Finite Group Databasebeta

← #6795197

Subgroups — #6795197

Order 512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36160C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32277C214yesmaximal, complemented
5#31349C214yesmaximal, complemented
6#29890C212yesmaximal, complemented
7#28711C212yesmaximal, complemented
8#27604C212yesmaximal, complemented
9#26299C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic, complemented
11#25704C212yesmaximal, complemented
12#25346C212yesmaximal, complemented
13#24557C212yesmaximal, complemented
14#24299C211yesmaximal, characteristic, complemented
15#22401C211yesmaximal, characteristic, complemented
16#21944C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic, complemented
18C2×#555C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#500C2212yescomplemented
22C2×#481C2211yescharacteristic
23C2×#476C2212yescomplemented
24C2×#436C2212yes—
25C2×#407C2212yes—
26#2857C2211yescharacteristic
27#2847C2212yes—
28#2563—21no—
29#2563—21no—
30#2563—22no—
31#2563C2212yes—
32#2563C2214yes—
33#2563C2211yescharacteristic
34#2563C2212yes—
35#2537C2211yescharacteristic
36#2485C2212yes—
37#2472—22no—
38#2459—22no—
39#2420—21no—
40#2414C2212yes—
41#2408—21no—
42C2×#393—22no—
43#2363—21no—
44#2354C2211yescharacteristic
45#2352—21no—
46C4×#121C2211yescharacteristic
47C2×#390—21no—
48C2×#382—21no—
49#2280C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2077C2214yescomplemented
54#2077C2214yescomplemented
55#2075C2214yescomplemented
56#2075C2214yescomplemented
57#2075C2214yescomplemented
58#2058C2212yescomplemented
59#2057C2214yescomplemented
60#2033C2214yescomplemented
61#2033C2214yescomplemented
62#2033C2214yescomplemented
63#2023C2212yescomplemented
64#1990C2212yescomplemented
65#1987C2212yescomplemented
66#1987C2212yescomplemented
67#1954C2212yes—
68#1947C2212yescomplemented
69#1926C2212yescomplemented
70#1921C2214yescomplemented
71#1919C2212yescomplemented
72#1895C2214yescomplemented
73#1874C2211yescharacteristic
74#1874C2212yes—
75#1870C2214yes—
76#1854C2214yescomplemented
77#1854C2214yescomplemented
78#1854C2214yescomplemented
79#1853C2211yescharacteristic
80#1853C2212yes—
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1825C2214yescomplemented
85#1824C2211yescharacteristic
86#1824C2212yes—
87#1809C2214yescomplemented
88#1809C2214yescomplemented
89#1809C2214yescomplemented
90#1797C2211yescharacteristic
91#1779C2212yes—
92C4×#106C2212yescomplemented
93C4×#102C2212yescomplemented
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4D411yescharacteristic
97C24×C4C2312yes—
98C22×M16C2314yes—
99#555—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#124—22no—
107C2×C22⋊Q8—21no—
108C2×#121—21no—
109C2×#121—21no—
110C2×#121C2311yescharacteristic
111C2×#121C2312yes—
112C2×#121C2312yes—
113C2×#120—22no—
114C2×C4×D4—21no—
115C2×C4×D4—44no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D412yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4D412yes—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136#500—44no—
137#481—24no—
138#476—44no—
139#436—44no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2312yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150#407—28no—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C4×C8C2312yes—
158#397—22no—
159#394—22no—
160#394—22no—
161#394—22no—
162#393—24no—
163#393—24no—
164#393—24no—
165#393—24no—
166#393—22no—
167#392—22no—
168#391—21no—
169#390—22no—
170#390—22no—
171#390—24no—
172#390—21no—
173#390—21no—
174#390—22no—
175#390C2311yescharacteristic
176#389—21no—
177#389C2312yes—
178#389C2311yescharacteristic
179#388—22no—
180#388—22no—
181#388—22no—
182#388—21no—
183#387—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—21no—
188#384C2311yescharacteristic
189#382—22no—
190#382—22no—
191#382—24no—
192#382—21no—
193#381—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—21no—
199#380—21no—
200#380C2314yes—
201#380C2314yes—
202#380C2314yes—
203#380C2314yes—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#380C2311yescharacteristic
210#379—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95—22no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2312yes—
235#336C2312yes—
236C25C2×D412yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—82no—
242C22×Q8—82no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—24no—
247C22×D4—42no—
248C22×D4—42no—
249C22×D4C2×D411yescharacteristic
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D412yes—
252C22×D4C2411yescharacteristic
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2412yes—
267C23×C4C2412yes—
268C23×C4C2411yescharacteristic
269C2×M16—28no—
270C2×M16—44no—
271C2×M16—44no—
272C22×C8C2414yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281#124—48no—
282C22⋊Q8—82no—
283#121—22no—
284#121—22no—
285#121—44no—
286#121—44no—
287#121—44no—
288#121—44no—
289#120—84no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—44no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298#117—44no—
299#117—44no—
300#117—44no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—84no—
348C2×C22⋊C4—84no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—44no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—44no—
365#107—44no—
366#107—44no—
367#106—24no—
368#106—42no—
369#106—44no—
370C4⋊C8—24no—
371#102—24no—
372#102—42no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—22no—
376C22⋊C8—28no—
377C22⋊C8—44no—
378C22⋊C8—44no—
379C22⋊C8#4114yes—
380C4×C8—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—21no—
394#95—22no—
395#95—22no—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—21no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24C22×D412yes—
405C24C22×D412yes—
406C24C22×D411yescharacteristic
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—81no—
415C2×Q8—81no—
416C2×Q8—816no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—416no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—88no—
485C22×C4—88no—
486C22×C4#13714yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C2×#4112yes—
495C22×C42−1+412yes—
496C22×C4C2511yescharacteristic
497M16—88no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8C2×#4114yes—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—81no—
531C4⋊C4—82no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—81no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—44no—
558C42—44no—
559C42—42no—
560C42—44no—
561C42—82no—
562C42H(5,𝔽2)14yes—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—48no—
582C23—48no—
583C23—48no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23#54712yes—
596C23C2×#13712yes—
597C23#57612yes—
598C23C23×D411yescharacteristic
599Q8—88no—
600Q8—88no—
601Q8—162no—
602D4—48no—
603D4—44no—
604D4—44no—
605D4—42no—
606D4—42no—
607D4—84no—
608D4—164no—
609D4—164no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—28no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—44no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—88no—
663C2×C4—84no—
664C2×C4#53412yes—
665C2×C4#53412yes—
666C2×C4#54712yes—
667C2×C4C2×#13712yes—
668C2×C4C2×#13712yes—
669C2×C4#57512yes—
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)12yes—
672C2×C4C2×2−1+411yescharacteristic
673C2×C4#58512yes—
674C8—24no—
675C8—44no—
676C8—42no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—816no—
698C22—81no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22#335714yes—
706C22C2×#53411yescharacteristic
707C22C2×#54711yescharacteristic
708C22#348511yescharacteristic
709C22C22×#13711yescharacteristic
710C22C2×#57511yescharacteristic
711C22C2×#57611yescharacteristic
712C22#358511yescharacteristic
713C4—22no—
714C4—22no—
715C4—22no—
716C4—42no—
717C4—42no—
718C4—41no—
719C4—81no—
720C4—82no—
721C4—82no—
722C4—81no—
723C4—81no—
724C4—82no—
725C4—84no—
726C4—82no—
727C4#319614yes—
728C4#351412yes—
729C4#356112yes—
730C2—24no—
731C2—82no—
732C2—82no—
733C2—161no—
734C2#3891811yescharacteristic
735C2C2×#319611yescharacteristic
736C2#6075611yescharacteristic
737C2C2×#335711yescharacteristic
738C2#6125611yescharacteristic
739C2#6171911yescharacteristic
740C2#6174011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36160C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2563C2semidirect
3. #33149C2semidirect
4. #32277C2semidirect
5. #31349C2semidirect
6. #29890C2semidirect
7. #28711C2semidirect
8. #27604C2semidirect
9. #26299C2semidirect
10. #25813C2semidirect
11. #25704C2semidirect
12. #25346C2semidirect
13. #24557C2semidirect
14. #24299C2semidirect
15. #22401C2semidirect
16. #21944C2semidirect
17. C4×#481C2semidirect
21. C2×#500C22semidirect
23. C2×#476C22semidirect
53. #2077C22semidirect
54. #2077C22semidirect
55. #2075C22semidirect
56. #2075C22semidirect
57. #2075C22semidirect
58. #2058C22semidirect
59. #2057C22semidirect
60. #2033C22semidirect
61. #2033C22semidirect
62. #2033C22semidirect
63. #2023C22semidirect
64. #1990C22semidirect
65. #1987C22semidirect
66. #1987C22semidirect
68. #1947C22semidirect
69. #1926C22semidirect
70. #1921C22semidirect
71. #1919C22semidirect
72. #1895C22semidirect
76. #1854C22semidirect
77. #1854C22semidirect
78. #1854C22semidirect
84. #1825C22semidirect
87. #1809C22semidirect
88. #1809C22semidirect
89. #1809C22semidirect
92. C4×#106C22semidirect
93. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#361602
C2×#25632
#331492
#322772
#313492
#298902
#287112
#276042
#262992
#258132
#257042
#253462
#245572
#242992
#224012
#219442
C4×#4812