24no—
758C2—42no—
759C2—42no—
760C2—162no—
761C2#3743911yescharacteristic
762C2C2×#319311yescharacteristic
763C2#6075111yescharacteristic
764C2#6080011yescharacteristic
765C2#6080311yescharacteristic
766C2C2×#325411yescharacteristic
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. #32723C2semidirect
4. #30248C2semidirect
5. #30199C2semidirect
6. #29953C2semidirect
7. #29923C2semidirect
8. #29713C2semidirect
9. #29663C2semidirect
10. #27677C2semidirect
11. #24759C2semidirect
12. #24387C2semidirect
13. #24209C2semidirect
14. #20626C2semidirect
21. C2×#461C22semidirect
55. #2099C22semidirect
56. #2095C22semidirect
57. #2093C22semidirect
58. #2072C22semidirect
59. #2068C22semidirect
60. #2055C22semidirect
61. #2043C22semidirect
63. #2024C22semidirect
64. #2014C22semidirect
65. #2013C22semidirect
67. #1981C22semidirect
68. #1947C22semidirect
72. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#28572
#327232
#302482
#301992
#299532
#299232
#297132
#296632
#276772
#247592
#243872
#242092
#206262