Finite Group Databasebeta

← #6436929

Subgroups — #6436929

Order 512.

2068 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#29939C212yesmaximal, complemented
4#29938C212yesmaximal, complemented
5#29921C212yesmaximal, complemented
6#29920C212yesmaximal, complemented
7#29907C212yesmaximal, complemented
8#29906C212yesmaximal, complemented
9#29891C214yesmaximal, complemented
10#23761C214yesmaximal, complemented
11#23760C214yesmaximal, complemented
12#21395C212yesmaximal, complemented
13#21365C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C2×#525—21no—
17C22×#117—21no—
18C23×C42C2211yescharacteristic
19C2×#432C2212yes—
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2856C2211yescharacteristic
23#2856C2211yescharacteristic
24#2854—21no—
25#2854C2212yes—
26#2850—21no—
27#2850—21no—
28#2847—21no—
29#2847C2212yes—
30C4×#121—21no—
31C4×#121—21no—
32C4×#121C2212yes—
33C4×#121C2212yes—
34#2297—21no—
35#2285—21no—
36#2285—21no—
37#2285—21no—
38#2285—21no—
39#2285C2211yescharacteristic
40#2285C2212yes—
41#2285C2212yes—
42#2285C2212yes—
43#2285C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#382—21no—
47C2×#380—21no—
48C2×#380C2212yes—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51#2100C2214yescomplemented
52#2099C2214yescomplemented
53#2094C2214yescomplemented
54#2093C2214yescomplemented
55#2084C2214yescomplemented
56#2083C2214yescomplemented
57#2077C2218yes—
58#2076C2218yes—
59#2071C2214yescomplemented
60#2071C2214yescomplemented
61#2034C2218yes—
62#2033C2218yes—
63#1987C2214yescomplemented
64#1982C2214yescomplemented
65#1948C2214yescomplemented
66#1947C2214yescomplemented
67#1874C2214yes—
68#1874C2214yes—
69#1853C2214yes—
70#1852C2214yes—
71#1817C2212yes—
72#1797C2212yes—
73#1797C2212yes—
74#1779C2212yes—
75#1776C2218yescomplemented
76#1762C2218yescomplemented
77#1749C2212yes—
78#1749C2212yes—
79C22×#41—21no—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83#532—28no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#124—21no—
87#525—48no—
88C2×#123—22no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×#121—21no—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121C2312yes—
97C2×#121C2312yes—
98C2×#121C2312yes—
99C2×#120—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—21no—
103C2×C4×D4C2312yes—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117—21no—
109C2×#117—48no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—22no—
117C22×C42—28no—
118C22×C42—22no—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#432—28no—
132C2×#107C2318yes—
133C2×#106C2318yes—
134C2×C4⋊C8C2314yes—
135C2×#102C2318yes—
136C2×#102C2318yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2314yes—
141#391—21no—
142#391—21no—
143#390—21no—
144#390—21no—
145#390—21no—
146#390—21no—
147#390C2312yes—
148#390C2312yes—
149#389—21no—
150#389—21no—
151#389C2312yes—
152#389C2312yes—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#386—22no—
158#386—22no—
159#386—22no—
160#386—22no—
161#386—22no—
162#386—22no—
163#384—21no—
164#384—21no—
165#384—21no—
166#384—21no—
167#384C2312yes—
168#384C2312yes—
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2312yes—
172#384C2312yes—
173#382—21no—
174#382—22no—
175#382—22no—
176#382—22no—
177#382—22no—
178#380—21no—
179#380—21no—
180#380—21no—
181#380—22no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380—22no—
187#380C2312yes—
188#380C2314yes—
189#380C2312yes—
190#380C2314yes—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C2×#95—22no—
205C2×#95—22no—
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#340C2318yes—
210#336C2314yes—
211#336C2314yes—
212C25C2×D411yescharacteristic
213C2×#41—44no—
214C2×#41—44no—
215C2×#41—44no—
216C22×Q8—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C2412yes—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C22×C8C2418yes—
254C4⋊Q8—24no—
255#127—24no—
256#124—44no—
257#123—48no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260#121—28no—
261#121—44no—
262#121—44no—
263#121—44no—
264#121—44no—
265#121—44no—
266#121—44no—
267#121—44no—
268#120—48no—
269C4×D4—28no—
270C4×D4—48no—
271C4×D4—48no—
272C4×D4—44no—
273#117—48no—
274#117—48no—
275#117—48no—
276#117—44no—
277#117—48no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—48no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—48no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42#4118yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—48no—
353#106—48no—
354C4⋊C8—28no—
355#102—48no—
356#102—48no—
357C22⋊C8—28no—
358C22⋊C8—44no—
359C22⋊C8—44no—
360#97—28no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—41no—
378C24—41no—
379C24C22×D411yescharacteristic
380C24C22×D411yescharacteristic
381C24C22×D411yescharacteristic
382#41—816no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C22×C4—232no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4#12314yes—
470C22×C4#12314yes—
471C22×C4AGL(1,ℤ/8)14yes—
472C22×C4AGL(1,ℤ/8)14yes—
473C22×C4#13714yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4114yes—
486C22×C4C2511yescharacteristic
487C2×C8—28no—
488C2×C8—28no—
489C2×C8—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—48no—
540C42—44no—
541C42—48no—
542C42#12314yes—
543C42#12314yes—
544C42C2×#4114yes—
545C42C2×#4114yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—28no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23C2×#12312yes—
577C23C2×AGL(1,ℤ/8)12yes—
578C23C2×AGL(1,ℤ/8)12yes—
579C23C2×#13712yes—
580C23#57512yes—
581C23C23×D411yescharacteristic
582Q8—84no—
583D4—48no—
584D4—84no—
585D4—88no—
586D4—88no—
587D4—88no—
588D4—84no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4—88no—
647C2×C4C2×#12312yes—
648C2×C4C2×#12312yes—
649C2×C4C2×#12312yes—
650C2×C4C2×#12312yes—
651C2×C4C2×#12312yes—
652C2×C4C2×AGL(1,ℤ/8)12yes—
653C2×C4C2×AGL(1,ℤ/8)12yes—
654C2×C4C2×AGL(1,ℤ/8)12yes—
655C2×C4C2×AGL(1,ℤ/8)12yes—
656C2×C4C2×#13712yes—
657C2×C4C2×#13712yes—
658C2×C4C22×#4112yes—
659C2×C4#58512yes—
660C2×C4#58512yes—
661C8—48no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—416no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22#309414yes—
691C22C22×#12311yescharacteristic
692C22#344911yescharacteristic
693C22#344911yescharacteristic
694C22C22×AGL(1,ℤ/8)11yescharacteristic
695C22C22×AGL(1,ℤ/8)11yescharacteristic
696C22C22×#13711yescharacteristic
697C22C2×#57511yescharacteristic
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—41no—
707C4—41no—
708C4—84no—
709C4—84no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4#309214yes—
714C4#345514yes—
715C2—24no—
716C2—42no—
717C2—42no—
718C2—162no—
719C2#3463511yescharacteristic
720C2C2×#309211yescharacteristic
721C2C2×#309411yescharacteristic
722C2#6096211yescharacteristic
723C2#6096311yescharacteristic
724C2#6096811yescharacteristic
725C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
3. #29939C2semidirect
4. #29938C2semidirect
5. #29921C2semidirect
6. #29920C2semidirect
7. #29907C2semidirect
8. #29906C2semidirect
9. #29891C2semidirect
10. #23761C2semidirect
11. #23760C2semidirect
12. #21395C2semidirect
13. #21365C2semidirect
14. #20715C2semidirect
51. #2100C22semidirect
52. #2099C22semidirect
53. #2094C22semidirect
54. #2093C22semidirect
55. #2084C22semidirect
56. #2083C22semidirect
59. #2071C22semidirect
60. #2071C22semidirect
63. #1987C22semidirect
64. #1982C22semidirect
65. #1948C22semidirect
66. #1947C22semidirect
75. #1776C22semidirect
76. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#22852
#299392
#299382
#299212
#299202
#299072
#299062
#298912
#237612
#237602
#213952
#213652
#207152