Finite Group Databasebeta

← #6563131

Subgroups — #6563131

Order 512.

2059 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36449C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic, complemented
3#30127C212yesmaximal, complemented
4#29922C211yesmaximal, characteristic, complemented
5#29498C212yesmaximal, complemented
6#28162C212yesmaximal, complemented
7#26533C212yesmaximal, complemented
8#25717C211yesmaximal, characteristic, complemented
9#25367C212yesmaximal, complemented
10#25172C214yesmaximal, complemented
11#24917C214yesmaximal, complemented
12#24208C211yesmaximal, characteristic
13#23806C212yesmaximal, complemented
14#23754C211yesmaximal, characteristic, complemented
15#23190C212yesmaximal, complemented
16#23001C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C2×#518—21no—
20C2×#492C2212yes—
21C2×#463C2214yescomplemented
22C2×#459C2212yes—
23C2×#450C2214yes—
24C2×#447C2212yes—
25C2×#442C2214yescomplemented
26C2×#440C2214yescomplemented
27#2863C2212yescomplemented
28#2857C2211yescharacteristic
29#2578—22no—
30#2571C2211yescharacteristic, complemented
31#2547C2211yescharacteristic, complemented
32#2530—22no—
33#2495—21no—
34#2485—21no—
35#2427C2212yes—
36#2421C2212yes—
37#2366—21no—
38#2366—21no—
39#2366C2212yes—
40#2366C2212yes—
41#2365C2211yescharacteristic, complemented
42#2352—21no—
43D4×C4⋊C4C2211yescharacteristic, complemented
44#2345—21no—
45D4×C22⋊C4—22no—
46D4×C22⋊C4C2214yes—
47#2328—22no—
48#2309—21no—
49C4×#127C2212yes—
50C4×#127C2212yes—
51C4×#124C2212yescomplemented
52C4×#121C2211yescharacteristic
53#2285C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2099C2211yescharacteristic, complemented
56#2096C2212yescomplemented
57#2093C2211yescharacteristic, complemented
58#2092C2214yescomplemented
59#2083C2211yescharacteristic, complemented
60#2075C2212yes—
61#2072C2212yescomplemented
62#2071C2211yescharacteristic, complemented
63#2065C2214yescomplemented
64#2055C2214yescomplemented
65#2041C2212yes—
66#2035C2214yes—
67#2033C2212yes—
68#2014C2212yes—
69#2014C2212yes—
70#2012C2214yescomplemented
71#2011C2212yes—
72#1982C2212yes—
73#1981C2212yescomplemented
74#1973C2214yescomplemented
75#1957C2212yes—
76#1948C2212yescomplemented
77#1939C2212yescomplemented
78#1936C2214yescomplemented
79#1924C2214yes—
80#1880C2214yescomplemented
81#1879C2214yescomplemented
82#1877C2214yescomplemented
83#1877C2214yescomplemented
84#1864C2214yescomplemented
85#1862C2214yescomplemented
86#1831C2212yescomplemented
87#1804C2214yescomplemented
88#1797C2211yescharacteristic
89#1786C2212yes—
90C4×#107C2212yescomplemented
91#1770C2212yescomplemented
92C4×#102C2212yescomplemented
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yescomplemented
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100C22×D8C2314yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127—28no—
103C2×#127—21no—
104C2×#127—21no—
105C2×#127C2312yes—
106C2×#127C2312yes—
107C2×#124C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×#121C2311yescharacteristic
110#518—48no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2314yes—
121C2×C4×D4C2314yes—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C2×#117D411yescharacteristic
126C22×C4⋊C4—21no—
127C22×C22⋊C4—24no—
128C22×C42C2311yescharacteristic
129#492—28no—
130#463—48no—
131#459—28no—
132#450—48no—
133#447—44no—
134#442—48no—
135#440—48no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#400—22no—
155#400—22no—
156#397—22no—
157#397—22no—
158#397—22no—
159#394—22no—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#390—22no—
164#390—22no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2314yes—
171#390C2311yescharacteristic
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389C2312yes—
177#389C2311yescharacteristic
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—21no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#386—21no—
189#386—22no—
190#386—24no—
191#386—24no—
192#386—24no—
193#386—24no—
194#385—22no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#383—22no—
199#382—21no—
200#382—21no—
201#382C2312yes—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380—21no—
208#380—22no—
209#380C2311yescharacteristic
210#380C2312yes—
211#379—24no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225#376D412yes—
226C43C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237C25—22no—
238C22×Q8C2412yes—
239C22×D4—28no—
240C22×D4—28no—
241C22×D4—28no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4—44no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—82no—
254C22×D4—82no—
255C22×D4C2414yes—
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—22no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—28no—
267C2×SD16—44no—
268C2×SD16—44no—
269C2×SD16—44no—
270C2×D8—28no—
271C2×D8—28no—
272C2×D8—44no—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—24no—
277#127—28no—
278#127—28no—
279#127—28no—
280#127—24no—
281#127#4118yes—
282#124—24no—
283#124—42no—
284C22⋊Q8—44no—
285#121—22no—
286#121—41no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—48no—
290C4×D4—48no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—48no—
297C4×D4—48no—
298C4×D4—44no—
299C4×D4—84no—
300C4×D4—84no—
301#117—24no—
302#117—42no—
303#117—44no—
304C2×C4⋊C4—21no—
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—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—416no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—42no—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—44no—
367#103—44no—
368#102—28no—
369#102—24no—
370#102—24no—
371#102—28no—
372#102—44no—
373#102—44no—
374#102—44no—
375C22⋊C8—24no—
376C22⋊C8—42no—
377#97—28no—
378C4×C8—24no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—28no—
399C24—21no—
400C24—24no—
401C24—44no—
402C24—44no—
403C24—82no—
404C24—82no—
405C24—82no—
406C24—82no—
407C24C22×D411yescharacteristic
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—21no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—432no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—88no—
439C2×D4—88no—
440C2×D4—816no—
441C2×D4—81no—
442C2×D4C2×#4118yes—
443C2×D4H(5,𝔽2)12yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—84no—
467C22×C4—82no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—82no—
472C22×C4AGL(1,ℤ/8)14yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2511yescharacteristic
483SD16—416no—
484SD16—88no—
485D8—416no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8H(5,𝔽2)14yes—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—82no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—82no—
539C42—22no—
540C42—22no—
541C42—24no—
542C42—24no—
543C42—22no—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—42no—
548C42—42no—
549C42—44no—
550C42—42no—
551C42—82no—
552C42C2×#4114yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—28no—
560C23—22no—
561C23—416no—
562C23—44no—
563C23—41no—
564C23—41no—
565C23—44no—
566C23—41no—
567C23—44no—
568C23—44no—
569C23—41no—
570C23—41no—
571C23—44no—
572C23—41no—
573C23—44no—
574C23—88no—
575C23—88no—
576C23—164no—
577C23—164no—
578C23—164no—
579C23—164no—
580C23C2×AGL(1,ℤ/8)12yes—
581C23C23×D411yescharacteristic
582Q8—44no—
583Q8—44no—
584D4—28no—
585D4—24no—
586D4—28no—
587D4—416no—
588D4—416no—
589D4—416no—
590D4—42no—
591D4—44no—
592D4—44no—
593D4—88no—
594D4—88no—
595D4—84no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—84no—
643C2×C4—168no—
644C2×C4—168no—
645C2×C4—164no—
646C2×C4#52512yes—
647C2×C4#52512yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C2×AGL(1,ℤ/8)12yes—
650C2×C4#57512yes—
651C2×C4#57712yes—
652C2×C4C22×#4112yes—
653C2×C4C2×H(5,𝔽2)12yes—
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4#58511yescharacteristic
656C8—24no—
657C8—44no—
658C8—42no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—42no—
667C22—42no—
668C22—48no—
669C22—48no—
670C22—42no—
671C22—48no—
672C22—84no—
673C22—84no—
674C22—81no—
675C22—81no—
676C22—84no—
677C22—81no—
678C22—81no—
679C22—84no—
680C22—1616no—
681C22C2×#52511yescharacteristic
682C22#348811yescharacteristic
683C22#349811yescharacteristic
684C22C22×AGL(1,ℤ/8)11yescharacteristic
685C22C2×#57511yescharacteristic
686C22C2×#57711yescharacteristic
687C22#358511yescharacteristic
688C4—22no—
689C4—22no—
690C4—22no—
691C4—42no—
692C4—42no—
693C4—41no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—81no—
698C4—81no—
699C4—82no—
700C4—81no—
701C4—82no—
702C4—82no—
703C4#311414yes—
704C4#351312yes—
705C4#351912yes—
706C2—42no—
707C2—82no—
708C2—88no—
709C2#3599511yescharacteristic
710C2C2×#311411yescharacteristic
711C2#6057911yescharacteristic
712C2#6195911yescharacteristic
713C2#6198611yescharacteristic
714C2#6242111yescharacteristic
715C2#6245511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#24208C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36449C2semidirect
2. #34210C2semidirect
3. #30127C2semidirect
4. #29922C2semidirect
5. #29498C2semidirect
6. #28162C2semidirect
7. #26533C2semidirect
8. #25717C2semidirect
9. #25367C2semidirect
10. #25172C2semidirect
11. #24917C2semidirect
13. #23806C2semidirect
14. #23754C2semidirect
15. #23190C2semidirect
16. #23001C2semidirect
17. #22636C2semidirect
21. C2×#463C22semidirect
25. C2×#442C22semidirect
26. C2×#440C22semidirect
27. #2863C22semidirect
30. #2571C22semidirect
31. #2547C22semidirect
41. #2365C22semidirect
43. D4×C4⋊C4C22semidirect
51. C4×#124C22semidirect
55. #2099C22semidirect
56. #2096C22semidirect
57. #2093C22semidirect
58. #2092C22semidirect
59. #2083C22semidirect
61. #2072C22semidirect
62. #2071C22semidirect
63. #2065C22semidirect
64. #2055C22semidirect
70. #2012C22semidirect
73. #1981C22semidirect
74. #1973C22semidirect
76. #1948C22semidirect
77. #1939C22semidirect
78. #1936C22semidirect
80. #1880C22semidirect
81. #1879C22semidirect
82. #1877C22semidirect
83. #1877C22semidirect
84. #1864C22semidirect
85. #1862C22semidirect
86. #1831C22semidirect
87. #1804C22semidirect
90. C4×#107C22semidirect
91. #1770C22semidirect
92. C4×#102C22semidirect
94. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#364492
#342102
#301272
#299222
#294982
#281622
#265332
#257172
#253672
#251722
#249172
#242082
#238062
#237542
#231902
#230012
#226362