Finite Group Databasebeta

← #416151

Subgroups — #416151

Order 512.

2228 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2856C211yesmaximal, characteristic
2C2×#1957C211yesmaximal, characteristic, complemented
3C2×#1957C211yesmaximal, characteristic, complemented
4C2×#1834C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
6#14686C211yesmaximal, characteristic, complemented
7#11189C214yesmaximal, complemented
8#10969C214yesmaximal, complemented
9#10841C211yesmaximal, characteristic, complemented
10C22×#127C2211yescharacteristic
11C22×#121—21no—
12C22×#121—21no—
13C22×#121C2212yes—
14C23×C42C2211yescharacteristic
15C22×#102C2214yes—
16C22×#102C2214yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2856—24no—
21#2856—24no—
22C2×#390—21no—
23C2×#390—21no—
24C2×#390C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×#380—21no—
30C2×#380C2212yes—
31#1957—24no—
32#1957—24no—
33#1834—24no—
34C4×C22⋊C8—24no—
35C2×#340C2214yes—
36C2×#340C2214yes—
37C2×#336C2212yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C23×D4—22no—
42C23×D4—21no—
43C23×D4—21no—
44C23×D4C2312yes—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#127—22no—
51C2×#127—22no—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127D414yes—
56C2×#121—28no—
57C2×#121—28no—
58C2×#121—24no—
59C2×#121—24no—
60C2×#121—28no—
61C2×#121—44no—
62C2×#121—44no—
63C2×#121—44no—
64C2×#121—44no—
65C2×#121—44no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#102—28no—
90C2×#102—28no—
91C2×#102—28no—
92C2×#102—44no—
93C2×#102—44no—
94C2×#102—44no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8D414yescomplemented
104#390—28no—
105#390—28no—
106#390—24no—
107#390—24no—
108#384—22no—
109#384—22no—
110#384—22no—
111#384—22no—
112#384—22no—
113#384D414yes—
114#380—44no—
115#380—44no—
116#380—44no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120#340—28no—
121#340—44no—
122#336—24no—
123C25—22no—
124C25C2×D411yescharacteristic
125C22×D4—28no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—24no—
136C22×D4—24no—
137C22×D4—44no—
138C22×D4—44no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4C2×D414yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4D814yes—
181C23×C4SD1614yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4#4114yes—
188C23×C4C2411yescharacteristic
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8C2×D414yes—
202C22×C8#4114yes—
203#127—28no—
204#127—28no—
205#127—44no—
206#127—44no—
207#121—416no—
208#121—416no—
209#121—432no—
210#121—432no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C2×D414yes—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274C2×C42#4112yes—
275#102—416no—
276#102—416no—
277C22⋊C8—216no—
278C4×C8—28no—
279C4×C8—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—24no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24#12412yes—
311C24C2×D812yes—
312C24C2×SD1612yes—
313C24C22×D411yescharacteristic
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—28no—
317C2×D4—28no—
318C2×D4—416no—
319C2×D4—416no—
320C2×D4—416no—
321C2×D4—42no—
322C2×D4—44no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—88no—
340C2×D4—88no—
341C2×D4—88no—
342C2×D4—88no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4#12012yes—
421C22×C4#12012yes—
422C22×C4#12112yes—
423C22×C4#12112yes—
424C22×C4#12312yes—
425C22×C4#12312yes—
426C22×C4#12412yes—
427C22×C4C2×D812yes—
428C22×C4C2×D812yes—
429C22×C4C2×SD1612yes—
430C22×C4C2×SD1612yes—
431C22×C4#13512yes—
432C22×C4#13512yes—
433C22×C4AGL(1,ℤ/8)12yes—
434C22×C4AGL(1,ℤ/8)12yes—
435C22×C4AGL(1,ℤ/8)12yes—
436C22×C4#13712yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)12yes—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C4⋊C4—28no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—88no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—44no—
482C42—44no—
483C42#12114yes—
484C42#12114yes—
485C42#12314yes—
486C42#12414yes—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
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—
553C23—82no—
554C23—82no—
555C23—82no—
556C23#48614yes—
557C23C2×#12011yescharacteristic
558C23C2×#12111yescharacteristic
559C23C2×#12311yescharacteristic
560C23C2×#12411yescharacteristic
561C23#53211yescharacteristic
562C23#53511yescharacteristic
563C23#53511yescharacteristic
564C23C22×D811yescharacteristic
565C23C22×SD1611yescharacteristic
566C23C2×#13511yescharacteristic
567C23C2×#13511yescharacteristic
568C23C2×AGL(1,ℤ/8)11yescharacteristic
569C23C2×AGL(1,ℤ/8)11yescharacteristic
570C23C2×AGL(1,ℤ/8)11yescharacteristic
571C23C2×#13711yescharacteristic
572D4—48no—
573D4—48no—
574D4—832no—
575D4—84no—
576D4—84no—
577D4—816no—
578D4—816no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4#39414yes—
628C2×C4#39414yes—
629C2×C4#44714yes—
630C2×C4#44914yes—
631C2×C4#45014yes—
632C2×C4#45214yes—
633C2×C4#46514yes—
634C2×C4#46614yes—
635C2×C4#46814yes—
636C2×C4#46914yes—
637C2×C4#48014yes—
638C2×C4#48114yes—
639C2×C4#48214yes—
640C2×C4#48214yes—
641C2×C4C2×#12111yescharacteristic
642C2×C4C2×#12111yescharacteristic
643C2×C4C2×#12311yescharacteristic
644C2×C4#52512yes—
645C2×C4C2×#12411yescharacteristic
646C2×C4#52912yes—
647C2×C4#53512yes—
648C2×C4#54012yes—
649C2×C4#54612yes—
650C2×C4#54612yes—
651C2×C4#54612yes—
652C2×C4#55012yes—
653C2×C4#55312yes—
654C2×C4#55312yes—
655C2×C4#55312yes—
656C2×C4#55512yes—
657C8—44no—
658C8—44no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—162no—
682C22—162no—
683C22—162no—
684C22—162no—
685C22—162no—
686C22—162no—
687C22—162no—
688C22—162no—
689C22C2×#39411yescharacteristic
690C22#243111yescharacteristic
691C22#243511yescharacteristic
692C22#250011yescharacteristic
693C22#250411yescharacteristic
694C22#261911yescharacteristic
695C22#285011yescharacteristic
696C22C2×#44711yescharacteristic
697C22C2×#45011yescharacteristic
698C22C2×#44911yescharacteristic
699C22C2×#45211yescharacteristic
700C22C2×#46511yescharacteristic
701C22C2×#46611yescharacteristic
702C22C2×#46811yescharacteristic
703C22C2×#46911yescharacteristic
704C22C2×#48011yescharacteristic
705C22C2×#48211yescharacteristic
706C22C2×#48211yescharacteristic
707C22C2×#48111yescharacteristic
708C22C2×#48611yescharacteristic
709C22#313111yescharacteristic
710C22#313811yescharacteristic
711C22#314111yescharacteristic
712C22#316711yescharacteristic
713C22#316911yescharacteristic
714C22#318211yescharacteristic
715C22#318411yescharacteristic
716C22#320511yescharacteristic
717C22#320511yescharacteristic
718C22#320611yescharacteristic
719C22#320711yescharacteristic
720C22#320711yescharacteristic
721C22#320711yescharacteristic
722C22#320811yescharacteristic
723C22#321011yescharacteristic
724C4—42no—
725C4—42no—
726C4—42no—
727C4—84no—
728C4—162no—
729C4—162no—
730C4#203318yes—
731C4#203418yes—
732C4#241412yes—
733C4#242012yes—
734C4#249012yes—
735C4#253512yes—
736C2—44no—
737C2—164no—
738C2#1521911yescharacteristic
739C2C2×#203311yescharacteristic
740C2C2×#203411yescharacteristic
741C2#2559711yescharacteristic
742C2#2559911yescharacteristic
743C2#2585511yescharacteristic
744C2#2585711yescharacteristic
745C2#2789011yescharacteristic
746C2#2789111yescharacteristic
747C2#2797311yescharacteristic
748C2#2797411yescharacteristic
749C2#2980611yescharacteristic
750C2#2980711yescharacteristic
751C2#3304211yescharacteristic
752C2#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
2. C2×#1957C2semidirect
3. C2×#1957C2semidirect
4. C2×#1834C2semidirect
5. C2×C4×C22⋊C8C2semidirect
6. #14686C2semidirect
7. #11189C2semidirect
8. #10969C2semidirect
9. #10841C2semidirect
103. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
C2×#19572
C2×#18342
C2×C4×C22⋊C82
#146862
#111892
#109692
#108412