Finite Group Databasebeta

← #6800241

Subgroups — #6800241

Order 512.

1867 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36544C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic, complemented
3#32956C211yesmaximal, characteristic, complemented
4#32581C214yesmaximal, complemented
5#32554C214yesmaximal, complemented
6#29972C212yesmaximal, complemented
7#29110C212yesmaximal, complemented
8#28194C212yesmaximal, complemented
9#28114C212yesmaximal
10#27960C211yesmaximal, characteristic, complemented
11#27644C211yesmaximal, characteristic, complemented
12#25477C212yesmaximal, complemented
13#24993C212yesmaximal, complemented
14#24327C212yesmaximal, complemented
15#22510C211yesmaximal, characteristic, complemented
16#22081C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic, complemented
19C22×C22⋊Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#480C2212yes—
23C2×#476C2212yes—
24C2×#447C2211yescharacteristic
25C2×#446C2212yescomplemented
26C2×C4×D8C2212yes—
27#2857C2211yescharacteristic
28#2851C2212yes—
29#2584C2212yescomplemented
30#2580C2212yescomplemented
31#2567C2211yescharacteristic, complemented
32#2538—21no—
33#2538—21no—
34#2538—22no—
35#2538C2212yes—
36#2538C2212yes—
37#2538C2214yes—
38#2538C2211yescharacteristic, complemented
39#2465—22no—
40#2449—22no—
41#2423—21no—
42#2409—21no—
43C2×#396—22no—
44#2360—21no—
45#2355C2211yescharacteristic, complemented
46D4×C4⋊C4—21no—
47C4×#121C2212yes—
48C4×#121C2211yescharacteristic
49C2×#390C2212yes—
50C2×#389C2212yes—
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53#2101C2214yescomplemented
54#2099C2214yescomplemented
55#2088C2214yescomplemented
56#2087C2214yescomplemented
57#2083C2212yescomplemented
58#2073C2214yes—
59#2071C2212yescomplemented
60#2067C2214yes—
61#2059C2214yescomplemented
62#2054C2214yes—
63#2025C2214yes—
64#1990C2212yescomplemented
65#1985C2214yescomplemented
66#1984C2212yescomplemented
67#1947C2212yescomplemented
68#1943C2214yescomplemented
69#1936C2214yes—
70#1936C2214yes—
71#1936C2214yescomplemented
72#1925C2214yes—
73#1921C2214yescomplemented
74#1879C2214yes—
75#1879C2212yes—
76#1874C2211yescharacteristic
77#1864C2214yes—
78#1864C2214yescomplemented
79#1862C2212yes—
80#1853C2212yes—
81#1852C2211yescharacteristic
82#1834C2212yescomplemented
83#1823C2212yes—
84#1823C2212yes—
85#1823C2212yes—
86#1802C2212yes—
87#1802C2212yescomplemented
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1786C2211yescharacteristic
91#1786C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×D8C2312yes—
99#554—44no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—48no—
110C2×#121C2311yescharacteristic
111C2×C4×Q8—44no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D412yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—44no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
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#480—28no—
141#476—44no—
142#447—24no—
143#446—44no—
144C4×D8—44no—
145C2×#107C2314yes—
146C2×#107C2312yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2311yescharacteristic
160C2×#97C2314yes—
161C2×C4×C8C2312yes—
162#399—22no—
163#398—22no—
164#396—24no—
165#396—24no—
166#396—24no—
167#396—24no—
168#396—22no—
169#395—22no—
170#395—22no—
171#393—22no—
172#391—21no—
173#391—21no—
174#391—21no—
175#390—22no—
176#390—22no—
177#390C2312yes—
178#390C2314yes—
179#390C2311yescharacteristic
180#390C2314yes—
181#389—22no—
182#389—22no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#389C2314yes—
189#389C2312yes—
190#389C2311yescharacteristic
191#389C2312yes—
192#389C2314yes—
193#388—22no—
194#388—21no—
195#387—22no—
196#387—22no—
197#387—22no—
198#386—22no—
199#386—21no—
200#385—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384C2311yescharacteristic
208#384C2312yes—
209#384C2312yes—
210#382C2312yes—
211#380—21no—
212#380—21no—
213#380—22no—
214#380—22no—
215#380—24no—
216#380C2311yescharacteristic
217#380C2312yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C2×#95—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236#336C2312yes—
237C25—21no—
238C22×Q8—24no—
239C22×Q8—21no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2×D412yes—
245C22×D4—21no—
246C22×D4—24no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—42no—
251C22×D4C2411yescharacteristic
252C22×D4C2412yes—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2411yescharacteristic
267C2×D8—24no—
268C2×D8—24no—
269C2×D8—42no—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2414yes—
273C4⋊Q8—24no—
274#125—44no—
275C22⋊Q8—48no—
276C22⋊Q8—44no—
277C22⋊Q8—416no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—44no—
281C22⋊Q8—82no—
282#121—22no—
283#121#4114yes—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—42no—
294C4×D4—42no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—84no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—82no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—48no—
358C2×C22⋊C4—84no—
359C2×C42—24no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—44no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371#107—28no—
372#107—42no—
373#106—44no—
374C4⋊C8—24no—
375C4⋊C8—44no—
376#102—28no—
377#102—28no—
378#102—24no—
379#102—24no—
380#102—42no—
381#102—42no—
382#102—44no—
383C22⋊C8—22no—
384C22⋊C8—24no—
385C22⋊C8—22no—
386#97—44no—
387C4×C8—24no—
388#95—24no—
389#95—24no—
390#95—21no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401C24—24no—
402C24—21no—
403C24—22no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24C22×D411yescharacteristic
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—416no—
421C2×Q8—42no—
422C2×Q8—81no—
423C2×Q8—81no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—48no—
435C2×D4—48no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C2×#4112yes—
441C2×D42−1+414yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—816no—
481C22×C4AGL(1,ℤ/8)14yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4112yes—
492C22×C42−1+412yes—
493C22×C4C2511yescharacteristic
494D8—48no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—81no—
537C4⋊C4—82no—
538C4⋊C4C2×#4112yes—
539C22⋊C4—24no—
540C22⋊C4—44no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—81no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—82no—
554C42—22no—
555C42—22no—
556C42—24no—
557C42—22no—
558C42—24no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—82no—
563C42H(5,𝔽2)14yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—24no—
570C23—48no—
571C23—48no—
572C23—48no—
573C23—41no—
574C23—42no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—41no—
579C23—42no—
580C23—41no—
581C23—41no—
582C23—42no—
583C23—42no—
584C23—41no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23#54812yes—
590C23C2×AGL(1,ℤ/8)12yes—
591C23#57612yes—
592C23C23×D411yescharacteristic
593Q8—48no—
594Q8—44no—
595Q8—44no—
596Q8—162no—
597D4—24no—
598D4—24no—
599D4—24no—
600D4—48no—
601D4—48no—
602D4—42no—
603D4—42no—
604D4—42no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—88no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—88no—
656C2×C4—84no—
657C2×C4—88no—
658C2×C4—164no—
659C2×C4#53412yes—
660C2×C4#53412yes—
661C2×C4#54812yes—
662C2×C4C2×AGL(1,ℤ/8)12yes—
663C2×C4C2×AGL(1,ℤ/8)12yes—
664C2×C4#57512yes—
665C2×C4C22×#4111yescharacteristic
666C2×C4C2×H(5,𝔽2)12yes—
667C2×C4C2×2−1+412yes—
668C2×C4C2×2−1+411yescharacteristic
669C8—44no—
670C8—42no—
671C8—42no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—42no—
679C22—44no—
680C22—42no—
681C22—44no—
682C22—44no—
683C22—42no—
684C22—816no—
685C22—82no—
686C22—81no—
687C22—81no—
688C22—82no—
689C22—82no—
690C22—81no—
691C22—82no—
692C22—81no—
693C22#336614yes—
694C22C2×#53411yescharacteristic
695C22C2×#54811yescharacteristic
696C22#348611yescharacteristic
697C22C22×AGL(1,ℤ/8)11yescharacteristic
698C22C2×#57511yescharacteristic
699C22C2×#57611yescharacteristic
700C22#358611yescharacteristic
701C4—22no—
702C4—22no—
703C4—22no—
704C4—42no—
705C4—42no—
706C4—41no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—81no—
711C4—82no—
712C4—82no—
713C4—84no—
714C4—81no—
715C4—82no—
716C4#319414yes—
717C4#352012yes—
718C4#353812yes—
719C2—24no—
720C2—82no—
721C2—84no—
722C2#3912111yescharacteristic
723C2C2×#319411yescharacteristic
724C2#6076211yescharacteristic
725C2C2×#336611yescharacteristic
726C2#6132311yescharacteristic
727C2#6182711yescharacteristic
728C2#6183511yescharacteristic
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. #36544C2semidirect
2. C2×#2538C2semidirect
3. #32956C2semidirect
4. #32581C2semidirect
5. #32554C2semidirect
6. #29972C2semidirect
7. #29110C2semidirect
8. #28194C2semidirect
10. #27960C2semidirect
11. #27644C2semidirect
12. #25477C2semidirect
13. #24993C2semidirect
14. #24327C2semidirect
15. #22510C2semidirect
16. #22081C2semidirect
18. C2×#554C22semidirect
25. C2×#446C22semidirect
29. #2584C22semidirect
30. #2580C22semidirect
31. #2567C22semidirect
38. #2538C22semidirect
45. #2355C22semidirect
53. #2101C22semidirect
54. #2099C22semidirect
55. #2088C22semidirect
56. #2087C22semidirect
57. #2083C22semidirect
59. #2071C22semidirect
61. #2059C22semidirect
64. #1990C22semidirect
65. #1985C22semidirect
66. #1984C22semidirect
67. #1947C22semidirect
68. #1943C22semidirect
71. #1936C22semidirect
73. #1921C22semidirect
78. #1864C22semidirect
82. #1834C22semidirect
87. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#365442
C2×#25382
#329562
#325812
#325542
#299722
#291102
#281942
#281142
#279602
#276442
#254772
#249932
#243272
#225102
#220812
C4×#4472