Finite Group Databasebeta

← #360499

Subgroups — #360499

Order 512.

3522 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C8C211yesmaximal, characteristic
2C22×C4×C22⋊C4C211yesmaximal, characteristic
3C2×#1850C212yesmaximal, complemented
4C2×#1797C212yesmaximal, complemented
5C2×#1779C211yesmaximal, characteristic
6#11042C214yesmaximal, complemented
7#10714C212yesmaximal, complemented
8#10712C212yesmaximal, complemented
9C25×C4C2211yescharacteristic
10C24×C8C2212yes—
11C23×C22⋊C4C411yescharacteristic
12C23×C22⋊C4C411yescharacteristic
13C23×C42C2212yes—
14C22×C4⋊C8C2214yescomplemented
15C22×C22⋊C8—28no—
16C22×C22⋊C8—24no—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23C2×C4×C22⋊C4—28no—
24C2×C4×C22⋊C4C412yes—
25C2×C4×C22⋊C4C411yescharacteristic
26C2×C4×C22⋊C4C411yescharacteristic
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C22×#95C412yes—
31#1850—28no—
32#1797—28no—
33#1779—24no—
34C2×#336C2214yescomplemented
35C2×#336C2214yescomplemented
36C2×#336C2214yescomplemented
37C26C2×C411yescharacteristic
38C24×C4—24no—
39C24×C4—24no—
40C24×C4—24no—
41C24×C4—24no—
42C24×C4—44no—
43C24×C4C2×C412yes—
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C412yes—
48C24×C4D414yes—
49C24×C4Q814yes—
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C23×C8—24no—
54C23×C8—28no—
55C23×C8—24no—
56C23×C8—24no—
57C23×C8—44no—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C23×C8C2314yes—
61C22×C22⋊C4—24no—
62C22×C22⋊C4—24no—
63C22×C22⋊C4—24no—
64C22×C22⋊C4—24no—
65C22×C22⋊C4—24no—
66C22×C22⋊C4C818yescomplemented
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C42—24no—
74C22×C42—24no—
75C22×C42—28no—
76C22×C42—44no—
77C22×C42C2×C412yes—
78C22×C42C2×C412yes—
79C22×C42C2312yes—
80C22×C42C2312yes—
81C2×C4⋊C8—28no—
82C2×C4⋊C8—28no—
83C2×C4⋊C8—44no—
84C2×C22⋊C8—232no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—28no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—44no—
98C2×C22⋊C8D418yes—
99C4×C22⋊C4—24no—
100C4×C22⋊C4—24no—
101C4×C22⋊C4—24no—
102C4×C22⋊C4—28no—
103C4×C22⋊C4—28no—
104C4×C22⋊C4D418yes—
105C2×#95—28no—
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110#336—28no—
111#336—28no—
112#336—44no—
113C25—22no—
114C25—22no—
115C25—24no—
116C25—22no—
117C25—22no—
118C25—24no—
119C25—42no—
120C25—42no—
121C25C4⋊C414yes—
122C25C2×C814yes—
123C25M1614yes—
124C25C22×C411yescharacteristic
125C25C22×C411yescharacteristic
126C25C22×C411yescharacteristic
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—216no—
149C23×C4—216no—
150C23×C4—216no—
151C23×C4—28no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—48no—
162C23×C4—48no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—416no—
171C23×C4—416no—
172C23×C4—416no—
173C23×C4—416no—
174C23×C4C4⋊C414yes—
175C23×C4C4⋊C414yes—
176C23×C4C4⋊C414yes—
177C23×C4C2×C818yescomplemented
178C23×C4C22×C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C412yes—
181C23×C4C22×C412yes—
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C412yes—
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D414yes—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D414yes—
190C23×C4C2×Q812yes—
191C23×C4C2×Q812yes—
192C23×C4C2×Q812yes—
193C23×C4#4114yes—
194C23×C4#4114yes—
195C23×C4C2411yescharacteristic
196C22×C8—216no—
197C22×C8—24no—
198C22×C8—24no—
199C22×C8—24no—
200C22×C8—24no—
201C22×C8—24no—
202C22×C8—24no—
203C22×C8—22no—
204C22×C8—24no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—24no—
211C22×C8—24no—
212C22×C8—24no—
213C22×C8—416no—
214C22×C8—416no—
215C22×C8—48no—
216C22×C8—48no—
217C22×C8C2×D418yes—
218C2×C22⋊C4—232no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—232no—
231C2×C22⋊C4C22⋊C414yes—
232C2×C22⋊C4C22⋊C414yes—
233C2×C22⋊C4C2×C814yes—
234C2×C22⋊C4C2×C814yes—
235C2×C22⋊C4C2×C818yescomplemented
236C2×C22⋊C4M1614yes—
237C2×C22⋊C4M1614yes—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—416no—
250C2×C42—416no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C2×D418yes—
254C4⋊C8—416no—
255C22⋊C8—216no—
256C22⋊C8—216no—
257C22⋊C8—48no—
258C22⋊C8—44no—
259C22⋊C8—48no—
260C22⋊C8—44no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95C22⋊C418yes—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—22no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—28no—
284C24—28no—
285C24—28no—
286C24—28no—
287C24—28no—
288C24—28no—
289C24—28no—
290C24—28no—
291C24—216no—
292C24—216no—
293C24—44no—
294C24—44no—
295C24—48no—
296C24—48no—
297C24—48no—
298C24—48no—
299C24—48no—
300C24—48no—
301C24—48no—
302C24—48no—
303C24—48no—
304C24—48no—
305C24C4⋊C8116yescomplemented
306C24C2×C22⋊C412yes—
307C24C2×C4⋊C412yes—
308C24C2×C4⋊C412yes—
309C24C2×C4⋊C412yes—
310C24#11712yes—
311C24C4×D414yes—
312C24C22×C812yes—
313C24C22×C812yes—
314C24C22×C812yes—
315C24C2×M1612yes—
316C24C2×M1612yes—
317C24C2×M1612yes—
318C24C2×M1612yes—
319C24C2×M1612yes—
320C24#13114yes—
321C24C23×C411yescharacteristic
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—216no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—216no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—432no—
393C22×C4—432no—
394C22×C4—464no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4C2×C22⋊C414yes—
414C22×C4C2×C22⋊C412yes—
415C22×C4C2×C22⋊C414yes—
416C22×C4C2×C4⋊C412yes—
417C22×C4C2×C4⋊C412yes—
418C22×C4C2×C4⋊C412yes—
419C22×C4#11712yes—
420C22×C4C4×D414yes—
421C22×C4#12014yes—
422C22×C4#12118yes—
423C22×C4C22⋊Q818yes—
424C22×C4C22⋊Q818yes—
425C22×C4#12314yes—
426C22×C4C22×C814yes—
427C22×C4C22×C814yes—
428C22×C4C2×M1614yes—
429C22×C4C2×M1614yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×Q811yescharacteristic
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)14yes—
437C2×C8—28no—
438C2×C8—28no—
439C2×C8—28no—
440C2×C8—432no—
441C2×C8—44no—
442C2×C8—44no—
443C2×C8—44no—
444C2×C8—44no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—44no—
452C2×C8—44no—
453C2×C8—44no—
454C2×C8—44no—
455C22⋊C4—216no—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—216no—
460C22⋊C4C22⋊C8116yescomplemented
461C42—28no—
462C42—28no—
463C42—48no—
464C42—48no—
465C42—44no—
466C42—44no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—232no—
502C23—22no—
503C23—232no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—416no—
518C23—416no—
519C23—432no—
520C23—432no—
521C23—416no—
522C23—432no—
523C23—432no—
524C23—416no—
525C23—416no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23#38014yes—
535C23#38514yes—
536C23C2×C22⋊C814yes—
537C23#40714yes—
538C23C2×C4⋊C814yes—
539C23C2×C4⋊C814yes—
540C23C2×C4⋊C814yes—
541C23#42314yes—
542C23#42414yes—
543C23#43114yes—
544C23#43214yes—
545C23C8×D418yes—
546C23#43518yes—
547C23C22×C22⋊C411yescharacteristic
548C23C22×C4⋊C411yescharacteristic
549C23C2×#11711yescharacteristic
550C23C2×C4×D412yes—
551C23#51812yes—
552C23C23×C811yescharacteristic
553C23C22×M1611yescharacteristic
554C23C22×M1611yescharacteristic
555C23C22×M1611yescharacteristic
556C23C2×#13112yes—
557C23#56812yes—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—216no—
577C2×C4—216no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—416no—
591C2×C4—432no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—432no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4#38014yes—
613C2×C4#38514yes—
614C2×C4C2×C22⋊C818yescomplemented
615C2×C4C2×#12011yescharacteristic
616C2×C4C2×#12112yes—
617C2×C4C2×C22⋊Q812yes—
618C2×C4C2×C22⋊Q812yes—
619C2×C4C2×#12311yescharacteristic
620C2×C4#53412yes—
621C2×C4#54312yes—
622C2×C4D4214yes—
623C2×C4#54614yes—
624C2×C4#54614yes—
625C2×C4#55114yes—
626C8—48no—
627C8—82no—
628C8—82no—
629C22—28no—
630C22—28no—
631C22—28no—
632C22—28no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—28no—
639C22—28no—
640C22—28no—
641C22—28no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—464no—
653C22—44no—
654C22—44no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22#179718yescomplemented
662C22#185018yescomplemented
663C22C2×#38011yescharacteristic
664C22C2×#38511yescharacteristic
665C22#231511yescharacteristic
666C22D4×C22⋊C412yes—
667C22#234212yes—
668C22C22×C22⋊C811yescharacteristic
669C22C2×#40711yescharacteristic
670C22#288012yes—
671C22C22×C4⋊C811yescharacteristic
672C22C2×#42311yescharacteristic
673C22C2×#42411yescharacteristic
674C22#290511yescharacteristic
675C22C2×#43111yescharacteristic
676C22C2×#43211yescharacteristic
677C22#292412yes—
678C22C2×C8×D412yes—
679C22C2×#43512yes—
680C22D4×M1612yes—
681C22D4×M1612yes—
682C22#297212yes—
683C22#297312yes—
684C22#297412yes—
685C22#297412yes—
686C4—24no—
687C4—24no—
688C4—28no—
689C4—42no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4#241012yes—
694C4#241212yes—
695C4#243714yes—
696C2—28no—
697C2—28no—
698C2—44no—
699C2#1387711yescharacteristic
700C2C2×#179711yescharacteristic
701C2#2062811yescharacteristic
702C2C2×#185011yescharacteristic
703C2#2083711yescharacteristic
704C2#2193011yescharacteristic
705C2#2193211yescharacteristic
706C2D4×C22⋊C812yes—
707C2#2239912yes—
708C2#2255212yes—
709C2#2255412yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1850C2semidirect
4. C2×#1797C2semidirect
6. #11042C2semidirect
7. #10714C2semidirect
8. #10712C2semidirect
14. C22×C4⋊C8C22semidirect
34. C2×#336C22semidirect
35. C2×#336C22semidirect
36. C2×#336C22semidirect
66. C22×C22⋊C4C8semidirect
177. C23×C4C2×C8semidirect
235. C2×C22⋊C4C2×C8semidirect
305. C24C4⋊C8semidirect
460. C22⋊C4C22⋊C8semidirect
614. C2×C4C2×C22⋊C8semidirect
661. C22#1797semidirect
662. C22#1850semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C82
C22×C4×C22⋊C42
C2×#18502
C2×#17972
C2×#17792
#110422
#107142
#107122