Finite Group Databasebeta

← #6802674

Subgroups — #6802674

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36579C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic, complemented
3#32905C211yesmaximal, characteristic, complemented
4#32654C214yesmaximal, complemented
5#32605C214yesmaximal, complemented
6#29761C212yesmaximal, complemented
7#28303C212yesmaximal
8#28263C212yesmaximal, complemented
9#28123C212yesmaximal, complemented
10#27910C211yesmaximal, characteristic, complemented
11#27645C211yesmaximal, characteristic, complemented
12#25293C212yesmaximal, complemented
13#25057C212yesmaximal, complemented
14#24185C212yesmaximal, complemented
15Q8×C22⋊C8C211yesmaximal, characteristic, complemented
16#22079C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic, complemented
19C22×C22⋊Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#484C2212yes—
23C2×#484C2212yes—
24C2×#481C2212yes—
25C2×#461C2212yes—
26C2×#450C2212yes—
27C2×#447C2211yescharacteristic
28C2×C8×Q8C2212yescomplemented
29C2×C4×D8C2212yes—
30#2857C2211yescharacteristic
31#2850C2212yes—
32#2592C2212yescomplemented
33#2581C2212yescomplemented
34#2568C2211yescharacteristic, complemented
35#2538—21no—
36#2538—21no—
37#2538—22no—
38#2538C2214yes—
39#2538C2212yes—
40#2538C2212yes—
41#2538C2211yescharacteristic, complemented
42#2481—22no—
43#2469—22no—
44#2421—21no—
45#2409—21no—
46C2×#396—22no—
47#2362—21no—
48#2355C2211yescharacteristic, complemented
49#2352—21no—
50C4×C22⋊Q8C2212yes—
51C4×#121C2211yescharacteristic
52C2×#390C2212yes—
53C2×#389C2212yes—
54C2×#384C2211yescharacteristic
55C2×#380—21no—
56#2095C2214yescomplemented
57#2092C2214yes—
58#2085C2214yescomplemented
59#2084C2212yescomplemented
60#2071C2212yescomplemented
61#2067C2214yes—
62#2067C2214yescomplemented
63#2066C2214yescomplemented
64#2055C2214yes—
65#2042C2214yescomplemented
66#2012C2214yes—
67#1984C2214yescomplemented
68#1984C2212yescomplemented
69#1970C2214yes—
70#1968C2214yes—
71#1956C2212yescomplemented
72#1948C2212yescomplemented
73#1942C2214yescomplemented
74#1933C2214yescomplemented
75#1925C2214yes—
76#1921C2214yescomplemented
77#1880C2214yes—
78#1879C2212yes—
79#1877C2214yes—
80#1874C2211yescharacteristic
81#1872C2212yes—
82#1863C2214yescomplemented
83#1862C2212yes—
84#1852C2211yescharacteristic
85#1845C2212yescomplemented
86#1824C2212yes—
87#1802C2212yes—
88#1802C2212yescomplemented
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1786C2211yescharacteristic
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8D411yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2314yes—
99C22×D8C2312yes—
100#556—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#126—22no—
103C2×#124C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—48no—
110C2×C22⋊Q8C2312yes—
111C2×#121C2311yescharacteristic
112C2×C4×Q8—44no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D412yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—44no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2312yes—
127C2×#117—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4D412yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—21no—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140#484—28no—
141#484—44no—
142#481—44no—
143#461—44no—
144#450—44no—
145#447—24no—
146C8×Q8—44no—
147C4×D8—44no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#107C2312yes—
151C2×#106C2314yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2311yescharacteristic
165C2×C22⋊C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167C2×C4×C8C2314yes—
168#400—22no—
169#400—22no—
170#397—22no—
171#396—24no—
172#396—24no—
173#396—24no—
174#396—24no—
175#396—22no—
176#395—22no—
177#393—22no—
178#391—21no—
179#391—21no—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#390—22no—
184#390—22no—
185#390C2314yes—
186#390C2312yes—
187#390C2311yescharacteristic
188#390C2314yes—
189#389—21no—
190#389—21no—
191#389—22no—
192#389—22no—
193#389C2314yes—
194#389C2312yes—
195#389C2311yescharacteristic
196#389C2314yes—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—21no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#386—22no—
207#386—22no—
208#386—21no—
209#385—22no—
210#384—21no—
211#384—21no—
212#384C2311yescharacteristic
213#384C2312yes—
214#384C2312yes—
215#382—21no—
216#382C2312yes—
217#380—21no—
218#380—21no—
219#380—21no—
220#380—22no—
221#380—22no—
222#380—24no—
223#380C2311yescharacteristic
224C4×C4⋊C4—21no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—21no—
230C2×#95—22no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239#336C2312yes—
240C25—21no—
241C22×Q8—24no—
242C22×Q8—24no—
243C22×Q8—42no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2×D412yes—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2412yes—
248C22×D4—24no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—42no—
254C22×D4C2411yescharacteristic
255C22×D4C2412yes—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—44no—
262C23×C4—44no—
263C23×C4—44no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D412yes—
268C23×C4C2411yescharacteristic
269C2×SD16—28no—
270C2×SD16—44no—
271C2×SD16—44no—
272C2×D8—24no—
273C2×D8—24no—
274C2×D8—42no—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—24no—
279#126—48no—
280#124—44no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—48no—
284C22⋊Q8—416no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C22⋊Q8—48no—
288#121—22no—
289#121—22no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×Q8—44no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—44no—
305#117—44no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—48no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—44no—
360C2×C22⋊C4—48no—
361C2×C22⋊C4—84no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—44no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375#107—28no—
376#107—42no—
377#107—44no—
378#106—44no—
379C4⋊C8—24no—
380C4⋊C8—44no—
381#103—28no—
382#103—44no—
383#103—44no—
384#102—24no—
385#102—24no—
386#102—42no—
387#102—42no—
388#102—44no—
389#102—44no—
390C22⋊C8—24no—
391C22⋊C8—22no—
392C22⋊C8#4114yes—
393C4×C8—24no—
394C4×C8—44no—
395#95—24no—
396#95—24no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410C24—24no—
411C24—22no—
412C24—21no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24C22×D411yescharacteristic
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—416no—
433C2×Q8AGL(1,ℤ/8)14yes—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—48no—
445C2×D4—48no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D42−1+414yes—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—816no—
488C22×C4#13514yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4C2511yescharacteristic
501SD16—88no—
502D8—48no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8C2×#4114yes—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—21no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—48no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—81no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C22⋊C4—44no—
546C22⋊C4—48no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—42no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—48no—
556C22⋊C4—44no—
557C22⋊C4—82no—
558C22⋊C4—81no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C42—22no—
562C42—22no—
563C42—24no—
564C42—22no—
565C42—24no—
566C42—42no—
567C42—42no—
568C42—44no—
569C42—42no—
570C42—82no—
571C42H(5,𝔽2)14yes—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—24no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23#55012yes—
598C23C2×#13512yes—
599C23#57812yes—
600C23C23×D411yescharacteristic
601Q8—28no—
602Q8—48no—
603Q8—44no—
604Q8—44no—
605Q8—44no—
606D4—24no—
607D4—24no—
608D4—48no—
609D4—42no—
610D4—42no—
611D4—42no—
612D4—42no—
613D4—84no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—88no—
661C2×C4—84no—
662C2×C4—88no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—88no—
666C2×C4#53412yes—
667C2×C4#53412yes—
668C2×C4#55012yes—
669C2×C4C2×#13512yes—
670C2×C4C2×#13512yes—
671C2×C4C2×AGL(1,ℤ/8)14yes—
672C2×C4C2×AGL(1,ℤ/8)12yes—
673C2×C4C22×#4111yescharacteristic
674C2×C4C2×H(5,𝔽2)12yes—
675C2×C4C2×H(5,𝔽2)11yescharacteristic
676C2×C4C2×2−1+412yes—
677C8—24no—
678C8—42no—
679C8—44no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—816no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—81no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22—81no—
700C22—81no—
701C22#339314yes—
702C22C2×#53411yescharacteristic
703C22C2×#55011yescharacteristic
704C22#348611yescharacteristic
705C22C22×#13511yescharacteristic
706C22C22×AGL(1,ℤ/8)11yescharacteristic
707C22C2×#57811yescharacteristic
708C22#358511yescharacteristic
709C4—22no—
710C4—24no—
711C4—22no—
712C4—22no—
713C4—42no—
714C4—41no—
715C4—42no—
716C4—42no—
717C4—81no—
718C4—82no—
719C4—81no—
720C4—81no—
721C4—82no—
722C4—84no—
723C4—82no—
724C4#318814yes—
725C4#352012yes—
726C4#353612yes—
727C2—24no—
728C2—84no—
729C2—161no—
730C2#3913311yescharacteristic
731C2C2×#318811yescharacteristic
732C2#6076411yescharacteristic
733C2C2×#339311yescharacteristic
734C2#6140211yescharacteristic
735C2#6181911yescharacteristic
736C2#6183711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36579C2semidirect
2. C2×#2538C2semidirect
3. #32905C2semidirect
4. #32654C2semidirect
5. #32605C2semidirect
6. #29761C2semidirect
8. #28263C2semidirect
9. #28123C2semidirect
10. #27910C2semidirect
11. #27645C2semidirect
12. #25293C2semidirect
13. #25057C2semidirect
14. #24185C2semidirect
15. Q8×C22⋊C8C2semidirect
16. #22079C2semidirect
18. C2×#556C22semidirect
28. C2×C8×Q8C22semidirect
32. #2592C22semidirect
33. #2581C22semidirect
34. #2568C22semidirect
41. #2538C22semidirect
48. #2355C22semidirect
56. #2095C22semidirect
58. #2085C22semidirect
59. #2084C22semidirect
60. #2071C22semidirect
62. #2067C22semidirect
63. #2066C22semidirect
65. #2042C22semidirect
67. #1984C22semidirect
68. #1984C22semidirect
71. #1956C22semidirect
72. #1948C22semidirect
73. #1942C22semidirect
74. #1933C22semidirect
76. #1921C22semidirect
82. #1863C22semidirect
85. #1845C22semidirect
88. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#365792
C2×#25382
#329052
#326542
#326052
#297612
#283032
#282632
#281232
#279102
#276452
#252932
#250572
#241852
Q8×C22⋊C82
#220792
C4×#4472