Finite Group Databasebeta

← #6605589

Subgroups — #6605589

Order 512.

1951 conjugacy classes of subgroups.

The subgroup lattice has 710 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#32985C212yesmaximal, complemented
5#27948C211yesmaximal, characteristic, complemented
6#27681C211yesmaximal, characteristic, complemented
7#27602C212yesmaximal, complemented
8#26301C212yesmaximal, complemented
9#26139C214yesmaximal, complemented
10#25015C212yesmaximal, complemented
11#24946C214yesmaximal, complemented
12#24551C212yesmaximal, complemented
13#22657C212yesmaximal, complemented
14#22490C212yesmaximal, complemented
15D4×C22⋊C8C211yesmaximal, characteristic
16C4×#483C211yesmaximal, characteristic, complemented
17#20843C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C2×#485C2214yescomplemented
22C2×#483C2211yescharacteristic
23C2×C8×D4C2212yes—
24C2×#427C2214yescomplemented
25C22×C22⋊C8C2212yes—
26#2857C2211yescharacteristic
27#2848C2212yes—
28#2547C2211yescharacteristic
29#2538C2211yescharacteristic
30#2485C2212yes—
31#2480—22no—
32#2466—22no—
33#2447—21no—
34#2435—21no—
35#2415C2212yes—
36#2367—21no—
37#2360—21no—
38#2354—21no—
39#2354—21no—
40#2354—22no—
41#2354C2214yes—
42#2354C2211yescharacteristic
43#2354C2212yes—
44#2354C2212yes—
45C4×C22⋊Q8C2211yescharacteristic
46C2×#390—21no—
47C2×#388—22no—
48C2×#384C2211yescharacteristic
49#2280C2212yes—
50C2×#380C2212yes—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2100C2212yescomplemented
54#2095C2212yescomplemented
55#2084C2214yescomplemented
56#2077C2214yescomplemented
57#2077C2214yescomplemented
58#2076C2214yescomplemented
59#2035C2214yescomplemented
60#1990C2212yescomplemented
61#1988C2212yescomplemented
62#1954C2212yes—
63#1947C2212yescomplemented
64#1947C2212yescomplemented
65#1927C2212yescomplemented
66#1920C2212yescomplemented
67#1918C2214yescomplemented
68#1892C2214yescomplemented
69#1873C2214yescomplemented
70#1872C2212yes—
71#1872C2211yescharacteristic
72#1869C2214yes—
73#1854C2214yescomplemented
74#1854C2214yescomplemented
75#1854C2214yescomplemented
76#1854C2214yescomplemented
77#1853C2212yes—
78#1853C2212yes—
79#1853C2211yescharacteristic
80#1852C2212yes—
81#1852C2212yes—
82#1844C2212yescomplemented
83#1825C2214yescomplemented
84#1825C2214yescomplemented
85#1823C2211yescharacteristic
86#1823C2212yes—
87#1807C2214yescomplemented
88#1803C2212yescomplemented
89#1797C2211yescharacteristic
90#1779C2212yes—
91C4×#106C2212yescomplemented
92C4×#103C2212yescomplemented
93C4×C22⋊C8C2211yescharacteristic
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—48no—
119C2×#117C2312yes—
120C2×#117C2314yes—
121C2×#117C2314yes—
122C2×#117C2312yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4D412yes—
132C22×C42—22no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135#485—48no—
136#483—24no—
137C8×D4—28no—
138#427—48no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×C22⋊C8—28no—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C4×C8C2312yes—
157#397—22no—
158#397—22no—
159#397—22no—
160#396—22no—
161#394—22no—
162#394—22no—
163#391—21no—
164#391—21no—
165#391C2311yescharacteristic
166#390—22no—
167#390—24no—
168#390—22no—
169#390—21no—
170#390—21no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389C2312yes—
175#389C2311yescharacteristic
176#388—24no—
177#388—24no—
178#388—24no—
179#388—24no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—21no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—21no—
189#384—21no—
190#384—21no—
191#384C2312yes—
192#384C2312yes—
193#384C2311yescharacteristic
194#381—22no—
195#380—21no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2314yes—
200#380C2312yes—
201#380C2314yes—
202#380C2312yes—
203#380C2312yes—
204#380C2312yes—
205#380C2311yescharacteristic
206#379—22no—
207#379—22no—
208C4×C4⋊C4—24no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222C2×#95—21no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
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—28no—
247C23×C4—24no—
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—28no—
262C22×C8—44no—
263C22×C8C2414yes—
264C22×C8C2412yes—
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—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
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—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—48no—
342C2×C22⋊C4—84no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—48no—
357C2×C42—44no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—28no—
363#107—44no—
364#106—24no—
365#106—24no—
366#106—44no—
367#106—44no—
368C4⋊C8—24no—
369#103—24no—
370#103—24no—
371#103—44no—
372C22⋊C8—24no—
373C22⋊C8—28no—
374C22⋊C8—22no—
375C22⋊C8—28no—
376C22⋊C8—44no—
377C22⋊C8#4114yes—
378C4×C8—24no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24C22×D412yes—
402C24C22×D412yes—
403C24C22×D411yescharacteristic
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—416no—
419C2×D4—42no—
420C2×D4#12314yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—41no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4#13514yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4118yes—
473C22×C4C2×#4112yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—416no—
485C2×C8C2×#4114yes—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4#12314yes—
516C4⋊C4H(5,𝔽2)14yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4#12318yes—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—48no—
540C42—48no—
541C42—44no—
542C42—42no—
543C42—44no—
544C42—44no—
545C42—42no—
546C42—82no—
547C42C2×#4114yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
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—22no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—48no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23C2×#12314yes—
573C23#54712yes—
574C23C2×#13512yes—
575C23#57712yes—
576C23C23×D411yescharacteristic
577Q8—42no—
578Q8—42no—
579Q8—84no—
580D4—28no—
581D4—48no—
582D4—44no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4C2×#12312yes—
639C2×C4C2×#12314yes—
640C2×C4C2×#12312yes—
641C2×C4#54712yes—
642C2×C4C2×#13512yes—
643C2×C4C2×#13512yes—
644C2×C4#57512yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4112yes—
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4C2×2−1+411yescharacteristic
649C8—24no—
650C8—44no—
651C8—42no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—816no—
673C22#334014yes—
674C22C22×#12311yescharacteristic
675C22C2×#54711yescharacteristic
676C22#348411yescharacteristic
677C22C22×#13511yescharacteristic
678C22C2×#57511yescharacteristic
679C22C2×#57711yescharacteristic
680C22#358611yescharacteristic
681C4—22no—
682C4—22no—
683C4—22no—
684C4—24no—
685C4—41no—
686C4—44no—
687C4—42no—
688C4—82no—
689C4—81no—
690C4—82no—
691C4—81no—
692C4—82no—
693C4—81no—
694C4—81no—
695C4—84no—
696C4#309114yes—
697C4#348112yes—
698C4#352012yes—
699C2—24no—
700C2—44no—
701C2—82no—
702C2#3623711yescharacteristic
703C2C2×#309111yescharacteristic
704C2#6052211yescharacteristic
705C2C2×#334011yescharacteristic
706C2#6125711yescharacteristic
707C2#6156511yescharacteristic
708C2#6159611yescharacteristic
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. #32985C2semidirect
5. #27948C2semidirect
6. #27681C2semidirect
7. #27602C2semidirect
8. #26301C2semidirect
9. #26139C2semidirect
10. #25015C2semidirect
11. #24946C2semidirect
12. #24551C2semidirect
13. #22657C2semidirect
14. #22490C2semidirect
16. C4×#483C2semidirect
17. #20843C2semidirect
21. C2×#485C22semidirect
24. C2×#427C22semidirect
53. #2100C22semidirect
54. #2095C22semidirect
55. #2084C22semidirect
56. #2077C22semidirect
57. #2077C22semidirect
58. #2076C22semidirect
59. #2035C22semidirect
60. #1990C22semidirect
61. #1988C22semidirect
63. #1947C22semidirect
64. #1947C22semidirect
65. #1927C22semidirect
66. #1920C22semidirect
67. #1918C22semidirect
68. #1892C22semidirect
69. #1873C22semidirect
73. #1854C22semidirect
74. #1854C22semidirect
75. #1854C22semidirect
76. #1854C22semidirect
82. #1844C22semidirect
83. #1825C22semidirect
84. #1825C22semidirect
87. #1807C22semidirect
88. #1803C22semidirect
91. C4×#106C22semidirect
92. C4×#103C22semidirect
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
#329852
#279482
#276812
#276022
#263012
#261392
#250152
#249462
#245512
#226572
#224902
D4×C22⋊C82
C4×#4832
#208432