Finite Group Databasebeta

← #6802691

Subgroups — #6802691

Order 512.

1881 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36577C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic
3#32905C211yesmaximal, characteristic
4#32665C214yesmaximal, complemented
5#32621C214yesmaximal, complemented
6#29757C212yesmaximal, complemented
7#28311C212yesmaximal
8#28272C212yesmaximal, complemented
9#28123C212yesmaximal
10#27910C211yesmaximal, characteristic, complemented
11#27648C211yesmaximal, characteristic, complemented
12#25301C212yesmaximal, complemented
13#25056C212yesmaximal, complemented
14#24187C212yesmaximal, complemented
15#22510C211yesmaximal, characteristic, complemented
16#22085C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic, complemented
19C22×C22⋊Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#484C2212yes—
23C2×#484C2212yes—
24C2×#484C2212yes—
25C2×#461C2212yes—
26C2×#450C2211yescharacteristic
27C2×#450C2212yes—
28C2×#446C2212yescomplemented
29C2×C4×SD16C2212yes—
30#2857C2211yescharacteristic
31#2857C2212yes—
32#2582C2211yescharacteristic, complemented
33#2582C2212yescomplemented
34#2568C2212yescomplemented
35#2568C2211yescharacteristic, complemented
36#2538—21no—
37#2538—21no—
38#2538—22no—
39#2538C2214yes—
40#2538C2212yes—
41#2538C2212yes—
42#2453—21no—
43#2453—22no—
44#2442—22no—
45#2442—21no—
46C2×#396—22no—
47#2351—21no—
48Q8×C4⋊C4C2211yescharacteristic, complemented
49Q8×C22⋊C4—21no—
50C4×C22⋊Q8C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52C2×#390C2212yes—
53C2×#389C2212yes—
54C2×#384C2211yescharacteristic
55C2×#380—21no—
56#2095C2214yescomplemented
57#2093C2214yes—
58#2083C2212yescomplemented
59#2083C2214yescomplemented
60#2072C2212yescomplemented
61#2068C2214yescomplemented
62#2068C2214yescomplemented
63#2067C2214yes—
64#2055C2214yes—
65#2043C2214yescomplemented
66#2013C2214yes—
67#1985C2214yescomplemented
68#1984C2212yescomplemented
69#1973C2214yes—
70#1973C2214yes—
71#1956C2212yescomplemented
72#1947C2212yescomplemented
73#1943C2214yescomplemented
74#1935C2214yescomplemented
75#1925C2214yes—
76#1918C2214yescomplemented
77#1880C2214yes—
78#1877C2212yes—
79#1877C2214yes—
80#1872C2211yescharacteristic
81#1872C2212yes—
82#1865C2214yescomplemented
83#1864C2212yes—
84#1853C2211yescharacteristic
85#1845C2212yescomplemented
86#1824C2212yes—
87#1803C2212yescomplemented
88#1802C2212yes—
89#1797C2211yescharacteristic
90#1797C2212yes—
91#1786C2211yescharacteristic
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C22×SD16C2314yes—
100#557—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#125—21no—
104C2×#125—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—48no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C22⋊Q8C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—44no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D412yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—44no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4D412yes—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4—21no—
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#484—28no—
140#484—44no—
141#484—44no—
142#461—44no—
143#450—24no—
144#450—44no—
145#446—44no—
146C4×SD16—44no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#106C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×C22⋊C8C2311yescharacteristic
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2311yescharacteristic
165C2×#97C2314yes—
166C2×C4×C8C2312yes—
167#400—22no—
168#400—22no—
169#400—22no—
170#398—22no—
171#398—22no—
172#398—22no—
173#396—24no—
174#396—24no—
175#396—24no—
176#396—24no—
177#391—21no—
178#391—22no—
179#391—21no—
180#391—22no—
181#391C2312yes—
182#391C2311yescharacteristic
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#390—22no—
187#390—22no—
188#390C2314yes—
189#390C2314yes—
190#389—21no—
191#389—22no—
192#389—21no—
193#389—22no—
194#389—21no—
195#389—22no—
196#389—21no—
197#389—22no—
198#389—22no—
199#389—22no—
200#389C2312yes—
201#389C2314yes—
202#389C2311yescharacteristic
203#389C2314yes—
204#387—21no—
205#387—22no—
206#385—21no—
207#385—22no—
208#384—21no—
209#384—21no—
210#384C2311yescharacteristic
211#384C2312yes—
212#384C2312yes—
213#384C2312yes—
214#382—21no—
215#382—22no—
216#380—22no—
217#380—21no—
218#380—21no—
219#380—22no—
220#380—22no—
221#380—22no—
222#380—24no—
223#380C2311yescharacteristic
224C4×C4⋊C4—21no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
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×Q8C2411yescharacteristic
249C22×D4—24no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—42no—
254C22×D4C2412yes—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4C2411yescharacteristic
267C2×SD16—24no—
268C2×SD16—28no—
269C2×SD16—24no—
270C2×SD16—42no—
271C2×SD16—44no—
272C2×SD16—44no—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—24no—
277C4⋊Q8—44no—
278#125—48no—
279#125—44no—
280C22⋊Q8—22no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—22no—
284C22⋊Q8—48no—
285C22⋊Q8—416no—
286C22⋊Q8—48no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—42no—
293C4×Q8—42no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—42no—
300C4×D4—42no—
301C4×D4—44no—
302C4×D4—44no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
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—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—48no—
332C2×C4⋊C4—82no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2414yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—48no—
357C2×C22⋊C4—84no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372#107—28no—
373#107—44no—
374#107—44no—
375#106—42no—
376C4⋊C8—24no—
377C4⋊C8—44no—
378#103—28no—
379#103—42no—
380#103—44no—
381#103—44no—
382#103—44no—
383#102—24no—
384#102—24no—
385#102—42no—
386#102—44no—
387C22⋊C8—24no—
388C22⋊C8—22no—
389C22⋊C8#4114yes—
390#97—44no—
391C4×C8—24no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—21no—
396#95—22no—
397#95—21no—
398#95—21no—
399#95—22no—
400#95—21no—
401#95—22no—
402#95—21no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409C24—24no—
410C24—22no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24C22×D411yescharacteristic
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—42no—
434C2×Q8—416no—
435C2×Q8AGL(1,ℤ/8)14yes—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—48no—
443C2×D4—48no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D42−1+414yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—42no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—816no—
481C22×C4#13714yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4C2511yescharacteristic
494SD16—48no—
495SD16—88no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8C2×#4114yes—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—81no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—81no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C42—22no—
554C42—22no—
555C42—24no—
556C42—22no—
557C42—24no—
558C42—42no—
559C42—42no—
560C42—44no—
561C42—42no—
562C42—44no—
563C42—42no—
564C42H(5,𝔽2)14yes—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—24no—
570C23—48no—
571C23—48no—
572C23—48no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23#55012yes—
584C23C2×#13712yes—
585C23#57812yes—
586C23C23×D411yescharacteristic
587Q8—28no—
588Q8—48no—
589Q8—44no—
590Q8—42no—
591Q8—44no—
592Q8—44no—
593Q8—42no—
594D4—24no—
595D4—24no—
596D4—48no—
597D4—42no—
598D4—42no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—88no—
649C2×C4—84no—
650C2×C4—88no—
651C2×C4—84no—
652C2×C4—88no—
653C2×C4#53412yes—
654C2×C4#53412yes—
655C2×C4#55012yes—
656C2×C4C2×AGL(1,ℤ/8)14yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4C2×#13712yes—
659C2×C4C2×#13712yes—
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)12yes—
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×2−1+412yes—
664C8—24no—
665C8—42no—
666C8—44no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—816no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22#339814yes—
682C22C2×#53411yescharacteristic
683C22C2×#55011yescharacteristic
684C22#348611yescharacteristic
685C22C22×AGL(1,ℤ/8)11yescharacteristic
686C22C22×#13711yescharacteristic
687C22C2×#57811yescharacteristic
688C22#358511yescharacteristic
689C4—22no—
690C4—24no—
691C4—22no—
692C4—22no—
693C4—42no—
694C4—41no—
695C4—42no—
696C4—42no—
697C4—82no—
698C4—81no—
699C4—81no—
700C4—81no—
701C4—82no—
702C4—84no—
703C4—81no—
704C4—82no—
705C4#319314yes—
706C4#352012yes—
707C4#353612yes—
708C2—24no—
709C2—84no—
710C2#3913311yescharacteristic
711C2C2×#319311yescharacteristic
712C2#6076411yescharacteristic
713C2C2×#339811yescharacteristic
714C2#6140211yescharacteristic
715C2#6182611yescharacteristic
716C2#6183711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #36577C2semidirect
4. #32665C2semidirect
5. #32621C2semidirect
6. #29757C2semidirect
8. #28272C2semidirect
10. #27910C2semidirect
11. #27648C2semidirect
12. #25301C2semidirect
13. #25056C2semidirect
14. #24187C2semidirect
15. #22510C2semidirect
16. #22085C2semidirect
18. C2×#557C22semidirect
28. C2×#446C22semidirect
32. #2582C22semidirect
33. #2582C22semidirect
34. #2568C22semidirect
35. #2568C22semidirect
48. Q8×C4⋊C4C22semidirect
56. #2095C22semidirect
58. #2083C22semidirect
59. #2083C22semidirect
60. #2072C22semidirect
61. #2068C22semidirect
62. #2068C22semidirect
65. #2043C22semidirect
67. #1985C22semidirect
68. #1984C22semidirect
71. #1956C22semidirect
72. #1947C22semidirect
73. #1943C22semidirect
74. #1935C22semidirect
76. #1918C22semidirect
82. #1865C22semidirect
85. #1845C22semidirect
87. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#365772
C2×#25382
#329052
#326652
#326212
#297572
#283112
#282722
#281232
#279102
#276482
#253012
#250562
#241872
#225102
#220852
C4×#4502