Finite Group Databasebeta

← #154381

Subgroups — #154381

Order 512.

4454 conjugacy classes of subgroups.

The subgroup lattice has 773 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#126C211yesmaximal, characteristic
2C2×#1437C212yesmaximal
3C2×#1436C212yesmaximal, complemented
4C2×#1434C211yesmaximal, characteristic, complemented
5C2×C4×#95C211yesmaximal, characteristic, complemented
6#7557C212yesmaximal, complemented
7#7557C212yesmaximal, complemented
8#7555C212yesmaximal
9#7555C212yesmaximal
10C22×#126—24no—
11C22×#126—24no—
12C22×#126—28no—
13C22×#126—28no—
14C22×#126C418yescomplemented
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2214yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C42C2212yes—
23C23×C42C2211yescharacteristic
24C22×#95C2212yes—
25C22×#95C2212yes—
26C22×#95C2212yes—
27C22×#95C2214yes—
28C22×#95C2212yes—
29C22×#95C2214yes—
30C22×#95C2214yes—
31C22×#95C2212yes—
32#1437—28no—
33#1436—28no—
34#1434—24no—
35C4×#95—24no—
36C26C2311yescharacteristic
37C24×C4C2314yes—
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2314yes—
44C24×C4C2311yescharacteristic
45C2×#126—264no—
46C2×#126—264no—
47C2×#126—232no—
48C2×#126—232no—
49C2×#126—232no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4C2×C414yes—
62C22×C4⋊C4C2×C414yes—
63C22×C4⋊C4C2×C414yes—
64C22×C4⋊C4D418yes—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—24no—
72C22×C22⋊C4—24no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—24no—
76C22×C22⋊C4—28no—
77C22×C22⋊C4—28no—
78C22×C22⋊C4—28no—
79C22×C22⋊C4—28no—
80C22×C22⋊C4—28no—
81C22×C22⋊C4—44no—
82C22×C22⋊C4—44no—
83C22×C22⋊C4C2×C414yes—
84C22×C22⋊C4C2×C414yes—
85C22×C22⋊C4C2×C414yes—
86C22×C22⋊C4D418yes—
87C22×C42—24no—
88C22×C42—24no—
89C22×C42—24no—
90C22×C42—28no—
91C22×C42—24no—
92C22×C42—24no—
93C22×C42—44no—
94C22×C42C2×C414yes—
95C22×C42D414yes—
96C22×C42D414yes—
97C2×#95—24no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95—28no—
101C2×#95—28no—
102C2×#95—28no—
103C2×#95—28no—
104C2×#95—28no—
105C2×#95—28no—
106C2×#95—24no—
107C2×#95—24no—
108C2×#95—28no—
109C2×#95—24no—
110C2×#95—28no—
111C2×#95—24no—
112C2×#95—28no—
113C2×#95—28no—
114C2×#95—44no—
115C2×#95—44no—
116C2×#95—44no—
117C25—21no—
118C25—21no—
119C25—22no—
120C25—24no—
121C25—24no—
122C25—24no—
123C25—24no—
124C25—22no—
125C25—22no—
126C25—22no—
127C25—22no—
128C25C22×C412yes—
129C25C2×D414yes—
130C25C2411yescharacteristic
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4C22×C412yes—
187C23×C4C22×C412yes—
188C23×C4C22×C412yes—
189C23×C4C22×C412yes—
190C23×C4C22×C412yes—
191C23×C4C22×C412yes—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D414yes—
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4#4118yes—
204C23×C4#4118yes—
205C23×C4#4118yes—
206C23×C4#4118yes—
207#126—2256no—
208C2×C4⋊C4—216no—
209C2×C4⋊C4—216no—
210C2×C4⋊C4—216no—
211C2×C4⋊C4—216no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—216no—
221C2×C4⋊C4—216no—
222C2×C4⋊C4C22⋊C4116yescomplemented
223C2×C22⋊C4—216no—
224C2×C22⋊C4—216no—
225C2×C22⋊C4—216no—
226C2×C22⋊C4—216no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—216no—
236C2×C22⋊C4—216no—
237C2×C22⋊C4—232no—
238C2×C22⋊C4—416no—
239C2×C22⋊C4—416no—
240C2×C22⋊C4—416no—
241C2×C22⋊C4—416no—
242C2×C22⋊C4—416no—
243C2×C22⋊C4—416no—
244C2×C22⋊C4C22⋊C4116yes—
245C2×C42—28no—
246C2×C42—28no—
247C2×C42—216no—
248C2×C42—216no—
249C2×C42—216no—
250C2×C42—416no—
251C2×C42—416no—
252C2×C42—48no—
253C2×C42—48no—
254C2×C42C22⋊C4116yescomplemented
255#95—216no—
256#95—216no—
257#95—232no—
258#95—216no—
259#95—416no—
260#95—416no—
261#95—48no—
262#95—48no—
263C24—22no—
264C24—22no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24—24no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—28no—
274C24—28no—
275C24—28no—
276C24—28no—
277C24—28no—
278C24—28no—
279C24—42no—
280C24—42no—
281C24—44no—
282C24—42no—
283C24—42no—
284C24—44no—
285C24—44no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—42no—
298C24—42no—
299C24—44no—
300C24—44no—
301C24—44no—
302C24—44no—
303C24—44no—
304C24—44no—
305C24—44no—
306C24—44no—
307C24—44no—
308C24—44no—
309C24—44no—
310C24—44no—
311C24—44no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24C2×C22⋊C414yes—
316C24C23×C411yescharacteristic
317C24C22×D411yescharacteristic
318C24C22×D411yescharacteristic
319C24C22×D412yes—
320C24C22×D412yes—
321C24C2×#4114yes—
322C24C2×#4114yes—
323C24C2×#4114yes—
324C24C2×#4114yes—
325C24H(5,𝔽2)12yes—
326C24H(5,𝔽2)12yes—
327C242−1+412yes—
328C242−1+412yes—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4C2×C22⋊C414yes—
483C22×C4C2×C22⋊C414yes—
484C22×C4C2×C22⋊C414yes—
485C22×C4C2×C22⋊C414yes—
486C22×C4C2×C22⋊C414yes—
487C22×C4C2×C22⋊C414yes—
488C22×C4C2×C22⋊C414yes—
489C22×C4C2×C22⋊C414yes—
490C22×C4#12018yes—
491C22×C4#12018yes—
492C22×C4#126116yescomplemented
493C4⋊C4—232no—
494C4⋊C4—232no—
495C4⋊C4—232no—
496C22⋊C4—232no—
497C22⋊C4—232no—
498C22⋊C4—232no—
499C22⋊C4—464no—
500C42—232no—
501C42—432no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—28no—
506C23—48no—
507C23—42no—
508C23—42no—
509C23—48no—
510C23—44no—
511C23—44no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—44no—
517C23—44no—
518C23—48no—
519C23—48no—
520C23—44no—
521C23—44no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—48no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—48no—
534C23—48no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—84no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23—82no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23C22×C22⋊C411yescharacteristic
556C23C22×C22⋊C411yescharacteristic
557C23C22×C22⋊C411yescharacteristic
558C23C4×#4112yes—
559C23C4×#4112yes—
560C23C4×#4112yes—
561C23C4×#4112yes—
562C23#52012yes—
563C23#52012yes—
564C23C2×#12012yes—
565C23C2×#12012yes—
566C23#52512yes—
567C23#52512yes—
568C23#52512yes—
569C23#52512yes—
570C23#52514yes—
571C23#52514yes—
572C23C2×#12614yes—
573C23#52914yes—
574C23#52914yes—
575C23#52914yes—
576C23#53214yes—
577C23#53212yes—
578C23#53212yes—
579C23#53212yes—
580C23#53212yes—
581C23#53214yes—
582C23#53511yescharacteristic
583C23#53511yescharacteristic
584C23#53611yescharacteristic
585C23#53611yescharacteristic
586C23#53712yes—
587C23#53712yes—
588C23#53814yes—
589C23#53914yes—
590C23#54114yes—
591C23#54214yes—
592C23#54614yes—
593C23#54614yes—
594C23#54614yes—
595C23#54714yes—
596C23#54714yes—
597C23#54714yes—
598C23#54814yes—
599C23#55014yes—
600C23#55114yes—
601C23#55114yes—
602C23#55214yes—
603C23#55214yes—
604C23#55314yes—
605C23#55314yes—
606C23#55614yes—
607C23#55614yes—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—216no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—216no—
619C2×C4—216no—
620C2×C4—216no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4—48no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—48no—
642C2×C4—48no—
643C2×C4—48no—
644C2×C4—48no—
645C2×C4—48no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—48no—
649C2×C4—48no—
650C2×C4—48no—
651C2×C4—48no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—48no—
657C2×C4—48no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—48no—
666C2×C4—48no—
667C2×C4—48no—
668C2×C4—48no—
669C2×C4—48no—
670C2×C4—48no—
671C2×C4—48no—
672C2×C4—48no—
673C2×C4—48no—
674C2×C4—48no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4#37918yes—
684C22—416no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—88no—
689C22—88no—
690C22—88no—
691C22—88no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—88no—
697C22C2×#37911yescharacteristic
698C22#229712yes—
699C22#229712yes—
700C22#229711yescharacteristic
701C22#229711yescharacteristic
702C22#229711yescharacteristic
703C22#229711yescharacteristic
704C22C4×#12614yes—
705C22#231111yescharacteristic
706C22#231111yescharacteristic
707C22#233414yes—
708C22#233912yes—
709C22#233912yes—
710C22#233912yes—
711C22#234112yes—
712C22#237314yes—
713C22#237314yes—
714C22#238814yes—
715C22#238814yes—
716C22#240412yes—
717C22#240612yes—
718C22#240714yes—
719C22#243014yes—
720C22#243112yes—
721C22#243212yes—
722C22#246614yes—
723C22#246714yes—
724C22#246814yes—
725C22#246914yes—
726C22#247814yes—
727C22#247914yes—
728C22#248014yes—
729C22#248114yes—
730C22#249614yes—
731C22#251214yes—
732C22#251914yes—
733C22#252114yes—
734C22#255314yes—
735C22#255414yes—
736C22#255914yes—
737C22#256014yes—
738C22#260114yes—
739C22#260114yes—
740C22#260312yes—
741C22#260312yes—
742C22#260412yes—
743C22#260412yes—
744C22#263612yes—
745C22#263712yes—
746C22#266212yes—
747C22#266412yes—
748C22#285214yes—
749C22#286014yes—
750C4—28no—
751C4—44no—
752C4—44no—
753C4—416no—
754C4—44no—
755C4—44no—
756C4—44no—
757C4—88no—
758C2—84no—
759C2#1385111yescharacteristic
760C2#1393711yescharacteristic
761C2#1395611yescharacteristic
762C2#1410114yes—
763C2#1410214yes—
764C2#1416611yescharacteristic
765C2#1416611yescharacteristic
766C2#1429111yescharacteristic
767C2#1471111yescharacteristic
768C2#1491714yes—
769C2#1493014yes—
770C2#1493114yes—
771C2#1507214yes—
subgroupG/H
center, Frattini, socle, cosocleC25C24
commutatorC24C23×C4
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1436C2semidirect
4. C2×#1434C2semidirect
5. C2×C4×#95C2semidirect
6. #7557C2semidirect
7. #7557C2semidirect
14. C22×#126C4semidirect
222. C2×C4⋊C4C22⋊C4semidirect
254. C2×C42C22⋊C4semidirect
492. C22×C4#126semidirect

Maximal subgroups

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

subgroupindex
C23×#1262
C2×#14372
C2×#14362
C2×#14342
C2×C4×#952
#75572
#75552