Finite Group Databasebeta

← #415882

Subgroups — #415882

Order 512.

2572 conjugacy classes of subgroups.

The subgroup lattice has 793 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#486C211yesmaximal, characteristic, complemented
2C2×#2857C211yesmaximal, characteristic, complemented
3C2×#2856C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1957C211yesmaximal, characteristic, complemented
6C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
7#11066C214yesmaximal, complemented
8#10969C214yesmaximal, complemented
9#10841C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12C22×C22⋊Q8—21no—
13C22×#121—21no—
14C22×#121—21no—
15C22×#121C2212yes—
16C23×C42C2211yescharacteristic
17C2×#486—24no—
18C2×#486—24no—
19C2×#486—24no—
20C22×#102C2214yes—
21C22×#102C2214yes—
22C22×C22⋊C8C2212yes—
23C22×C4×C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2857—24no—
26#2857—24no—
27#2856—24no—
28#2856—24no—
29C2×#391—21no—
30C2×#390—21no—
31C2×#390—21no—
32C2×#390C2212yes—
33C2×#389—21no—
34C2×#389C2212yes—
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#380—21no—
38C2×#380C2212yes—
39#1990—24no—
40#1957—24no—
41C4×C22⋊C8—24no—
42C2×#340C2214yes—
43C2×#340C2214yes—
44C2×#336C2212yes—
45C23×Q8—21no—
46C23×D4—22no—
47C23×D4—21no—
48C23×D4—21no—
49C23×D4C2312yes—
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D414yescomplemented
61C2×#127—22no—
62C2×#127—22no—
63C2×#127—22no—
64C2×#127—22no—
65C2×#127—22no—
66C2×#127D414yes—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—44no—
69C2×C22⋊Q8—44no—
70C2×#121—28no—
71C2×#121—28no—
72C2×#121—24no—
73C2×#121—24no—
74C2×#121—28no—
75C2×#121—44no—
76C2×#121—44no—
77C2×#121—44no—
78C2×#121—44no—
79C2×#121—44no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4—22no—
93C22×C42—22no—
94C22×C42—22no—
95C22×C42—22no—
96C22×C42—22no—
97C22×C42—22no—
98C22×C42—22no—
99C22×C42D411yescharacteristic
100C22×C42D411yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103#486—232no—
104C2×#102—28no—
105C2×#102—28no—
106C2×#102—28no—
107C2×#102—28no—
108C2×#102—28no—
109C2×#102—44no—
110C2×C22⋊C8—24no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—24no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118C2×C4×C8D414yescomplemented
119#391—28no—
120#390—28no—
121#390—28no—
122#390—24no—
123#390—24no—
124#389—28no—
125#389—24no—
126#389—24no—
127#384—22no—
128#384—22no—
129#384D414yes—
130#384D414yescomplemented
131#380—28no—
132#380—24no—
133#380—24no—
134C2×#95—22no—
135C2×#95—22no—
136#340—28no—
137#340—44no—
138#336—24no—
139C25—22no—
140C25C2×D411yescharacteristic
141C22×Q8—24no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×D4—28no—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—24no—
156C22×D4—22no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—44no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4C2×D414yes—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4D814yes—
204C23×C4SD1614yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4#4114yes—
211C23×C4C2411yescharacteristic
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8C2×D414yes—
225C22×C8#4114yes—
226C4⋊Q8—28no—
227C4⋊Q8—28no—
228C4⋊Q8—44no—
229C4⋊Q8—44no—
230#127—28no—
231#127—28no—
232#127—44no—
233#127—44no—
234C22⋊Q8—432no—
235#121—416no—
236#121—416no—
237#121—432no—
238#121—432no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42#4112yes—
305#102—232no—
306#102—416no—
307C22⋊C8—216no—
308C4×C8—28no—
309C4×C8—44no—
310#95—28no—
311#95—28no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—24no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24#12412yes—
340C24C2×D812yes—
341C24C2×SD1612yes—
342C24C22×D411yescharacteristic
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×Q8—84no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—88no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—84no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4#12012yes—
457C22×C4#12012yes—
458C22×C4#12012yes—
459C22×C4#12012yes—
460C22×C4#12112yes—
461C22×C4#12112yes—
462C22×C4#12412yes—
463C22×C4C2×D812yes—
464C22×C4C2×D812yes—
465C22×C4C2×SD1612yes—
466C22×C4C2×SD1612yes—
467C22×C4#13512yes—
468C22×C4#13512yes—
469C22×C4AGL(1,ℤ/8)12yes—
470C22×C4AGL(1,ℤ/8)12yes—
471C22×C4AGL(1,ℤ/8)12yes—
472C22×C4#13712yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4111yescharacteristic
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—416no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—416no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—44no—
520C42—44no—
521C42#12014yes—
522C42#12114yes—
523C42#12114yes—
524C42#12714yes—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—82no—
568C23—82no—
569C23—82no—
570C23—82no—
571C23—82no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23—82no—
577C23—82no—
578C23—82no—
579C23—82no—
580C23—82no—
581C23—82no—
582C23—82no—
583C23—82no—
584C23—82no—
585C23—82no—
586C23—82no—
587C23—82no—
588C23—82no—
589C23—82no—
590C23—82no—
591C23—82no—
592C23—82no—
593C23—82no—
594C23#48614yescomplemented
595C23C2×#12011yescharacteristic
596C23C2×#12011yescharacteristic
597C23C2×#12111yescharacteristic
598C23C2×#12411yescharacteristic
599C23#53211yescharacteristic
600C23#53511yescharacteristic
601C23#53611yescharacteristic
602C23C22×D811yescharacteristic
603C23C22×SD1611yescharacteristic
604C23C2×#13511yescharacteristic
605C23C2×#13511yescharacteristic
606C23C2×AGL(1,ℤ/8)11yescharacteristic
607C23C2×AGL(1,ℤ/8)11yescharacteristic
608C23C2×AGL(1,ℤ/8)11yescharacteristic
609C23C2×#13711yescharacteristic
610Q8—816no—
611D4—48no—
612D4—48no—
613D4—832no—
614D4—84no—
615D4—816no—
616D4—816no—
617D4—84no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—416no—
643C2×C4—416no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—82no—
660C2×C4—82no—
661C2×C4—82no—
662C2×C4—82no—
663C2×C4—82no—
664C2×C4—82no—
665C2×C4—82no—
666C2×C4#39214yes—
667C2×C4#39214yes—
668C2×C4#44714yes—
669C2×C4#44714yes—
670C2×C4#44814yes—
671C2×C4#44914yes—
672C2×C4#44914yes—
673C2×C4#44914yes—
674C2×C4#45014yes—
675C2×C4#45214yes—
676C2×C4#46514yes—
677C2×C4#46614yes—
678C2×C4#46814yes—
679C2×C4#46914yes—
680C2×C4C2×#12011yescharacteristic
681C2×C4C2×#12111yescharacteristic
682C2×C4C2×#12111yescharacteristic
683C2×C4#52512yes—
684C2×C4#52912yes—
685C2×C4C2×#12711yescharacteristic
686C2×C4#53712yes—
687C2×C4#54212yes—
688C2×C4#54612yes—
689C2×C4#54612yes—
690C2×C4#54612yes—
691C2×C4#54712yes—
692C2×C4#54812yes—
693C2×C4#54812yes—
694C2×C4#54812yes—
695C2×C4#55012yes—
696C8—44no—
697C8—44no—
698C22—24no—
699C22—24no—
700C22—24no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—84no—
714C22—84no—
715C22—84no—
716C22—84no—
717C22—84no—
718C22—84no—
719C22—84no—
720C22—162no—
721C22—162no—
722C22—162no—
723C22—162no—
724C22—162no—
725C22—162no—
726C22—162no—
727C22—162no—
728C22C2×#39211yescharacteristic
729C22#243111yescharacteristic
730C22#243211yescharacteristic
731C22#243511yescharacteristic
732C22#243611yescharacteristic
733C22#262011yescharacteristic
734C22#285011yescharacteristic
735C22C2×#44711yescharacteristic
736C22C2×#44711yescharacteristic
737C22C2×#45011yescharacteristic
738C22C2×#44811yescharacteristic
739C22C2×#44911yescharacteristic
740C22C2×#44911yescharacteristic
741C22C2×#44911yescharacteristic
742C22C2×#45211yescharacteristic
743C22C2×#46511yescharacteristic
744C22C2×#46611yescharacteristic
745C22C2×#46811yescharacteristic
746C22C2×#46911yescharacteristic
747C22C2×#48611yescharacteristic
748C22#313111yescharacteristic
749C22#313811yescharacteristic
750C22#314111yescharacteristic
751C22#316711yescharacteristic
752C22#316911yescharacteristic
753C22#318211yescharacteristic
754C22#318411yescharacteristic
755C22#320511yescharacteristic
756C22#320711yescharacteristic
757C22#320811yescharacteristic
758C22#321011yescharacteristic
759C22#322611yescharacteristic
760C22#322711yescharacteristic
761C22#322811yescharacteristic
762C22#322811yescharacteristic
763C4—42no—
764C4—42no—
765C4—42no—
766C4—84no—
767C4—84no—
768C4—84no—
769C4#199918yes—
770C4#200018yes—
771C4#240712yes—
772C4#242612yes—
773C4#242712yes—
774C4#254012yes—
775C2—44no—
776C2—164no—
777C2#1521511yescharacteristic
778C2C2×#199911yescharacteristic
779C2C2×#200011yescharacteristic
780C2#2559811yescharacteristic
781C2#2559911yescharacteristic
782C2#2567111yescharacteristic
783C2#2567311yescharacteristic
784C2#2585511yescharacteristic
785C2#2585711yescharacteristic
786C2#2590311yescharacteristic
787C2#2590511yescharacteristic
788C2#2984211yescharacteristic
789C2#2984311yescharacteristic
790C2#3304211yescharacteristic
791C2#3304811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2856C2
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#486C2semidirect
2. C2×#2857C2semidirect
4. C2×#1990C2semidirect
5. C2×#1957C2semidirect
6. C2×C4×C22⋊C8C2semidirect
7. #11066C2semidirect
8. #10969C2semidirect
9. #10841C2semidirect
60. C2×C4⋊Q8D4semidirect
118. C2×C4×C8D4semidirect
130. #384D4semidirect
594. C23#486semidirect

Maximal subgroups

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

subgroupindex
C22×#4862
C2×#28572
C2×#28562
C2×#19902
C2×#19572
C2×C4×C22⋊C82
#110662
#109692
#108412