Finite Group Databasebeta

← #366988

Subgroups — #366988

Order 512.

1986 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#460C211yesmaximal, characteristic
2C2×#1982C211yesmaximal, characteristic
3C2×#1968C212yesmaximal, complemented
4C2×#1948C211yesmaximal, characteristic, complemented
5C2×#1803C211yesmaximal, characteristic, complemented
6C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
7#14713C211yesmaximal, characteristic
8C4×#391C211yesmaximal, characteristic, complemented
9#11121C212yesmaximal, complemented
10#11023C212yesmaximal, complemented
11#10925C212yesmaximal, complemented
12C23×SD16C2212yes—
13C22×#127C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#460—24no—
16C2×#460—24no—
17C2×#460C418yescomplemented
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#102C2212yes—
25C22×C4×C8C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#390—22no—
29C2×#390—22no—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#1982—24no—
36#1968—28no—
37#1948—24no—
38#1803—24no—
39C4×C4⋊C8—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1436—22no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4×#95—22no—
47C23×Q8C2311yescharacteristic
48C23×D4—21no—
49C23×D4—21no—
50C23×D4C2312yes—
51C22×SD16—24no—
52C22×SD16—24no—
53C22×SD16—24no—
54C22×SD16—24no—
55C22×SD16—42no—
56C22×SD16—42no—
57C22×SD16C2×C418yes—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×#127—22no—
61C2×#127—22no—
62C2×#127—22no—
63C2×#127—22no—
64C2×#127—22no—
65C2×#127C2×C414yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8C2×C414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—24no—
80C22×C22⋊C4—24no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42D412yes—
84C22×C42D412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#460—232no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—42no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8C2×C414yes—
95C2×C4⋊C8D414yes—
96C2×C4⋊C8D414yes—
97C2×C4⋊C8D414yes—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—42no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—42no—
108C2×#102—42no—
109C2×#102C2×C418yes—
110C2×C4×C8—24no—
111C2×C4×C8—24no—
112C2×C4×C8—42no—
113#391—22no—
114#391—22no—
115#391—22no—
116#391D414yescomplemented
117#390—24no—
118#390—24no—
119#390—42no—
120#390—42no—
121#390—48no—
122#390—48no—
123#384—22no—
124#384—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C43—22no—
133C43—22no—
134#340—24no—
135#340—24no—
136#336—24no—
137C25—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8C22×C412yes—
150C22×Q8C2×D412yes—
151C22×Q8#4112yes—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—22no—
155C22×D4—22no—
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—44no—
168C22×D4—44no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4C22×C414yes—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2411yescharacteristic
186C2×SD16—216no—
187C2×SD16—216no—
188C2×SD16—48no—
189C2×SD16—48no—
190C2×SD16—48no—
191C2×SD16—48no—
192C2×SD16—48no—
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×C8—22no—
202C22×C8—22no—
203C22×C8C22×C414yes—
204C22×C8C2×D414yes—
205C22×C8C2×D414yes—
206C22×C8C2×D414yes—
207#127—28no—
208#127—28no—
209#127—44no—
210#127—44no—
211C4×Q8—28no—
212C4×Q8—28no—
213C4×Q8—44no—
214C4×Q8—44no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
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—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4#4112yes—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—48no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—44no—
272C2×C42SD1618yescomplemented
273C2×C42C22×C412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42#4112yes—
279C2×C42#4112yes—
280C2×C42#4112yes—
281#106—48no—
282C4⋊C8—28no—
283C4⋊C8C22⋊C4116yescomplemented
284#103—48no—
285#102—216no—
286#102—48no—
287#102—48no—
288C4×C8—48no—
289#95—28no—
290#95—28no—
291#95—44no—
292#95—42no—
293#95—44no—
294C24—24no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—44no—
302C24—44no—
303C24—44no—
304C24—44no—
305C24C22×D411yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8C4×D414yes—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—416no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—416no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—84no—
371C2×D4—84no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
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—44no—
410C22×C4—44no—
411C22×C4C4×D412yes—
412C22×C4C4×D412yes—
413C22×C4#12112yes—
414C22×C4#12114yes—
415C22×C4#12112yes—
416C22×C4#12712yes—
417C22×C4#12712yes—
418C22×C4C2×SD1614yes—
419C22×C4C2×SD1614yes—
420C22×C4C2×SD1614yes—
421C22×C4#13514yes—
422C22×C4AGL(1,ℤ/8)14yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C4H(5,𝔽2)11yescharacteristic
436C22×C42−1+411yescharacteristic
437SD16—432no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8C2×C22⋊C418yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—48no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C22⋊C4—816no—
472C22⋊C4—816no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—216no—
478C42—48no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—48no—
484C42—84no—
485C42C2×C22⋊C414yes—
486C42C4×D414yes—
487C42#12114yes—
488C42#12414yes—
489C23—44no—
490C23—48no—
491C23—48no—
492C23—44no—
493C23—44no—
494C23—84no—
495C23—84no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—82no—
510C23—82no—
511C23—84no—
512C23C2×C4×D411yescharacteristic
513C23C2×#12111yescharacteristic
514C23C2×#12111yescharacteristic
515C23#52511yescharacteristic
516C23C2×#12711yescharacteristic
517C23#53511yescharacteristic
518C23#53711yescharacteristic
519C23C22×SD1612yes—
520C23C2×#13512yes—
521C23C2×AGL(1,ℤ/8)12yes—
522C23#57512yes—
523Q8—24no—
524Q8—24no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531D4—44no—
532D4—44no—
533D4—44no—
534D4—44no—
535D4—816no—
536D4—82no—
537D4—82no—
538D4—816no—
539D4—82no—
540D4—82no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—82no—
585C2×C4#39014yes—
586C2×C4#39014yes—
587C2×C4C4×SD1618yes—
588C2×C4#46018yescomplemented
589C2×C4#46518yes—
590C2×C4#49214yes—
591C2×C4#49514yes—
592C2×C4C22×C22⋊C411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C2×C4×D411yescharacteristic
595C2×C4C4×#4111yescharacteristic
596C2×C4#51811yescharacteristic
597C2×C4#52011yescharacteristic
598C2×C4C2×#12111yescharacteristic
599C2×C4C2×#12411yescharacteristic
600C2×C4#52911yescharacteristic
601C2×C4#53211yescharacteristic
602C2×C4#54011yescharacteristic
603C2×C4#54211yescharacteristic
604C2×C4D4211yescharacteristic
605C2×C4D4211yescharacteristic
606C2×C4D4211yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#54611yescharacteristic
610C2×C4#54711yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4#54811yescharacteristic
613C2×C4#54811yescharacteristic
614C2×C4#55011yescharacteristic
615C2×C4#55011yescharacteristic
616C2×C4#55011yescharacteristic
617C2×C4#55311yescharacteristic
618C2×C4#55311yescharacteristic
619C2×C4#55511yescharacteristic
620C8—28no—
621C8—44no—
622C22—88no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—88no—
627C22—162no—
628C22—162no—
629C22—164no—
630C22—164no—
631C22—162no—
632C22—162no—
633C22C2×#39011yescharacteristic
634C22C4×#12111yescharacteristic
635C22#232411yescharacteristic
636C22#241411yescharacteristic
637C22#241611yescharacteristic
638C22#254011yescharacteristic
639C22#254111yescharacteristic
640C22C2×C4×SD1612yes—
641C22C4×AGL(1,ℤ/8)12yes—
642C22C2×#46012yes—
643C22#303512yes—
644C22C2×#46512yes—
645C22#305512yes—
646C22#310312yes—
647C22#310612yes—
648C22C2×#49211yescharacteristic
649C22C2×#49511yescharacteristic
650C22#315111yescharacteristic
651C22#315311yescharacteristic
652C22#320812yes—
653C22#321212yes—
654C22#324612yes—
655C22#324712yes—
656C4—28no—
657C4—24no—
658C4—44no—
659C4—44no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—162no—
664C4#196818yescomplemented
665C4D4×C22⋊C411yescharacteristic
666C4#234111yescharacteristic
667C4#236111yescharacteristic
668C4#236311yescharacteristic
669C4#242611yescharacteristic
670C4#242811yescharacteristic
671C4#249011yescharacteristic
672C4#249111yescharacteristic
673C2—164no—
674C2#1399811yescharacteristic
675C2C2×#196811yescharacteristic
676C2#2123611yescharacteristic
677C2C4×#46012yes—
678C2#2214712yes—
679C2#2465312yes—
680C2#2475312yes—
681C2#2837111yescharacteristic
682C2#2837411yescharacteristic
683C2#2843911yescharacteristic
684C2#2844111yescharacteristic
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×#1968C2semidirect
4. C2×#1948C2semidirect
5. C2×#1803C2semidirect
6. C2×C4×C4⋊C8C2semidirect
8. C4×#391C2semidirect
9. #11121C2semidirect
10. #11023C2semidirect
11. #10925C2semidirect
17. C2×#460C4semidirect
116. #391D4semidirect
272. C2×C42SD16semidirect
283. C4⋊C8C22⋊C4semidirect
588. C2×C4#460semidirect
664. C4#1968semidirect

Maximal subgroups

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

subgroupindex
C22×#4602
C2×#19822
C2×#19682
C2×#19482
C2×#18032
C2×C4×C4⋊C82
#147132
C4×#3912
#111212
#110232
#109252