Finite Group Databasebeta

← #6795009

Subgroups — #6795009

Order 512.

1911 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36193C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3#32956C211yesmaximal, characteristic
4#32428C214yesmaximal, complemented
5#32344C214yesmaximal, complemented
6#29908C212yesmaximal, complemented
7#28530C212yesmaximal, complemented
8#27727C212yesmaximal, complemented
9#27585C212yesmaximal
10#25811C211yesmaximal, characteristic, complemented
11#25451C212yesmaximal, complemented
12#24678C212yesmaximal, complemented
13#24501C212yesmaximal
14#24301C211yesmaximal, characteristic, complemented
15#22599C211yesmaximal, characteristic, complemented
16#21981C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#498C2212yescomplemented
23C2×#475C2212yescomplemented
24C2×#476C2212yes—
25C2×#450C2211yescharacteristic
26C2×C4×SD16C2212yes—
27C2×#431C2212yes—
28#2857C2211yescharacteristic
29#2855C2212yes—
30#2567C2211yescharacteristic, complemented
31#2552C2211yescharacteristic, complemented
32#2542—21no—
33#2542—21no—
34#2542—22no—
35#2542C2212yes—
36#2542C2214yes—
37#2542C2212yes—
38#2520—22no—
39#2511C2212yescomplemented
40#2508—21no—
41#2507C2212yescomplemented
42#2442—21no—
43#2438—22no—
44C2×#399—22no—
45#2370—21no—
46#2369C2211yescharacteristic, complemented
47#2355C2211yescharacteristic, complemented
48#2344—21no—
49C4×C22⋊Q8C2211yescharacteristic
50C2×#390C2212yes—
51#2285C2212yes—
52C2×#384C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2099C2214yescomplemented
55#2096C2214yescomplemented
56#2094C2214yes—
57#2084C2214yescomplemented
58#2072C2214yescomplemented
59#2071C2214yes—
60#2065C2214yes—
61#2058C2212yescomplemented
62#2058C2214yescomplemented
63#2055C2214yes—
64#2024C2212yescomplemented
65#1990C2212yescomplemented
66#1988C2212yescomplemented
67#1968C2214yes—
68#1948C2214yescomplemented
69#1948C2214yescomplemented
70#1939C2212yescomplemented
71#1934C2214yes—
72#1926C2214yes—
73#1926C2214yes—
74#1916C2214yescomplemented
75#1877C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1864C2212yes—
79#1864C2214yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2212yes—
83#1853C2212yes—
84#1853C2211yescharacteristic
85#1844C2214yescomplemented
86#1834C2214yescomplemented
87#1817C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90#1786C2212yes—
91#1786C2211yescharacteristic
92C8×C4⋊C4C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—48no—
114C2×C22⋊Q8—48no—
115C2×C22⋊Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—44no—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×#117—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#498—44no—
136#476—44no—
137#475—44no—
138#450—24no—
139C4×SD16—44no—
140#431—28no—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2311yescharacteristic
158C2×C4×C8C2314yes—
159C2×C4×C8C2312yes—
160#400—22no—
161#400—22no—
162#399—24no—
163#399—24no—
164#399—24no—
165#399—24no—
166#398—22no—
167#398—22no—
168#398—22no—
169#395—22no—
170#391—21no—
171#391—22no—
172#391C2312yes—
173#391C2312yes—
174#391C2311yescharacteristic
175#390—22no—
176#390—22no—
177#390C2314yes—
178#390C2314yes—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389C2312yes—
186#389C2311yescharacteristic
187#389C2312yes—
188#389C2312yes—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—21no—
194#387—22no—
195#387—22no—
196#385—22no—
197#385—21no—
198#384—22no—
199#384—22no—
200#384C2312yes—
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2314yes—
204#384C2314yes—
205#382—21no—
206#382—22no—
207#381—22no—
208#380—21no—
209#380—22no—
210#380C2312yes—
211#380C2311yescharacteristic
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—21no—
226C2×#95D412yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C25—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2411yescharacteristic
246C22×D4—24no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—42no—
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276#125—44no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8—48no—
280C22⋊Q8—416no—
281C22⋊Q8—416no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284C22⋊Q8—48no—
285C22⋊Q8—82no—
286C4×Q8—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—42no—
290C4×D4—42no—
291C4×D4—44no—
292C4×D4—44no—
293#117—84no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4SD1618yescomplemented
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—48no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362#106—42no—
363#106—44no—
364#106—44no—
365#106—44no—
366#106—44no—
367C4⋊C8—28no—
368C4⋊C8—24no—
369#103—42no—
370#103—44no—
371#102—28no—
372#102—24no—
373#102—24no—
374#102—42no—
375#102—44no—
376C22⋊C8—24no—
377C22⋊C8—22no—
378C22⋊C8#4114yes—
379C4×C8—28no—
380C4×C8—24no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—24no—
401C24—22no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24C22×D411yescharacteristic
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—81no—
421C2×Q8—81no—
422C2×Q8—816no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D42−1+414yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—41no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—816no—
467C22×C4C2×SD1618yes—
468C22×C4C2×SD1614yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C42−1+412yes—
480C22×C4C2511yescharacteristic
481SD16—48no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8C2×#4114yes—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—82no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—81no—
523C4⋊C4—82no—
524C4⋊C4C2×SD1618yes—
525C4⋊C4C2×SD1614yes—
526C4⋊C4C2×SD1614yes—
527C4⋊C4AGL(1,ℤ/8)14yes—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C42—22no—
538C42—22no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—42no—
547C42—42no—
548C42—82no—
549C42—82no—
550C42H(5,𝔽2)14yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—24no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23#54712yes—
570C23C22×SD1612yes—
571C23#57712yes—
572C23C23×D411yescharacteristic
573Q8—42no—
574Q8—42no—
575Q8—88no—
576Q8—88no—
577Q8—162no—
578D4—24no—
579D4—24no—
580D4—48no—
581D4—42no—
582D4—42no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—216no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—88no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4#53512yes—
636C2×C4#53512yes—
637C2×C4#54712yes—
638C2×C4C22×SD1614yes—
639C2×C4C22×SD1614yes—
640C2×C4C22×SD1612yes—
641C2×C4C22×SD1612yes—
642C2×C4C2×AGL(1,ℤ/8)14yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)12yes—
646C2×C4C2×2−1+412yes—
647C2×C4C2×2−1+411yescharacteristic
648C8—24no—
649C8—44no—
650C8—42no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—816no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22#333514yes—
666C22C2×#53511yescharacteristic
667C22C2×#54711yescharacteristic
668C22#348611yescharacteristic
669C22C23×SD1611yescharacteristic
670C22C22×AGL(1,ℤ/8)11yescharacteristic
671C22C2×#57711yescharacteristic
672C22#358611yescharacteristic
673C4—22no—
674C4—24no—
675C4—24no—
676C4—24no—
677C4—22no—
678C4—22no—
679C4—42no—
680C4—41no—
681C4—81no—
682C4—81no—
683C4—82no—
684C4—84no—
685C4—81no—
686C4—82no—
687C4—81no—
688C4—82no—
689C4#320814yes—
690C4#352012yes—
691C4#353812yes—
692C2—24no—
693C2—84no—
694C2#3891711yescharacteristic
695C2C2×#320811yescharacteristic
696C2#6081311yescharacteristic
697C2C2×#333511yescharacteristic
698C2#6125211yescharacteristic
699C2#6181811yescharacteristic
700C2#6184311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #36193C2semidirect
4. #32428C2semidirect
5. #32344C2semidirect
6. #29908C2semidirect
7. #28530C2semidirect
8. #27727C2semidirect
10. #25811C2semidirect
11. #25451C2semidirect
12. #24678C2semidirect
14. #24301C2semidirect
15. #22599C2semidirect
16. #21981C2semidirect
22. C2×#498C22semidirect
23. C2×#475C22semidirect
30. #2567C22semidirect
31. #2552C22semidirect
39. #2511C22semidirect
41. #2507C22semidirect
46. #2369C22semidirect
47. #2355C22semidirect
54. #2099C22semidirect
55. #2096C22semidirect
57. #2084C22semidirect
58. #2072C22semidirect
61. #2058C22semidirect
62. #2058C22semidirect
64. #2024C22semidirect
65. #1990C22semidirect
66. #1988C22semidirect
68. #1948C22semidirect
69. #1948C22semidirect
70. #1939C22semidirect
74. #1916C22semidirect
85. #1844C22semidirect
86. #1834C22semidirect
92. C8×C4⋊C4C22semidirect
321. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#361932
C2×#25422
#329562
#324282
#323442
#299082
#285302
#277272
#275852
#258112
#254512
#246782
#245012
#243012
#225992
#219812
C4×#4502