Finite Group Databasebeta

← #6640634

Subgroups — #6640634

Order 512.

1840 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic, complemented
3#32980C212yesmaximal, complemented
4#32279C214yesmaximal, complemented
5#32051C214yesmaximal, complemented
6#31933C214yesmaximal, complemented
7#31835C214yesmaximal, complemented
8#29594C212yesmaximal, complemented
9#29103C211yesmaximal, characteristic, complemented
10#28301C211yesmaximal, characteristic, complemented
11#28117C212yesmaximal, complemented
12#27961C211yesmaximal, characteristic, complemented
13#27906C211yesmaximal, characteristic, complemented
14#27651C212yesmaximal, complemented
15#21902C211yesmaximal, characteristic, complemented
16C2×#538—21no—
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#481C2212yes—
20C2×#481C2212yes—
21C2×#480C2212yes—
22C2×#447C2212yes—
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2847C2212yes—
26#2639—21no—
27#2617—21no—
28#2559—22no—
29#2557—22no—
30#2500—22no—
31#2500—22no—
32#2500C2214yes—
33#2500C2214yes—
34#2496C2212yescomplemented
35#2491C2212yescomplemented
36#2428C2212yescomplemented
37#2422C2212yescomplemented
38C2×#394—21no—
39C2×#394—21no—
40#2325—21no—
41C4×#121C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#388—21no—
45C2×#388—21no—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2100C2214yescomplemented
50#2094C2214yescomplemented
51#2093C2212yescomplemented
52#2085C2214yescomplemented
53#2073C2214yescomplemented
54#2071C2212yescomplemented
55#2065C2214yescomplemented
56#2054C2214yescomplemented
57#2034C2214yes—
58#2034C2214yes—
59#2023C2212yescomplemented
60#2011C2212yescomplemented
61#1999C2214yes—
62#1999C2214yes—
63#1990C2212yescomplemented
64#1970C2214yescomplemented
65#1957C2212yescomplemented
66#1949C2214yescomplemented
67#1934C2214yescomplemented
68#1926C2214yescomplemented
69#1923C2214yescomplemented
70#1923C2214yescomplemented
71#1879C2214yescomplemented
72#1875C2214yescomplemented
73#1875C2214yescomplemented
74#1854C2214yescomplemented
75#1854C2214yescomplemented
76#1844C2212yescomplemented
77#1834C2212yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1804C2214yescomplemented
82#1797C2212yes—
83#1786C2212yes—
84#1779C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C22×D8C2314yes—
91#538—48no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#126—22no—
95C2×#124—22no—
96C2×#123—22no—
97C2×C22⋊Q8—21no—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121—22no—
101C2×#121—22no—
102C2×#121—44no—
103C2×#121C2312yes—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#121C2314yes—
107C2×#120—22no—
108C2×C4×D4—22no—
109C2×C4×D4C2312yes—
110C2×#117—22no—
111C2×#117—44no—
112C2×#117C2314yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
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×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#481—44no—
130#481—44no—
131#480—44no—
132#447—44no—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#397—22no—
151#397—22no—
152#394—22no—
153#394—22no—
154#394—24no—
155#394—24no—
156#394—24no—
157#394—24no—
158#394—22no—
159#393—22no—
160#392—22no—
161#391—21no—
162#390—21no—
163#390—21no—
164#390—22no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2314yes—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—24no—
178#388—24no—
179#388—24no—
180#388—24no—
181#387—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#382C2314yes—
196#380—21no—
197#380C2312yes—
198#380C2312yes—
199#380C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4C2312yes—
205C4×C22⋊C4C2312yes—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C25—21no—
220C25C2×D411yescharacteristic
221C22×Q8—21no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—24no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4—44no—
233C22×D4—82no—
234C22×D4—82no—
235C22×D4C2412yes—
236C22×D4C2412yes—
237C23×C4—24no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C2×D8—44no—
253C2×D8—44no—
254C2×D8—44no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C4⋊Q8—42no—
258#127—42no—
259#126—48no—
260#124—48no—
261#123—48no—
262C22⋊Q8—44no—
263#121—24no—
264#121—24no—
265#121—48no—
266#121—48no—
267#121—48no—
268#121—44no—
269#121—44no—
270#121—44no—
271#121—44no—
272#120—48no—
273C4×D4—44no—
274C4×D4—48no—
275#117—48no—
276#117—48no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
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—24no—
318C2×C22⋊C4—24no—
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—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—44no—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—44no—
345#106—44no—
346#106—44no—
347C4⋊C8—44no—
348#102—44no—
349#102—44no—
350#102—44no—
351#102—44no—
352#102—44no—
353#102—44no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356C22⋊C8—44no—
357C22⋊C8—44no—
358C4×C8—42no—
359C4×C8—42no—
360#95—24no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—41no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24C22×D412yes—
389C24C22×D411yescharacteristic
390C2×Q8—42no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—48no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—88no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4—84no—
418C2×D4—84no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—816no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—416no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4#13514yes—
465C22×C4AGL(1,ℤ/8)14yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4H(5,𝔽2)14yes—
475C22×C4C2511yescharacteristic
476D8—88no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—81no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—82no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—41no—
531C42—41no—
532C42—42no—
533C42—48no—
534C42—44no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23#53512yes—
575C23C2×#13512yes—
576C23C2×AGL(1,ℤ/8)12yes—
577C23C23×D411yescharacteristic
578Q8—162no—
579D4—44no—
580D4—44no—
581D4—44no—
582D4—44no—
583D4—84no—
584D4—816no—
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—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—164no—
639C2×C4#53512yes—
640C2×C4C2×#13512yes—
641C2×C4C2×#13512yes—
642C2×C4C2×AGL(1,ℤ/8)12yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×H(5,𝔽2)12yes—
645C2×C4#58512yes—
646C2×C4#58512yes—
647C8—44no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—88no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—168no—
678C22#320714yes—
679C22C2×#53511yescharacteristic
680C22#348511yescharacteristic
681C22#348511yescharacteristic
682C22C22×#13511yescharacteristic
683C22C22×AGL(1,ℤ/8)11yescharacteristic
684C22#358511yescharacteristic
685C22#358511yescharacteristic
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4#353314yes—
701C2—24no—
702C2—82no—
703C2—162no—
704C2—162no—
705C2#3695611yescharacteristic
706C2C2×#320711yescharacteristic
707C2#6080611yescharacteristic
708C2#6169911yescharacteristic
709C2#6170011yescharacteristic
710C2#6170711yescharacteristic
711C2#6172011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #35005C2semidirect
2. C2×#2500C2semidirect
3. #32980C2semidirect
4. #32279C2semidirect
5. #32051C2semidirect
6. #31933C2semidirect
7. #31835C2semidirect
8. #29594C2semidirect
9. #29103C2semidirect
10. #28301C2semidirect
11. #28117C2semidirect
12. #27961C2semidirect
13. #27906C2semidirect
14. #27651C2semidirect
15. #21902C2semidirect
34. #2496C22semidirect
35. #2491C22semidirect
36. #2428C22semidirect
37. #2422C22semidirect
49. #2100C22semidirect
50. #2094C22semidirect
51. #2093C22semidirect
52. #2085C22semidirect
53. #2073C22semidirect
54. #2071C22semidirect
55. #2065C22semidirect
56. #2054C22semidirect
59. #2023C22semidirect
60. #2011C22semidirect
63. #1990C22semidirect
64. #1970C22semidirect
65. #1957C22semidirect
66. #1949C22semidirect
67. #1934C22semidirect
68. #1926C22semidirect
69. #1923C22semidirect
70. #1923C22semidirect
71. #1879C22semidirect
72. #1875C22semidirect
73. #1875C22semidirect
74. #1854C22semidirect
75. #1854C22semidirect
76. #1844C22semidirect
77. #1834C22semidirect
81. #1804C22semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#25002
#329802
#322792
#320512
#319332
#318352
#295942
#291032
#283012
#281172
#279612
#279062
#276512
#219022