Finite Group Databasebeta

← #6592129

Subgroups — #6592129

Order 512.

1905 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36161C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3#33146C211yesmaximal, characteristic, complemented
4#32984C212yesmaximal, complemented
5#27961C211yesmaximal, characteristic, complemented
6#27648C211yesmaximal, characteristic, complemented
7#27604C212yesmaximal, complemented
8#26301C212yesmaximal, complemented
9#26211C214yesmaximal, complemented
10#24970C212yesmaximal, complemented
11#24948C214yesmaximal, complemented
12#24552C212yesmaximal, complemented
13#22707C212yesmaximal, complemented
14#22490C212yesmaximal, complemented
15D4×C22⋊C8C211yesmaximal, characteristic
16C4×#483C211yesmaximal, characteristic, complemented
17#20699C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C2×#484C2212yes—
22C2×#483C2211yescharacteristic
23C2×C8×D4C2212yes—
24C22×C22⋊C8C2212yes—
25#2857C2211yescharacteristic
26#2848C2212yes—
27#2547C2211yescharacteristic
28#2538C2211yescharacteristic
29#2485C2212yes—
30#2480—22no—
31#2466—22no—
32#2447—21no—
33#2435—21no—
34#2415C2212yes—
35#2367—21no—
36#2360—21no—
37#2354—21no—
38#2354—21no—
39#2354—22no—
40#2354C2211yescharacteristic
41#2354C2214yes—
42#2354C2212yes—
43#2354C2212yes—
44C4×C22⋊Q8C2211yescharacteristic
45C2×#390—21no—
46C2×#388—22no—
47C2×#384C2211yescharacteristic
48#2280C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yescomplemented
53#2083C2212yescomplemented
54#2077C2214yescomplemented
55#2076C2214yescomplemented
56#2075C2214yescomplemented
57#2072C2212yescomplemented
58#2036C2214yescomplemented
59#1990C2212yescomplemented
60#1988C2212yescomplemented
61#1954C2212yes—
62#1947C2212yescomplemented
63#1947C2212yescomplemented
64#1927C2212yescomplemented
65#1920C2212yescomplemented
66#1918C2214yescomplemented
67#1892C2214yescomplemented
68#1873C2214yescomplemented
69#1873C2214yescomplemented
70#1872C2211yescharacteristic
71#1869C2214yes—
72#1854C2214yescomplemented
73#1854C2214yescomplemented
74#1854C2214yescomplemented
75#1853C2211yescharacteristic
76#1852C2212yes—
77#1834C2212yescomplemented
78#1825C2214yescomplemented
79#1825C2214yescomplemented
80#1825C2214yescomplemented
81#1824C2212yes—
82#1824C2212yes—
83#1824C2212yes—
84#1823C2211yescharacteristic
85#1823C2212yes—
86#1808C2214yescomplemented
87#1803C2212yescomplemented
88#1797C2211yescharacteristic
89#1779C2212yes—
90C4×#106C2212yescomplemented
91C4×#103C2212yescomplemented
92C4×C22⋊C8C2211yescharacteristic
93#1735C2214yescomplemented
94#1735C2214yescomplemented
95C23×D4D411yescharacteristic
96C24×C4C2312yes—
97C23×C8C2314yes—
98#556—28no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#126—22no—
101C2×#124—21no—
102C2×#124—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C4×Q8—21no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×D4—28no—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D412yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C2×#117—22no—
118C2×#117—28no—
119C2×#117—44no—
120C2×#117C2312yes—
121C2×#117C2314yes—
122C2×#117C2314yes—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4D412yes—
133C22×C42—22no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136#484—44no—
137#483—24no—
138C8×D4—28no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×C22⋊C8—44no—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C4×C8C2312yes—
155#397—22no—
156#397—22no—
157#397—22no—
158#396—22no—
159#394—22no—
160#394—22no—
161#391—21no—
162#391—21no—
163#391C2311yescharacteristic
164#390—22no—
165#390—24no—
166#390—22no—
167#390—21no—
168#390—21no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389C2311yescharacteristic
173#389C2312yes—
174#388—24no—
175#388—24no—
176#388—24no—
177#388—24no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—21no—
183#385—22no—
184#385—22no—
185#385—22no—
186#385—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#381—22no—
193#380—21no—
194#380—22no—
195#380—22no—
196#380C2312yes—
197#380C2314yes—
198#380C2312yes—
199#380C2314yes—
200#380C2312yes—
201#380C2312yes—
202#380C2312yes—
203#380C2311yescharacteristic
204#379—22no—
205#379—22no—
206C4×C4⋊C4—24no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C2×#95—22no—
220C2×#95—21no—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236C25C2×D412yes—
237C22×Q8—21no—
238C22×Q8C2411yescharacteristic
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—42no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2×D412yes—
244C22×D4C2×D411yescharacteristic
245C23×C4—21no—
246C23×C4—24no—
247C23×C4—28no—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—44no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C22×C8—28no—
261C22×C8—44no—
262C22×C8—44no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267#126—48no—
268#124—44no—
269#124—44no—
270C22⋊Q8—28no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×D4—24no—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4#4118yes—
283#117—28no—
284#117—28no—
285#117—28no—
286#117—44no—
287#117—44no—
288#117—48no—
289#117—48no—
290#117—44no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4#4114yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—48no—
343C2×C22⋊C4—84no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—48no—
358C2×C42—44no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—44no—
364#107—44no—
365#106—24no—
366#106—42no—
367C4⋊C8—24no—
368#103—24no—
369#103—42no—
370#103—44no—
371C22⋊C8—28no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—24no—
375C22⋊C8—44no—
376C22⋊C8—44no—
377C4×C8—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393C24—24no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24C22×D412yes—
401C24C22×D412yes—
402C24C22×D411yescharacteristic
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—416no—
418C2×D4—42no—
419C2×D4#12314yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4#13514yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4118yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4C2511yescharacteristic
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—88no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4#12314yes—
515C4⋊C4C2×#4114yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4#12318yes—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—48no—
539C42—48no—
540C42—44no—
541C42—42no—
542C42—44no—
543C42—44no—
544C42—42no—
545C42—82no—
546C42C2×#4114yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23C2×#12314yes—
572C23#54612yes—
573C23C2×#13512yes—
574C23#57812yes—
575C23C23×D411yescharacteristic
576Q8—42no—
577Q8—42no—
578Q8—84no—
579D4—28no—
580D4—48no—
581D4—44no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4C2×#12312yes—
638C2×C4C2×#12314yes—
639C2×C4C2×#12312yes—
640C2×C4#54612yes—
641C2×C4C2×#13512yes—
642C2×C4C2×#13512yes—
643C2×C4#57512yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4112yes—
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4#58512yes—
648C8—42no—
649C8—44no—
650C8—42no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—816no—
672C22#330214yes—
673C22C22×#12311yescharacteristic
674C22C2×#54611yescharacteristic
675C22#348511yescharacteristic
676C22C22×#13511yescharacteristic
677C22C2×#57511yescharacteristic
678C22C2×#57811yescharacteristic
679C22#358511yescharacteristic
680C4—22no—
681C4—24no—
682C4—22no—
683C4—22no—
684C4—42no—
685C4—41no—
686C4—44no—
687C4—82no—
688C4—82no—
689C4—81no—
690C4—81no—
691C4—82no—
692C4—81no—
693C4—81no—
694C4—84no—
695C4#309114yes—
696C4#347512yes—
697C4#351312yes—
698C2—24no—
699C2—44no—
700C2—82no—
701C2#3615511yescharacteristic
702C2C2×#309111yescharacteristic
703C2#6052611yescharacteristic
704C2C2×#330211yescharacteristic
705C2#6116011yescharacteristic
706C2#6170011yescharacteristic
707C2#6173011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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
3. #33146C2semidirect
4. #32984C2semidirect
5. #27961C2semidirect
6. #27648C2semidirect
7. #27604C2semidirect
8. #26301C2semidirect
9. #26211C2semidirect
10. #24970C2semidirect
11. #24948C2semidirect
12. #24552C2semidirect
13. #22707C2semidirect
14. #22490C2semidirect
16. C4×#483C2semidirect
17. #20699C2semidirect
52. #2100C22semidirect
53. #2083C22semidirect
54. #2077C22semidirect
55. #2076C22semidirect
56. #2075C22semidirect
57. #2072C22semidirect
58. #2036C22semidirect
59. #1990C22semidirect
60. #1988C22semidirect
62. #1947C22semidirect
63. #1947C22semidirect
64. #1927C22semidirect
65. #1920C22semidirect
66. #1918C22semidirect
67. #1892C22semidirect
68. #1873C22semidirect
69. #1873C22semidirect
72. #1854C22semidirect
73. #1854C22semidirect
74. #1854C22semidirect
77. #1834C22semidirect
78. #1825C22semidirect
79. #1825C22semidirect
80. #1825C22semidirect
86. #1808C22semidirect
87. #1803C22semidirect
90. C4×#106C22semidirect
91. C4×#103C22semidirect
93. #1735C22semidirect
94. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#361612
C2×#23542
#331462
#329842
#279612
#276482
#276042
#263012
#262112
#249702
#249482
#245522
#227072
#224902
D4×C22⋊C82
C4×#4832
#206992