Finite Group Databasebeta

← #376089

Subgroups — #376089

Order 512.

2336 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#127C211yesmaximal, characteristic
2C2×#2014C211yesmaximal, characteristic
3C2×#1880C212yesmaximal, complemented
4C2×#1877C212yesmaximal, complemented
5C4×#391C211yesmaximal, characteristic, complemented
6#11267C211yesmaximal, characteristic, complemented
7#11050C211yesmaximal, characteristic, complemented
8#11008C211yesmaximal, characteristic
9#10931C212yesmaximal, complemented
10#10795C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal, complemented
12C23×SD16C2212yes—
13C22×#127C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4—21no—
16C22×C4×D4—21no—
17C22×C4×D4C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C4×#127—28no—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#390—21no—
26C2×#390—21no—
27C2×#390C2212yes—
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#2014C418yescomplemented
32#1880—28no—
33#1877—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41#1436—21no—
42#1436—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C4×#95—21no—
46C4×#95—21no—
47C23×Q8C2311yescharacteristic
48C23×D4—21no—
49C23×D4—21no—
50C23×D4C2312yes—
51C24×C4—22no—
52C22×SD16—24no—
53C22×SD16—24no—
54C22×SD16—24no—
55C22×SD16—24no—
56C22×SD16—42no—
57C22×SD16—42no—
58C22×SD16C2×C418yescomplemented
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×#127—24no—
62C2×#127—24no—
63C2×#127—22no—
64C2×#127C2×C414yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—24no—
68C2×C4×Q8—42no—
69C2×C4×D4—28no—
70C2×C4×D4—28no—
71C2×C4×D4—24no—
72C2×C4×D4—24no—
73C2×C4×D4—24no—
74C2×C4×D4—24no—
75C2×C4×D4—42no—
76C2×C4×D4—42no—
77C2×C4×D4—44no—
78C2×C4×D4—44no—
79C2×C4×D4—44no—
80C2×C4×D4—44no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—22no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C42—21no—
92C22×C42—21no—
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—42no—
108C2×#102—42no—
109C2×#102—42no—
110C2×#102—42no—
111C2×#102C2×C418yescomplemented
112#391—22no—
113#391—22no—
114#391C2×C414yescomplemented
115#391D414yescomplemented
116#390—28no—
117#390—28no—
118#390—24no—
119#390D418yes—
120#384—24no—
121C4×C4⋊C4—42no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C43—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#336—24no—
136#336—24no—
137#336—22no—
138#336C2×C414yescomplemented
139C25—22no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8C22×C412yes—
148C22×Q8C2×D412yes—
149C22×Q8#4112yes—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
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×D4C22×C414yes—
176C22×D4C2×D414yes—
177C22×D4#4114yes—
178C23×C4—21no—
179C23×C4—28no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2411yescharacteristic
197C2×SD16—216no—
198C2×SD16—216no—
199C2×SD16—48no—
200C2×SD16—48no—
201C2×SD16—48no—
202C2×SD16—48no—
203C2×SD16—48no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8C22×C414yes—
218#127—216no—
219#127—216no—
220C4×Q8—48no—
221C4×Q8—48no—
222C4×D4—48no—
223C4×D4—48no—
224C4×D4—48no—
225C4×D4—48no—
226C4×D4—432no—
227C4×D4—432no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4#4112yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42SD1614yescomplemented
289C2×C42C22×C412yes—
290C2×C42C2×D414yes—
291C2×C42C2×D412yes—
292C2×C42#4112yes—
293C2×C42#4112yes—
294#103—48no—
295#102—48no—
296#102—48no—
297#102—48no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—42no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—42no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24C22×D411yescharacteristic
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8C4×D414yescomplemented
339C2×D4—216no—
340C2×D4—216no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4C4×D418yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
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—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4C4×D412yes—
452C22×C4C4×D412yes—
453C22×C4#12012yes—
454C22×C4#12012yes—
455C22×C4#12112yes—
456C22×C4#12112yes—
457C22×C4C2×SD1612yes—
458C22×C4C2×SD1612yes—
459C22×C4C2×SD1612yes—
460C22×C4#13512yes—
461C22×C4AGL(1,ℤ/8)12yes—
462C22×C4AGL(1,ℤ/8)12yes—
463C22×C4AGL(1,ℤ/8)12yes—
464C22×C4#13712yes—
465C22×C4C23×C411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4112yes—
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4H(5,𝔽2)12yes—
474C22×C4H(5,𝔽2)12yes—
475SD16—432no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C42—28no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—24no—
512C42—24no—
513C42—28no—
514C42—44no—
515C42—44no—
516C42—48no—
517C42—84no—
518C23—28no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23—82no—
553C23C2×C4×D411yescharacteristic
554C23C2×#12011yescharacteristic
555C23C2×#12111yescharacteristic
556C23#52511yescharacteristic
557C23#52511yescharacteristic
558C23#53511yescharacteristic
559C23#53511yescharacteristic
560C23C22×SD1611yescharacteristic
561C23C2×#13511yescharacteristic
562C23C2×AGL(1,ℤ/8)11yescharacteristic
563C23C2×AGL(1,ℤ/8)11yescharacteristic
564C23C2×AGL(1,ℤ/8)11yescharacteristic
565C23C2×#13711yescharacteristic
566C23#57511yescharacteristic
567C23#57511yescharacteristic
568Q8—28no—
569Q8—44no—
570Q8—44no—
571Q8—44no—
572D4—28no—
573D4—28no—
574D4—432no—
575D4—432no—
576D4—44no—
577D4—44no—
578D4—44no—
579D4—44no—
580D4—44no—
581D4—44no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
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—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—832no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—82no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—82no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4C4×SD1614yes—
657C2×C4#44014yes—
658C2×C4#45014yes—
659C2×C4#46014yes—
660C2×C4C2×C4×D411yescharacteristic
661C2×C4C2×C4×D412yes—
662C2×C4C4×#4111yescharacteristic
663C2×C4#51812yes—
664C2×C4#52512yes—
665C2×C4#52911yescharacteristic
666C2×C4#53211yescharacteristic
667C2×C4#54012yes—
668C2×C4D4212yes—
669C2×C4#54612yes—
670C2×C4#54612yes—
671C2×C4#54612yes—
672C2×C4#54612yes—
673C2×C4#55112yes—
674C2×C4#55312yes—
675C2×C4#55312yes—
676C8—44no—
677C8—82no—
678C22—48no—
679C22—48no—
680C22—48no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—84no—
690C22—84no—
691C22—84no—
692C22—84no—
693C22C4×#12011yescharacteristic
694C22C4×#12111yescharacteristic
695C22#232811yescharacteristic
696C22#240411yescharacteristic
697C22#240811yescharacteristic
698C22#253511yescharacteristic
699C22#259811yescharacteristic
700C22C2×C4×SD1611yescharacteristic
701C22C2×#44011yescharacteristic
702C22C4×AGL(1,ℤ/8)11yescharacteristic
703C22C4×AGL(1,ℤ/8)11yescharacteristic
704C22C2×#45011yescharacteristic
705C22#300211yescharacteristic
706C22#300311yescharacteristic
707C22#300711yescharacteristic
708C22C2×#46011yescharacteristic
709C22#303511yescharacteristic
710C22#303711yescharacteristic
711C22#303811yescharacteristic
712C22#310111yescharacteristic
713C22#310311yescharacteristic
714C22#310511yescharacteristic
715C22#310611yescharacteristic
716C22#311011yescharacteristic
717C22#311211yescharacteristic
718C22#311411yescharacteristic
719C22#311511yescharacteristic
720C22#320611yescharacteristic
721C22#320711yescharacteristic
722C22#320811yescharacteristic
723C22#321011yescharacteristic
724C22#321311yescharacteristic
725C22#321411yescharacteristic
726C22#321711yescharacteristic
727C22#321911yescharacteristic
728C4—24no—
729C4—24no—
730C4—24no—
731C4—24no—
732C4—44no—
733C4—42no—
734C4—42no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—84no—
739C4#235812yes—
740C4#236412yes—
741C4#245712yes—
742C4#246412yes—
743C2—88no—
744C2#1417511yescharacteristic
745C2C4×#45011yescharacteristic
746C2#2145311yescharacteristic
747C2C4×#46011yescharacteristic
748C2#2150511yescharacteristic
749C2#2226811yescharacteristic
750C2#2227211yescharacteristic
751C2#2393111yescharacteristic
752C2#2393211yescharacteristic
753C2#2421111yescharacteristic
754C2#2421611yescharacteristic
755C2#2815711yescharacteristic
756C2#2816211yescharacteristic
757C2#2936611yescharacteristic
758C2#2936711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1880C2semidirect
4. C2×#1877C2semidirect
5. C4×#391C2semidirect
6. #11267C2semidirect
7. #11050C2semidirect
9. #10931C2semidirect
10. #10795C2semidirect
11. #10788C2semidirect
31. #2014C4semidirect
58. C22×SD16C2×C4semidirect
111. C2×#102C2×C4semidirect
114. #391C2×C4semidirect
115. #391D4semidirect
138. #336C2×C4semidirect
288. C2×C42SD16semidirect
338. C2×Q8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1272
C2×#20142
C2×#18802
C2×#18772
C4×#3912
#112672
#110502
#110082
#109312
#107952
#107882