Finite Group Databasebeta

← #6471244

Subgroups — #6471244

Order 512.

2328 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic, complemented
2C2×#2366C211yesmaximal, characteristic, complemented
3#33095C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic, complemented
5#32902C211yesmaximal, characteristic, complemented
6#29952C211yesmaximal, characteristic, complemented
7#29712C211yesmaximal, characteristic, complemented
8#29622C212yesmaximal, complemented
9#27079C214yesmaximal, complemented
10#26904C214yesmaximal, complemented
11#26769C214yesmaximal, complemented
12#26601C214yesmaximal, complemented
13#22046C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21452C212yesmaximal, complemented
16C2×#539—21no—
17C22×#127C2211yescharacteristic
18C22×C4×D4—21no—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#476C2212yescomplemented
22C2×#459C2212yescomplemented
23C2×#447C2212yes—
24C2×#447C2212yes—
25C2×#442C2214yescomplemented
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2854C2212yes—
29#2639—21no—
30#2618—21no—
31#2560—22no—
32#2558—22no—
33#2512C2212yescomplemented
34#2506C2212yescomplemented
35#2452C2212yescomplemented
36#2445C2212yescomplemented
37#2366—22no—
38#2366—22no—
39#2366C2214yes—
40#2366C2214yes—
41#2325—21no—
42C4×#121C2212yes—
43C2×#390—21no—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#386—21no—
47C2×#386—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2094C2212yescomplemented
51#2071C2212yescomplemented
52#2065C2214yescomplemented
53#2054C2214yescomplemented
54#2041C2214yes—
55#2041C2214yes—
56#2023C2212yescomplemented
57#2011C2214yes—
58#2011C2214yes—
59#2011C2212yescomplemented
60#1972C2214yescomplemented
61#1936C2214yescomplemented
62#1926C2214yescomplemented
63#1926C2214yescomplemented
64#1926C2214yescomplemented
65#1919C2214yescomplemented
66#1919C2214yescomplemented
67#1917C2214yescomplemented
68#1917C2214yescomplemented
69#1879C2214yescomplemented
70#1879C2214yescomplemented
71#1879C2214yescomplemented
72#1874C2212yes—
73#1874C2212yes—
74#1864C2214yescomplemented
75#1864C2214yescomplemented
76#1853C2212yes—
77#1853C2212yes—
78#1817C2212yes—
79#1802C2212yescomplemented
80#1802C2212yescomplemented
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1786C2212yes—
84#1786C2212yes—
85#1762C2214yescomplemented
86#1749C2212yes—
87C23×D4—21no—
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C24×C4—21no—
93C22×D8C2314yes—
94C22×D8C2314yes—
95#539—48no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127—24no—
98C2×#127—24no—
99C2×#127—22no—
100C2×#127C2314yes—
101C2×#127C2311yescharacteristic
102C2×#126—22no—
103C2×#125—22no—
104C2×#123—22no—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×#121—21no—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—44no—
118C2×C4×D4—44no—
119C2×C4×D4—48no—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2314yes—
123C2×C4×D4C2312yes—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#476—44no—
139#459—44no—
140#447—28no—
141#447—28no—
142#442—48no—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×#97C2314yes—
158#400—22no—
159#398—22no—
160#397—22no—
161#396—22no—
162#396—22no—
163#394—22no—
164#394—22no—
165#393—22no—
166#391—21no—
167#390—24no—
168#390—24no—
169#390—21no—
170#390—21no—
171#390—22no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#390C2314yes—
180#390C2312yes—
181#389—21no—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—22no—
191#386—22no—
192#386—22no—
193#386—22no—
194#386—24no—
195#386—24no—
196#386—24no—
197#386—24no—
198#385—22no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#380—21no—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25—21no—
224C25—21no—
225C22×Q8—21no—
226C22×D4—28no—
227C22×D4—28no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—24no—
232C22×D4—24no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—42no—
238C22×D4—42no—
239C22×D4—42no—
240C22×D4—42no—
241C22×D4—44no—
242C22×D4—44no—
243C22×D4—82no—
244C22×D4—82no—
245C22×D4#4114yes—
246C22×D4#4114yes—
247C22×D4C2412yes—
248C22×D4C2412yes—
249C22×D4C2412yes—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—82no—
261C23×C4—82no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×D8—28no—
270C2×D8—28no—
271C2×D8—44no—
272C2×D8—44no—
273C2×D8—44no—
274C2×D8—44no—
275C22×C8C2414yes—
276C22×C8C2414yes—
277C4⋊Q8—42no—
278#127—216no—
279#127—28no—
280#127—28no—
281#127—42no—
282#126—48no—
283#125—48no—
284#123—48no—
285#123—48no—
286C22⋊Q8—48no—
287C22⋊Q8—44no—
288#121—24no—
289#121—24no—
290#121—44no—
291#121—44no—
292C4×D4—28no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—44no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—48no—
302C4×D4—416no—
303C4×D4—416no—
304C4×D4—48no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
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—24no—
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—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—82no—
354C2×C22⋊C4—82no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—42no—
360C2×C42—42no—
361C2×C42—44no—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370#106—28no—
371#106—44no—
372C4⋊C8—42no—
373C4⋊C8—42no—
374#102—28no—
375#102—28no—
376#102—28no—
377#102—44no—
378#102—44no—
379#102—44no—
380#102—44no—
381#102—44no—
382C22⋊C8—24no—
383C22⋊C8—24no—
384#97—44no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—24no—
397C24—24no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24C22×D411yescharacteristic
414C2×Q8—42no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—48no—
434C2×D4—48no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—48no—
438C2×D4—432no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—48no—
442C2×D4—48no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—48no—
448C2×D4—42no—
449C2×D4—84no—
450C2×D4—84no—
451C2×D4—81no—
452C2×D4—81no—
453C2×D4—81no—
454C2×D4—81no—
455C2×D4—816no—
456C2×D4C2×#4114yes—
457C2×D4C2×#4114yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
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—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—44no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4—84no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4—816no—
500C22×C4—816no—
501C22×C4AGL(1,ℤ/8)14yes—
502C22×C4AGL(1,ℤ/8)14yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4C2511yescharacteristic
515D8—416no—
516D8—88no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—81no—
543C4⋊C4—81no—
544C4⋊C4H(5,𝔽2)14yes—
545C22⋊C4—48no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—48no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—84no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C42—24no—
566C42—22no—
567C42—22no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—41no—
572C42—41no—
573C42—42no—
574C42—48no—
575C42—44no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—24no—
581C23—24no—
582C23—48no—
583C23—48no—
584C23—48no—
585C23—48no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—48no—
591C23—48no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—84no—
608C23—84no—
609C23—84no—
610C23—84no—
611C23—84no—
612C23—84no—
613C23—84no—
614C23—84no—
615C23#52512yes—
616C23C2×AGL(1,ℤ/8)12yes—
617C23C2×AGL(1,ℤ/8)12yes—
618C23C23×D411yescharacteristic
619Q8—162no—
620D4—28no—
621D4—24no—
622D4—24no—
623D4—432no—
624D4—48no—
625D4—48no—
626D4—42no—
627D4—42no—
628D4—44no—
629D4—44no—
630D4—44no—
631D4—816no—
632D4—88no—
633D4—162no—
634D4—162no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—28no—
638C2×C4—22no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—22no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—21no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—48no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—88no—
681C2×C4—88no—
682C2×C4—82no—
683C2×C4—82no—
684C2×C4—816no—
685C2×C4—84no—
686C2×C4—164no—
687C2×C4#52512yes—
688C2×C4C2×AGL(1,ℤ/8)12yes—
689C2×C4C2×AGL(1,ℤ/8)14yes—
690C2×C4C2×AGL(1,ℤ/8)12yes—
691C2×C4C22×#4111yescharacteristic
692C2×C4C22×#4111yescharacteristic
693C2×C4C2×H(5,𝔽2)12yes—
694C2×C4#58512yes—
695C8—44no—
696C8—44no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—44no—
709C22—816no—
710C22—816no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22#310914yes—
727C22C2×#52511yescharacteristic
728C22#348411yescharacteristic
729C22#348711yescharacteristic
730C22C22×AGL(1,ℤ/8)11yescharacteristic
731C22C22×AGL(1,ℤ/8)11yescharacteristic
732C22#358511yescharacteristic
733C22#358511yescharacteristic
734C4—24no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—42no—
739C4—44no—
740C4—42no—
741C4—41no—
742C4—41no—
743C4—42no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4#345912yes—
749C4#350512yes—
750C2—24no—
751C2—84no—
752C2—84no—
753C2—162no—
754C2#3498711yescharacteristic
755C2C2×#310911yescharacteristic
756C2#6056411yescharacteristic
757C2#6157111yescharacteristic
758C2#6158711yescharacteristic
759C2#6189011yescharacteristic
760C2#6189111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35006C2semidirect
2. C2×#2366C2semidirect
3. #33095C2semidirect
4. #32956C2semidirect
5. #32902C2semidirect
6. #29952C2semidirect
7. #29712C2semidirect
8. #29622C2semidirect
9. #27079C2semidirect
10. #26904C2semidirect
11. #26769C2semidirect
12. #26601C2semidirect
13. #22046C2semidirect
14. #21916C2semidirect
15. #21452C2semidirect
21. C2×#476C22semidirect
22. C2×#459C22semidirect
25. C2×#442C22semidirect
33. #2512C22semidirect
34. #2506C22semidirect
35. #2452C22semidirect
36. #2445C22semidirect
50. #2094C22semidirect
51. #2071C22semidirect
52. #2065C22semidirect
53. #2054C22semidirect
56. #2023C22semidirect
59. #2011C22semidirect
60. #1972C22semidirect
61. #1936C22semidirect
62. #1926C22semidirect
63. #1926C22semidirect
64. #1926C22semidirect
65. #1919C22semidirect
66. #1919C22semidirect
67. #1917C22semidirect
68. #1917C22semidirect
69. #1879C22semidirect
70. #1879C22semidirect
71. #1879C22semidirect
74. #1864C22semidirect
75. #1864C22semidirect
79. #1802C22semidirect
80. #1802C22semidirect
85. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#23662
#330952
#329562
#329022
#299522
#297122
#296222
#270792
#269042
#267692
#266012
#220462
#219162
#214522