Finite Group Databasebeta

← #6640637

Subgroups — #6640637

Order 512.

1924 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3#32982C212yesmaximal, complemented
4#32282C214yesmaximal, complemented
5#32050C214yesmaximal, complemented
6#31932C214yesmaximal, complemented
7#31836C214yesmaximal, complemented
8#29596C212yesmaximal
9#29103C211yesmaximal, characteristic, complemented
10#28302C211yesmaximal, characteristic, complemented
11#28120C212yesmaximal, complemented
12#27960C211yesmaximal, characteristic, complemented
13#27905C211yesmaximal, characteristic, complemented
14#27647C212yesmaximal, complemented
15#21902C211yesmaximal, characteristic, complemented
16C22×#123—21no—
17C22×#123—21no—
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#483C2212yes—
21C2×#483C2212yes—
22C2×#483C2212yes—
23C2×#450C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2848C2212yes—
27#2646—21no—
28#2626—21no—
29#2589—22no—
30#2587—22no—
31#2501—22no—
32#2501—22no—
33#2501C2214yes—
34#2501C2214yes—
35#2497C2212yescomplemented
36#2492C2212yescomplemented
37#2429C2212yescomplemented
38#2423C2212yescomplemented
39C2×#399—21no—
40C2×#399—21no—
41#2329—21no—
42#2312—21no—
43C4×C22⋊Q8C2212yes—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2099C2214yescomplemented
50#2096C2214yescomplemented
51#2094C2212yescomplemented
52#2084C2214yescomplemented
53#2072C2214yescomplemented
54#2071C2212yescomplemented
55#2065C2214yescomplemented
56#2055C2214yescomplemented
57#2034C2214yes—
58#2034C2214yes—
59#2023C2212yescomplemented
60#2013C2212yescomplemented
61#2000C2214yes—
62#2000C2214yes—
63#1990C2212yescomplemented
64#1968C2214yescomplemented
65#1956C2212yescomplemented
66#1948C2214yescomplemented
67#1934C2214yescomplemented
68#1926C2214yescomplemented
69#1923C2214yescomplemented
70#1923C2214yescomplemented
71#1880C2214yescomplemented
72#1872C2214yescomplemented
73#1872C2214yescomplemented
74#1853C2214yescomplemented
75#1853C2214yescomplemented
76#1844C2212yescomplemented
77#1834C2212yescomplemented
78#1823C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1803C2214yescomplemented
82#1797C2212yes—
83#1786C2212yes—
84#1779C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×SD16C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#123—24no—
95C2×#123—24no—
96C2×#123—24no—
97C2×#123—24no—
98C2×#123—48no—
99C2×#123—48no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—22no—
105C2×#121—44no—
106C2×#121C2314yes—
107C2×C4×Q8C2312yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—44no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#483—44no—
129#483—44no—
130#483—44no—
131#450—44no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#400—22no—
150#399—22no—
151#399—24no—
152#399—24no—
153#399—24no—
154#399—24no—
155#398—22no—
156#398—22no—
157#396—22no—
158#396—22no—
159#393—22no—
160#391—21no—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#390—22no—
165#390—21no—
166#390—21no—
167#390C2314yes—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—22no—
188#384—21no—
189#384—21no—
190#384C2314yes—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#380—21no—
196#380—21no—
197#380—21no—
198#380C2312yes—
199#380C2312yes—
200#380C2312yes—
201#379—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4D412yes—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4C2312yes—
208C4×C22⋊C4C2312yes—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2312yes—
219#336C2312yes—
220C25—21no—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8C2412yes—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—24no—
227C22×D4—42no—
228C22×D4—42no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4C2412yes—
233C23×C4—24no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×SD16—44no—
252C2×SD16—44no—
253C2×SD16—44no—
254C22×C8C2414yes—
255C22×C8C2414yes—
256C4⋊Q8—42no—
257C4⋊Q8—42no—
258#123—416no—
259#123—416no—
260#123—416no—
261#123—416no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267#121—48no—
268#121—48no—
269C4×Q8—44no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4SD1618yescomplemented
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#106—44no—
344#106—44no—
345#106—44no—
346C4⋊C8—44no—
347#103—44no—
348#103—44no—
349#103—44no—
350#102—44no—
351#102—44no—
352#102—44no—
353C22⋊C8—24no—
354C22⋊C8—24no—
355C22⋊C8—44no—
356C22⋊C8—44no—
357C4×C8—42no—
358C4×C8—42no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—41no—
377C24—41no—
378C24—41no—
379C24—41no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24C2×SD1614yes—
388C24C22×D412yes—
389C24C22×D411yescharacteristic
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—42no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—42no—
407C2×D4—88no—
408C2×D4—816no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—816no—
450C22×C4C2×SD1614yes—
451C22×C4C2×SD1614yes—
452C22×C4AGL(1,ℤ/8)14yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4H(5,𝔽2)14yes—
462C22×C4C2511yescharacteristic
463SD16—88no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4C2×SD1618yes—
516C22⋊C4C2×SD1618yes—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—42no—
521C42—41no—
522C42—41no—
523C42—82no—
524C42—82no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—28no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23#53512yes—
557C23C22×SD1614yes—
558C23C22×SD1612yes—
559C23C2×AGL(1,ℤ/8)12yes—
560C23C23×D411yescharacteristic
561Q8—44no—
562Q8—44no—
563Q8—162no—
564D4—44no—
565D4—44no—
566D4—84no—
567D4—816no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—816no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4#53512yes—
620C2×C4C22×SD1614yes—
621C2×C4C22×SD1612yes—
622C2×C4C22×SD1612yes—
623C2×C4C2×AGL(1,ℤ/8)12yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×H(5,𝔽2)12yes—
626C2×C4#58512yes—
627C2×C4#58512yes—
628C8—44no—
629C8—44no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—416no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—168no—
652C22#320814yes—
653C22C2×#53511yescharacteristic
654C22#348511yescharacteristic
655C22#348511yescharacteristic
656C22C23×SD1611yescharacteristic
657C22C22×AGL(1,ℤ/8)11yescharacteristic
658C22#358611yescharacteristic
659C22#358611yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—24no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—42no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#353314yes—
676C2—24no—
677C2—44no—
678C2—162no—
679C2#3695611yescharacteristic
680C2C2×#320811yescharacteristic
681C2#6080711yescharacteristic
682C2#6169511yescharacteristic
683C2#6169511yescharacteristic
684C2#6170611yescharacteristic
685C2#6171711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
1. #35014C2semidirect
3. #32982C2semidirect
4. #32282C2semidirect
5. #32050C2semidirect
6. #31932C2semidirect
7. #31836C2semidirect
9. #29103C2semidirect
10. #28302C2semidirect
11. #28120C2semidirect
12. #27960C2semidirect
13. #27905C2semidirect
14. #27647C2semidirect
15. #21902C2semidirect
35. #2497C22semidirect
36. #2492C22semidirect
37. #2429C22semidirect
38. #2423C22semidirect
49. #2099C22semidirect
50. #2096C22semidirect
51. #2094C22semidirect
52. #2084C22semidirect
53. #2072C22semidirect
54. #2071C22semidirect
55. #2065C22semidirect
56. #2055C22semidirect
59. #2023C22semidirect
60. #2013C22semidirect
63. #1990C22semidirect
64. #1968C22semidirect
65. #1956C22semidirect
66. #1948C22semidirect
67. #1934C22semidirect
68. #1926C22semidirect
69. #1923C22semidirect
70. #1923C22semidirect
71. #1880C22semidirect
72. #1872C22semidirect
73. #1872C22semidirect
74. #1853C22semidirect
75. #1853C22semidirect
76. #1844C22semidirect
77. #1834C22semidirect
81. #1803C22semidirect
328. C2×C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#25012
#329822
#322822
#320502
#319322
#318362
#295962
#291032
#283022
#281202
#279602
#279052
#276472
#219022