Finite Group Databasebeta

← #6477868

Subgroups — #6477868

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic, complemented
3#32978C212yesmaximal, complemented
4#32957C211yesmaximal, characteristic, complemented
5#32903C211yesmaximal, characteristic, complemented
6#29969C211yesmaximal, characteristic, complemented
7#29755C211yesmaximal, characteristic, complemented
8#29593C212yesmaximal, complemented
9#27115C214yesmaximal, complemented
10#26947C214yesmaximal, complemented
11#26805C214yesmaximal, complemented
12#26677C214yesmaximal, complemented
13#22047C212yesmaximal, complemented
14#21906C211yesmaximal, characteristic, complemented
15#21580C212yesmaximal, complemented
16C2×#538—21no—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#481C2212yes—
21C2×#480C2214yes—
22C2×#477C2212yescomplemented
23C2×#460C2212yescomplemented
24C2×#448C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2847C2212yes—
28#2639—21no—
29#2617—21no—
30#2559—22no—
31#2557—22no—
32#2496C2212yes—
33#2491C2212yes—
34#2428C2212yes—
35#2422C2212yes—
36#2360—22no—
37#2360—22no—
38#2360C2214yes—
39#2360C2214yes—
40#2325—21no—
41C4×#121C2212yes—
42C2×#391C2211yescharacteristic
43C2×#389—21no—
44C2×#389C2211yescharacteristic
45C2×#387—21no—
46C2×#385—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2100C2212yescomplemented
51#2087C2214yescomplemented
52#2084C2212yescomplemented
53#2065C2214yescomplemented
54#2057C2212yescomplemented
55#2041C2212yescomplemented
56#2035C2214yes—
57#2035C2214yes—
58#2000C2214yes—
59#2000C2214yes—
60#1973C2214yescomplemented
61#1935C2214yescomplemented
62#1928C2214yescomplemented
63#1925C2214yescomplemented
64#1924C2214yes—
65#1924C2214yes—
66#1920C2214yescomplemented
67#1880C2214yescomplemented
68#1877C2214yescomplemented
69#1874C2212yes—
70#1874C2214yes—
71#1874C2212yes—
72#1863C2214yescomplemented
73#1852C2212yes—
74#1852C2212yes—
75#1852C2214yes—
76#1824C2212yes—
77#1823C2214yes—
78#1803C2212yescomplemented
79#1803C2212yescomplemented
80#1797C2211yescharacteristic
81#1797C2211yescharacteristic
82#1787C2212yes—
83#1779C2212yes—
84#1776C2214yescomplemented
85#1762C2214yescomplemented
86#1749C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C22×SD16C2314yes—
91#538—48no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#126—22no—
95C2×#124—22no—
96C2×#123—22no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8C2314yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121—22no—
105C2×#121C2312yes—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#120—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—44no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—44no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#481—28no—
134#480—48no—
135#477—44no—
136#460—44no—
137#448—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×#97C2314yes—
154#400—22no—
155#397—22no—
156#397—22no—
157#394—22no—
158#394—22no—
159#394—22no—
160#393—22no—
161#392—22no—
162#391—21no—
163#391—22no—
164#391C2314yes—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—24no—
173#389—24no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—24no—
182#387—24no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#385—24no—
189#385—24no—
190#385—22no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#380—21no—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—24no—
201C4×C22⋊C4—24no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4D412yes—
205C4×C22⋊C4C2312yes—
206C4×C22⋊C4C2312yes—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2312yes—
220#336C2312yes—
221#336C2314yes—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8#4114yes—
229C22×Q8C2412yes—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—21no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C2×SD16—28no—
258C2×SD16—44no—
259C2×SD16—44no—
260C22×C8C2414yes—
261C22×C8C2414yes—
262C4⋊Q8—42no—
263#127—42no—
264#126—48no—
265#124—48no—
266#123—48no—
267C22⋊Q8—216no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—44no—
271#121—24no—
272#121—24no—
273#121—48no—
274#121—44no—
275#121—44no—
276#121—44no—
277#121—44no—
278#120—48no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×D4—44no—
286C4×D4—48no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4#4114yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42—44no—
346C2×C42—82no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—28no—
354#107—44no—
355#106—28no—
356C4⋊C8—42no—
357C4⋊C8—42no—
358#103—28no—
359#103—44no—
360#103—44no—
361#102—28no—
362#102—28no—
363#102—44no—
364C22⋊C8—24no—
365C22⋊C8—24no—
366C22⋊C8—44no—
367C22⋊C8—44no—
368#97—44no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24AGL(1,ℤ/8)14yes—
392C24C22×D412yes—
393C24C22×D411yescharacteristic
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—48no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—48no—
404C2×Q8—42no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8C2×#4114yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—81no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—48no—
473C22×C4—42no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—84no—
477C22×C4—816no—
478C22×C4#13714yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2511yescharacteristic
490SD16—88no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C4⋊C4—24no—
498C4⋊C4—216no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—81no—
520C4⋊C4C2×#4114yes—
521C4⋊C4H(5,𝔽2)14yes—
522C22⋊C4—216no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—42no—
542C42—41no—
543C42—41no—
544C42—44no—
545C42—44no—
546C42—44no—
547C42—84no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—28no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23#52512yes—
573C23C2×AGL(1,ℤ/8)14yes—
574C23C2×AGL(1,ℤ/8)12yes—
575C23C2×#13712yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—48no—
580Q8—42no—
581Q8—42no—
582Q8—162no—
583D4—44no—
584D4—44no—
585D4—88no—
586D4—162no—
587D4—162no—
588D4—164no—
589D4—164no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—48no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—816no—
646C2×C4—84no—
647C2×C4—164no—
648C2×C4#52512yes—
649C2×C4C2×#13712yes—
650C2×C4C2×#13712yes—
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)12yes—
654C2×C4#58512yes—
655C8—44no—
656C8—44no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—416no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22#311214yes—
679C22C2×#52511yescharacteristic
680C22#348411yescharacteristic
681C22#348511yescharacteristic
682C22C22×AGL(1,ℤ/8)11yescharacteristic
683C22C22×#13711yescharacteristic
684C22#358511yescharacteristic
685C22#358611yescharacteristic
686C4—22no—
687C4—22no—
688C4—22no—
689C4—42no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—41no—
694C4—41no—
695C4—44no—
696C4—44no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4#346012yes—
702C4#350512yes—
703C2—24no—
704C2—44no—
705C2—162no—
706C2#3509111yescharacteristic
707C2C2×#311211yescharacteristic
708C2#6056511yescharacteristic
709C2#6156911yescharacteristic
710C2#6157411yescharacteristic
711C2#6170411yescharacteristic
712C2#6171911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35005C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2360C2semidirect
3. #32978C2semidirect
4. #32957C2semidirect
5. #32903C2semidirect
6. #29969C2semidirect
7. #29755C2semidirect
8. #29593C2semidirect
9. #27115C2semidirect
10. #26947C2semidirect
11. #26805C2semidirect
12. #26677C2semidirect
13. #22047C2semidirect
14. #21906C2semidirect
15. #21580C2semidirect
22. C2×#477C22semidirect
23. C2×#460C22semidirect
50. #2100C22semidirect
51. #2087C22semidirect
52. #2084C22semidirect
53. #2065C22semidirect
54. #2057C22semidirect
55. #2041C22semidirect
60. #1973C22semidirect
61. #1935C22semidirect
62. #1928C22semidirect
63. #1925C22semidirect
66. #1920C22semidirect
67. #1880C22semidirect
68. #1877C22semidirect
72. #1863C22semidirect
78. #1803C22semidirect
79. #1803C22semidirect
84. #1776C22semidirect
85. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#23602
#329782
#329572
#329032
#299692
#297552
#295932
#271152
#269472
#268052
#266772
#220472
#219062
#215802