Finite Group Databasebeta

← #6856256

Subgroups — #6856256

Order 512.

1971 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3#32902C211yesmaximal, characteristic
4#32446C214yesmaximal, complemented
5#31654C214yesmaximal, complemented
6#29718C212yesmaximal, complemented
7#28415C212yesmaximal, complemented
8#27978C212yesmaximal, complemented
9#27736C212yesmaximal, complemented
10#27678C212yesmaximal, complemented
11#25903C211yesmaximal, characteristic, complemented
12#25318C212yesmaximal, complemented
13#24678C212yesmaximal, complemented
14#24301C211yesmaximal, characteristic, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22046C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#486C2212yescomplemented
23C2×#475C2212yescomplemented
24C2×#459C2212yes—
25C2×#450C2211yescharacteristic
26C2×#447C2212yes—
27C2×C4×SD16C2212yes—
28#2856C2211yescharacteristic
29#2850C2212yes—
30#2597—22no—
31#2571C2211yescharacteristic, complemented
32#2548C2211yescharacteristic, complemented
33#2542—22no—
34#2542—21no—
35#2542—21no—
36#2542C2212yes—
37#2542C2212yes—
38#2542C2214yes—
39#2513—21no—
40#2513C2212yescomplemented
41#2504—21no—
42#2447C2212yescomplemented
43#2444—22no—
44C2×#399—22no—
45#2366C2211yescharacteristic, complemented
46#2361—21no—
47#2357—21no—
48#2355C2211yescharacteristic, complemented
49C4×C22⋊Q8C2211yescharacteristic
50C4×#121C2212yes—
51C2×#390C2212yes—
52C2×#384C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2100C2214yescomplemented
55#2094C2214yescomplemented
56#2073C2214yes—
57#2072C2214yescomplemented
58#2071C2214yescomplemented
59#2058C2212yescomplemented
60#2055C2214yes—
61#2054C2214yes—
62#2024C2212yescomplemented
63#2023C2214yescomplemented
64#2012C2214yes—
65#1990C2212yescomplemented
66#1988C2212yescomplemented
67#1972C2214yes—
68#1959C2214yescomplemented
69#1948C2214yescomplemented
70#1939C2212yescomplemented
71#1936C2214yes—
72#1926C2214yes—
73#1917C2214yescomplemented
74#1879C2214yes—
75#1879C2214yes—
76#1877C2212yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1864C2212yes—
83#1853C2212yes—
84#1853C2211yescharacteristic
85#1836C2214yescomplemented
86#1802C2212yes—
87#1802C2214yescomplemented
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1786C2211yescharacteristic
91#1770C2212yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C22×D8C2314yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#124C2312yes—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—48no—
114C2×C22⋊Q8—48no—
115C2×C22⋊Q8C2311yescharacteristic
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—44no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140#486—44no—
141#475—44no—
142#459—44no—
143#450—24no—
144#447—28no—
145C4×SD16—44no—
146C2×#106C2314yes—
147C2×#106C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2311yescharacteristic
161C2×C22⋊C8C2311yescharacteristic
162C2×#97C2314yes—
163C2×C4×C8C2312yes—
164#400—22no—
165#399—24no—
166#399—24no—
167#399—24no—
168#399—24no—
169#397—22no—
170#397—22no—
171#397—22no—
172#397—22no—
173#394—22no—
174#391C2312yes—
175#391C2311yescharacteristic
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—22no—
182#390—22no—
183#390C2312yes—
184#390C2314yes—
185#390C2314yes—
186#390C2312yes—
187#390C2312yes—
188#389—21no—
189#389C2311yescharacteristic
190#389C2312yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#387—21no—
197#387—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201#386—22no—
202#385—21no—
203#385—22no—
204#384—22no—
205#384—22no—
206#384C2311yescharacteristic
207#384C2314yes—
208#384C2314yes—
209#382—21no—
210#382—21no—
211#382C2312yes—
212#381—22no—
213#380C2311yescharacteristic
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C2×#95D412yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C25—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2411yescharacteristic
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—42no—
251C22×D4—42no—
252C22×D4#4114yes—
253C22×D4C2412yes—
254C22×D4C2412yes—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4—44no—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2411yescharacteristic
268C2×SD16—24no—
269C2×SD16—42no—
270C2×SD16—42no—
271C2×D8—28no—
272C2×D8—44no—
273C2×D8—44no—
274C22×C8C2414yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C4⋊Q8—24no—
278C4⋊Q8—24no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282#127—24no—
283#124—44no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—416no—
287C22⋊Q8—48no—
288C22⋊Q8—48no—
289C22⋊Q8—48no—
290C22⋊Q8—416no—
291C22⋊Q8—48no—
292#121—24no—
293#121—24no—
294#121—44no—
295#121—82no—
296C4×Q8—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—42no—
300C4×D4—42no—
301C4×D4—44no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—84no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—48no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4—48no—
359C2×C22⋊C4—48no—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—44no—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2412yes—
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373#106—42no—
374#106—44no—
375C4⋊C8—24no—
376C4⋊C8—44no—
377#103—42no—
378#103—44no—
379#102—28no—
380#102—24no—
381#102—42no—
382#102—42no—
383#102—44no—
384#102—44no—
385#102—44no—
386C22⋊C8—24no—
387C22⋊C8—22no—
388C22⋊C8—22no—
389#97—44no—
390C4×C8—24no—
391#95—21no—
392#95—22no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—24no—
407C24—22no—
408C24—22no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24C22×D411yescharacteristic
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—816no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—48no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—48no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—81no—
449C2×D4—81no—
450C2×D4C2×#4114yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—816no—
487C22×C4AGL(1,ℤ/8)14yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4112yes—
498C22×C42−1+412yes—
499C22×C4C2511yescharacteristic
500SD16—48no—
501D8—88no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—24no—
510C2×C8—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—42no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—48no—
536C4⋊C4—42no—
537C4⋊C4—81no—
538C4⋊C4—82no—
539C4⋊C4—81no—
540C4⋊C4AGL(1,ℤ/8)14yes—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—24no—
563C42—24no—
564C42—42no—
565C42—42no—
566C42—42no—
567C42—42no—
568C42—82no—
569C42H(5,𝔽2)14yes—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—24no—
575C23—48no—
576C23—48no—
577C23—42no—
578C23—48no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23#54812yes—
596C23C2×AGL(1,ℤ/8)12yes—
597C23#57712yes—
598C23C23×D411yescharacteristic
599Q8—42no—
600Q8—42no—
601Q8—88no—
602Q8—88no—
603D4—24no—
604D4—24no—
605D4—48no—
606D4—44no—
607D4—42no—
608D4—42no—
609D4—44no—
610D4—88no—
611D4—84no—
612D4—162no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—48no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—88no—
661C2×C4—84no—
662C2×C4—88no—
663C2×C4—88no—
664C2×C4—164no—
665C2×C4#53512yes—
666C2×C4#53512yes—
667C2×C4#54812yes—
668C2×C4C2×AGL(1,ℤ/8)14yes—
669C2×C4C2×AGL(1,ℤ/8)12yes—
670C2×C4C2×AGL(1,ℤ/8)12yes—
671C2×C4C2×AGL(1,ℤ/8)12yes—
672C2×C4C22×#4111yescharacteristic
673C2×C4C2×H(5,𝔽2)12yes—
674C2×C4C2×2−1+411yescharacteristic
675C2×C4#58512yes—
676C8—44no—
677C8—42no—
678C8—42no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—816no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22#337314yes—
701C22C2×#53511yescharacteristic
702C22C2×#54811yescharacteristic
703C22#348511yescharacteristic
704C22C22×AGL(1,ℤ/8)11yescharacteristic
705C22C22×AGL(1,ℤ/8)11yescharacteristic
706C22C2×#57711yescharacteristic
707C22#358611yescharacteristic
708C4—24no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—41no—
716C4—81no—
717C4—81no—
718C4—82no—
719C4—84no—
720C4—82no—
721C4—81no—
722C4—81no—
723C4#321714yes—
724C4#350512yes—
725C4#354112yes—
726C2—24no—
727C2—84no—
728C2—162no—
729C2#3972211yescharacteristic
730C2C2×#321711yescharacteristic
731C2#6081311yescharacteristic
732C2C2×#337311yescharacteristic
733C2#6131711yescharacteristic
734C2#6171711yescharacteristic
735C2#6172811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
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. #36587C2semidirect
2. C2×#2542C2semidirect
4. #32446C2semidirect
5. #31654C2semidirect
6. #29718C2semidirect
7. #28415C2semidirect
8. #27978C2semidirect
9. #27736C2semidirect
10. #27678C2semidirect
11. #25903C2semidirect
12. #25318C2semidirect
13. #24678C2semidirect
14. #24301C2semidirect
15. #22629C2semidirect
16. #22046C2semidirect
17. C4×#450C2semidirect
22. C2×#486C22semidirect
23. C2×#475C22semidirect
31. #2571C22semidirect
32. #2548C22semidirect
40. #2513C22semidirect
42. #2447C22semidirect
45. #2366C22semidirect
48. #2355C22semidirect
54. #2100C22semidirect
55. #2094C22semidirect
57. #2072C22semidirect
58. #2071C22semidirect
59. #2058C22semidirect
62. #2024C22semidirect
63. #2023C22semidirect
65. #1990C22semidirect
66. #1988C22semidirect
68. #1959C22semidirect
69. #1948C22semidirect
70. #1939C22semidirect
73. #1917C22semidirect
85. #1836C22semidirect
87. #1802C22semidirect
91. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#25422
#329022
#324462
#316542
#297182
#284152
#279782
#277362
#276782
#259032
#253182
#246782
#243012
#226292
#220462
C4×#4502