Finite Group Databasebeta

← #6476964

Subgroups — #6476964

Order 512.

2248 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#33174C212yesmaximal, complemented
4#33049C212yesmaximal, complemented
5#30325C214yesmaximal, complemented
6#28577C214yesmaximal, complemented
7#27973C211yesmaximal, characteristic, complemented
8#27950C211yesmaximal, characteristic, complemented
9#27892C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic, complemented
11#25718C211yesmaximal, characteristic, complemented
12#25597C212yesmaximal, complemented
13#24387C214yesmaximal, complemented
14#22047C214yesmaximal, complemented
15C4×#407C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#525—21no—
19C22×C22⋊Q8C2212yes—
20C23×C42C2211yescharacteristic
21C2×#492C2212yescomplemented
22C2×#487C2212yescomplemented
23C2×#468C2218yes—
24C2×#448C2214yes—
25C2×#407C2212yes—
26C2×C4×M16C2212yes—
27#2857—21no—
28#2857—21no—
29#2857—21no—
30#2857—21no—
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2212yes—
34#2857C2212yes—
35#2857C2212yes—
36#2857C2212yes—
37#2856C2211yescharacteristic
38#2856C2211yescharacteristic
39#2854—21no—
40#2854C2212yes—
41#2850—21no—
42#2850—21no—
43#2847—21no—
44#2847C2212yes—
45C4×#121—21no—
46C4×#121—21no—
47C4×#121C2212yes—
48C4×#121C2212yes—
49#2297—21no—
50C2×#391C2212yes—
51C2×#389—21no—
52C2×#389—21no—
53#2285C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×#380—21no—
56C2×#380—21no—
57#2083C2214yescomplemented
58#2071C2214yescomplemented
59#2043C2214yescomplemented
60#2036C2218yes—
61#2013C2214yescomplemented
62#2000C2218yes—
63#1990C2212yescomplemented
64#1973C2218yescomplemented
65#1957C2212yescomplemented
66#1956C2212yescomplemented
67#1956C2212yescomplemented
68#1935C2218yescomplemented
69#1928C2218yescomplemented
70#1894C2218yes—
71#1877C2218yescomplemented
72#1874C2214yes—
73#1852C2214yes—
74#1845C2212yescomplemented
75#1834C2212yescomplemented
76#1787C2214yes—
77#1749C2212yes—
78#1749C2212yes—
79C4×C22⋊C8C2211yescharacteristic
80C4×C22⋊C8C2211yescharacteristic
81C4×C22⋊C8C2212yes—
82#1728C2212yes—
83C22×#41—21no—
84C23×Q8C2312yes—
85C24×C4D411yescharacteristic
86C24×C4D411yescharacteristic
87C24×C4C2311yescharacteristic
88C22×SD16C2318yes—
89C22×M16C2314yes—
90#532—44no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#124—21no—
99#525—48no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—48no—
106C2×C22⋊Q8C2314yes—
107C2×C22⋊Q8C2314yes—
108C2×#121—21no—
109C2×#121—21no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×#121C2312yes—
115C2×#121C2312yes—
116C2×#120—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C2×#117—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—22no—
129C22×C42—44no—
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#492—44no—
143#487—44no—
144#468—416no—
145#448—216no—
146C2×#107C2318yes—
147C2×#103C2318yes—
148C2×#103C2318yes—
149C2×#103C2318yes—
150C2×#102C2318yes—
151#407—28no—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2312yes—
155C4×M16—44no—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2312yes—
159#391—21no—
160#391—21no—
161#391—22no—
162#391—22no—
163#391C2314yes—
164#391C2314yes—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#384—21no—
194#384—21no—
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#382—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—22no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380C2312yes—
212#380C2312yes—
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2318yes—
221#340C2318yes—
222#336C2312yes—
223#336C2312yes—
224#336C2314yes—
225C25C2×D411yescharacteristic
226C2×#41—44no—
227C2×#41—44no—
228C2×#41—44no—
229C22×Q8—21no—
230C22×Q8—22no—
231C22×Q8—28no—
232C22×Q8—44no—
233C22×Q8—44no—
234C22×Q8C2414yes—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—42no—
250C23×C4—42no—
251C23×C4—42no—
252C23×C4—42no—
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×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C2×SD16—216no—
272C2×SD16—48no—
273C2×SD16—48no—
274C2×M16—28no—
275C2×M16—28no—
276C2×M16—44no—
277C22×C8C2414yes—
278C22×C8C2414yes—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285#127—24no—
286#124—44no—
287#123—48no—
288C22⋊Q8—216no—
289C22⋊Q8—48no—
290C22⋊Q8—48no—
291C22⋊Q8—48no—
292C22⋊Q8—44no—
293C22⋊Q8—44no—
294#121—28no—
295#121—28no—
296#121—44no—
297#121—44no—
298#121—44no—
299#121—44no—
300#121—44no—
301#121—44no—
302#120—48no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—48no—
306C4×D4—48no—
307#117—44no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—48no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—48no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42—42no—
358C2×C42—42no—
359C2×C42—42no—
360C2×C42—42no—
361C2×C42—42no—
362C2×C42—42no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—48no—
380#103—216no—
381#103—48no—
382#103—48no—
383#102—48no—
384C22⋊C8—24no—
385C22⋊C8—28no—
386C22⋊C8#4118yescomplemented
387#97—44no—
388C4×C8—24no—
389C4×C8—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—41no—
407C24—41no—
408C24C22×D411yescharacteristic
409C24C22×D411yescharacteristic
410C24C22×D411yescharacteristic
411#41—816no—
412C2×Q8—28no—
413C2×Q8—24no—
414C2×Q8—28no—
415C2×Q8—24no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—816no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
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—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—416no—
496C22×C4—416no—
497C22×C4—416no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4AGL(1,ℤ/8)14yes—
508C22×C4#13714yes—
509C22×C4#13714yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C2×#4114yes—
522C22×C4H(5,𝔽2)14yes—
523C22×C4C2511yescharacteristic
524SD16—816no—
525M16—416no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8—24no—
531C2×C8C2×#4118yes—
532C4⋊C4—24no—
533C4⋊C4—28no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—48no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—48no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—48no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—48no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—44no—
583C42H(5,𝔽2)14yes—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—28no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23#53212yes—
615C23#53512yes—
616C23C2×AGL(1,ℤ/8)12yes—
617C23C2×#13712yes—
618C23C2×#13712yes—
619C23C23×D411yescharacteristic
620Q8—48no—
621Q8—48no—
622Q8—88no—
623Q8—84no—
624D4—48no—
625D4—88no—
626D4—88no—
627D4—88no—
628D4—84no—
629D4—84no—
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—28no—
641C2×C4—28no—
642C2×C4—28no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
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—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—88no—
686C2×C4—88no—
687C2×C4—88no—
688C2×C4#53212yes—
689C2×C4#53412yes—
690C2×C4#53412yes—
691C2×C4#53512yes—
692C2×C4C2×AGL(1,ℤ/8)12yes—
693C2×C4C2×AGL(1,ℤ/8)12yes—
694C2×C4C2×#13712yes—
695C2×C4C2×#13712yes—
696C2×C4C2×#13712yes—
697C2×C4C2×#13712yes—
698C2×C4#57512yes—
699C2×C4C22×#4112yes—
700C2×C4C2×H(5,𝔽2)12yes—
701C2×C4C2×H(5,𝔽2)12yes—
702C8—28no—
703C8—44no—
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—416no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—41no—
724C22—41no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22#318214yes—
733C22C2×#53211yescharacteristic
734C22C2×#53411yescharacteristic
735C22C2×#53511yescharacteristic
736C22C22×AGL(1,ℤ/8)11yescharacteristic
737C22C22×#13711yescharacteristic
738C22C22×#13711yescharacteristic
739C22C2×#57511yescharacteristic
740C4—22no—
741C4—22no—
742C4—22no—
743C4—22no—
744C4—22no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—41no—
749C4—84no—
750C4—82no—
751C4—82no—
752C4—84no—
753C4—82no—
754C4#319314yes—
755C4#319614yes—
756C4#351914yes—
757C2—24no—
758C2—42no—
759C2—42no—
760C2—162no—
761C2#3507811yescharacteristic
762C2C2×#318211yescharacteristic
763C2#6070111yescharacteristic
764C2C2×#319311yescharacteristic
765C2C2×#319611yescharacteristic
766C2#6080011yescharacteristic
767C2#6080311yescharacteristic
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. #33174C2semidirect
4. #33049C2semidirect
5. #30325C2semidirect
6. #28577C2semidirect
7. #27973C2semidirect
8. #27950C2semidirect
9. #27892C2semidirect
10. #25856C2semidirect
11. #25718C2semidirect
12. #25597C2semidirect
13. #24387C2semidirect
14. #22047C2semidirect
15. C4×#407C2semidirect
21. C2×#492C22semidirect
22. C2×#487C22semidirect
57. #2083C22semidirect
58. #2071C22semidirect
59. #2043C22semidirect
61. #2013C22semidirect
63. #1990C22semidirect
64. #1973C22semidirect
65. #1957C22semidirect
66. #1956C22semidirect
67. #1956C22semidirect
68. #1935C22semidirect
69. #1928C22semidirect
71. #1877C22semidirect
74. #1845C22semidirect
75. #1834C22semidirect
386. C22⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#28572
#331742
#330492
#303252
#285772
#279732
#279502
#278922
#258562
#257182
#255972
#243872
#220472
C4×#4072