Finite Group Databasebeta

← #6561797

Subgroups — #6561797

Order 512.

1925 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36258C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29921C211yesmaximal, characteristic, complemented
4#29910C212yesmaximal, complemented
5#27973C211yesmaximal, characteristic, complemented
6#27863C212yesmaximal, complemented
7#27644C211yesmaximal, characteristic, complemented
8#27585C212yesmaximal, complemented
9#26321C212yesmaximal, complemented
10#26219C214yesmaximal, complemented
11#24991C212yesmaximal, complemented
12#24882C214yesmaximal, complemented
13#23755C211yesmaximal, characteristic, complemented
14#23628C212yesmaximal, complemented
15#22715C212yesmaximal, complemented
16#22473C211yesmaximal, characteristic, complemented
17#22428C212yesmaximal, complemented
18C2×#546—21no—
19C2×D42C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#480C2212yes—
22C2×#436C2212yes—
23C2×#418C2214yes—
24#2856C2211yescharacteristic
25#2854C2212yes—
26#2548C2211yescharacteristic
27#2538C2211yescharacteristic
28#2426—21no—
29#2424—22no—
30#2416C2212yes—
31#2414—21no—
32#2411—22no—
33#2408C2212yes—
34#2355—21no—
35#2355C2212yescomplemented
36#2354—21no—
37#2354C2212yes—
38#2352—22no—
39#2352C2214yescomplemented
40D4×C4⋊C4C2211yescharacteristic
41D4×C22⋊C4—21no—
42#2314—22no—
43#2311—21no—
44C4×#121C2211yescharacteristic
45C2×#390C2212yes—
46#2285C2212yescomplemented
47#2285C2211yescharacteristic, complemented
48#2285C2212yescomplemented
49C2×#384C2211yescharacteristic, complemented
50#2274—21no—
51#2101C2214yescomplemented
52#2100C2211yescharacteristic, complemented
53#2094C2211yescharacteristic
54#2094C2212yescomplemented
55#2092C2214yescomplemented
56#2084C2211yescharacteristic, complemented
57#2083C2212yescomplemented
58#2077C2212yescomplemented
59#2073C2214yes—
60#2073C2214yescomplemented
61#2071C2211yescharacteristic
62#2071C2212yes—
63#2071C2212yescomplemented
64#2071C2212yescomplemented
65#2071C2212yescomplemented
66#2034C2212yes—
67#2034C2214yescomplemented
68#1984C2212yescomplemented
69#1982C2212yes—
70#1957C2212yes—
71#1948C2212yescomplemented
72#1947C2212yescomplemented
73#1926C2214yes—
74#1926C2214yescomplemented
75#1925C2214yescomplemented
76#1920C2214yescomplemented
77#1895C2214yes—
78#1875C2214yes—
79#1874C2214yescomplemented
80#1870C2214yescomplemented
81#1854C2214yescomplemented
82#1853C2214yescomplemented
83#1834C2212yescomplemented
84#1823C2212yescomplemented
85#1817C2212yescomplemented
86#1809C2214yescomplemented
87#1802C2212yes—
88#1797C2211yescharacteristic, complemented
89#1786C2214yes—
90C4×#106C2212yescomplemented
91C4×#102C2212yes—
92C4×C22⋊C8C2211yescharacteristic, complemented
93#1736C2214yescomplemented
94#1728C2212yescomplemented
95C22×#41D411yescharacteristic
96C23×D4—21no—
97C23×D4C2312yes—
98C22×M16C2314yes—
99#546—48no—
100D42—28no—
101C2×#127C2311yescharacteristic
102C2×#124—21no—
103C2×#123—22no—
104C2×C22⋊Q8—21no—
105C2×#121—22no—
106C2×#121—21no—
107C2×#121—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×#121C2311yescharacteristic
111C2×#121C2312yes—
112C2×#120—22no—
113C2×#120—22no—
114C4×#41—28no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D412yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117—21no—
125C2×#117—22no—
126C2×#117D412yes—
127C2×#117D411yescharacteristic
128C2×#117C2314yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#480—44no—
138#436—28no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144#418—48no—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C4×C8C2312yes—
156#396—22no—
157#396—22no—
158#394—22no—
159#394—22no—
160#394—22no—
161#392—22no—
162#391—22no—
163#390—22no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#390C2311yescharacteristic
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#389—21no—
175#389C2311yescharacteristic
176#389C2312yes—
177#388—24no—
178#388—24no—
179#388—22no—
180#388—22no—
181#388—21no—
182#387—24no—
183#387—24no—
184#386—24no—
185#386—22no—
186#386—21no—
187#386—22no—
188#385—22no—
189#384—21no—
190#384—21no—
191#384C2312yes—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382—21no—
197#382—22no—
198#382C2314yes—
199#380—22no—
200#380—22no—
201#380—21no—
202#380—22no—
203#380—21no—
204#380C2312yes—
205#380C2314yes—
206#380C2311yescharacteristic
207#380C2312yes—
208#379—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—24no—
219C4×C22⋊C4—21no—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C25—22no—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—42no—
238C22×Q8C2×D411yescharacteristic
239C22×D4—24no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—28no—
247C22×D4—44no—
248C22×D4—42no—
249C22×D4—44no—
250C22×D4—44no—
251C22×D4—44no—
252C22×D4C2×D412yes—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C22×D4C2411yescharacteristic
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264C23×C4C2412yes—
265C2×M16—28no—
266C2×M16—44no—
267C2×M16—44no—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271#127—24no—
272#124—44no—
273#123—48no—
274C22⋊Q8—44no—
275#121—22no—
276#121—28no—
277#121—41no—
278#121—48no—
279#121—44no—
280#121—44no—
281#121—44no—
282#121—82no—
283#120—48no—
284#120—48no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×D4—24no—
288C4×D4—28no—
289C4×D4—28no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4#4118yes—
294#117—24no—
295#117—28no—
296#117—44no—
297#117—44no—
298#117—44no—
299#117—84no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—48no—
350C2×C22⋊C4—84no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—28no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—44no—
368#106—42no—
369#106—44no—
370#106—42no—
371C4⋊C8—24no—
372#102—24no—
373#102—28no—
374#102—44no—
375#102—42no—
376#102—44no—
377#102—44no—
378#102—44no—
379C22⋊C8—28no—
380C22⋊C8—42no—
381C22⋊C8—42no—
382C4×C8—24no—
383#95—24no—
384#95—24no—
385#95—21no—
386#95—21no—
387#95—24no—
388#95—22no—
389#95—24no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—22no—
394C24—21no—
395C24—22no—
396C24—44no—
397C24—44no—
398C24—44no—
399C24—44no—
400C24—82no—
401C24—82no—
402C24—82no—
403C24C22×D412yes—
404C24C22×D411yescharacteristic
405#41—416no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—416no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—88no—
435C2×D4—82no—
436C2×D4—81no—
437C2×D4—81no—
438C2×D4—81no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—816no—
442C2×D4#12314yes—
443C2×D4H(5,𝔽2)12yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4AGL(1,ℤ/8)14yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4118yes—
488C22×C4C2511yescharacteristic
489M16—88no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—81no—
527C4⋊C4#12314yes—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—44no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4#12318yes—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—24no—
554C42—22no—
555C42—24no—
556C42—42no—
557C42—44no—
558C42—84no—
559C42—84no—
560C42C2×#4114yes—
561C23—22no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—21no—
566C23—22no—
567C23—21no—
568C23—22no—
569C23—21no—
570C23—22no—
571C23—22no—
572C23—21no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—88no—
588C23—88no—
589C23—88no—
590C23—88no—
591C23—164no—
592C23—164no—
593C23—164no—
594C23C2×#12314yes—
595C23C2×AGL(1,ℤ/8)12yes—
596C23C23×D411yescharacteristic
597Q8—44no—
598D4—28no—
599D4—24no—
600D4—48no—
601D4—44no—
602D4—42no—
603D4—44no—
604D4—88no—
605D4—84no—
606D4—88no—
607D4—84no—
608D4—162no—
609D4—164no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—84no—
657C2×C4—164no—
658C2×C4—164no—
659C2×C4C2×#12314yes—
660C2×C4C2×#12312yes—
661C2×C4C2×#12312yes—
662C2×C4C2×AGL(1,ℤ/8)12yes—
663C2×C4C2×AGL(1,ℤ/8)12yes—
664C2×C4#57512yes—
665C2×C4#57712yes—
666C2×C4C22×#4112yes—
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4#58512yes—
669C2×C4#58511yescharacteristic
670C8—44no—
671C8—42no—
672C8—42no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—42no—
680C22—41no—
681C22—42no—
682C22—42no—
683C22—41no—
684C22—41no—
685C22—42no—
686C22—41no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—1616no—
702C22C22×#12311yescharacteristic
703C22#348811yescharacteristic
704C22#350011yescharacteristic
705C22C22×AGL(1,ℤ/8)11yescharacteristic
706C22C2×#57511yescharacteristic
707C22C2×#57711yescharacteristic
708C22#358511yescharacteristic
709C4—22no—
710C4—22no—
711C4—22no—
712C4—24no—
713C4—42no—
714C4—42no—
715C4—41no—
716C4—84no—
717C4—81no—
718C4—82no—
719C4—81no—
720C4—82no—
721C4—81no—
722C4—82no—
723C4#309414yes—
724C4#351312yes—
725C4#352112yes—
726C2—42no—
727C2—44no—
728C2—82no—
729C2—162no—
730C2#3597611yescharacteristic
731C2C2×#309411yescharacteristic
732C2#6052511yescharacteristic
733C2#6196111yescharacteristic
734C2#6199611yescharacteristic
735C2#6253911yescharacteristic
736C2#6259211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36258C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34188C2semidirect
3. #29921C2semidirect
4. #29910C2semidirect
5. #27973C2semidirect
6. #27863C2semidirect
7. #27644C2semidirect
8. #27585C2semidirect
9. #26321C2semidirect
10. #26219C2semidirect
11. #24991C2semidirect
12. #24882C2semidirect
13. #23755C2semidirect
14. #23628C2semidirect
15. #22715C2semidirect
16. #22473C2semidirect
17. #22428C2semidirect
35. #2355C22semidirect
39. #2352C22semidirect
46. #2285C22semidirect
47. #2285C22semidirect
48. #2285C22semidirect
49. C2×#384C22semidirect
51. #2101C22semidirect
52. #2100C22semidirect
54. #2094C22semidirect
55. #2092C22semidirect
56. #2084C22semidirect
57. #2083C22semidirect
58. #2077C22semidirect
60. #2073C22semidirect
63. #2071C22semidirect
64. #2071C22semidirect
65. #2071C22semidirect
67. #2034C22semidirect
68. #1984C22semidirect
71. #1948C22semidirect
72. #1947C22semidirect
74. #1926C22semidirect
75. #1925C22semidirect
76. #1920C22semidirect
79. #1874C22semidirect
80. #1870C22semidirect
81. #1854C22semidirect
82. #1853C22semidirect
83. #1834C22semidirect
84. #1823C22semidirect
85. #1817C22semidirect
86. #1809C22semidirect
88. #1797C22semidirect
90. C4×#106C22semidirect
92. C4×C22⋊C8C22semidirect
93. #1736C22semidirect
94. #1728C22semidirect

Maximal subgroups

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

subgroupindex
#362582
#341882
#299212
#299102
#279732
#278632
#276442
#275852
#263212
#262192
#249912
#248822
#237552
#236282
#227152
#224732
#224282