Finite Group Databasebeta

← #195868

Subgroups — #195868

Order 512.

1948 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic, complemented
4C2×#1486C212yesmaximal, complemented
5#9076C214yesmaximal, complemented
6#9072C214yesmaximal, complemented
7#7901C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431—21no—
13C2×#431—21no—
14C2×#431—28no—
15C2×#431C2212yes—
16C2×#431C2212yescomplemented
17C2×#423C2212yes—
18C22×C4⋊C8—21no—
19C2×#407—21no—
20C22×C22⋊C8—21no—
21C22×C4×C8C2212yescomplemented
22C4×#121—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121C412yes—
27C4×#121C412yes—
28C4×#121C2212yes—
29C4×#121C2212yes—
30C2×#390C411yescharacteristic
31C2×#389C411yescharacteristic
32C2×#388—21no—
33C2×#386—21no—
34C2×#380C411yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37#1797—21no—
38#1797—21no—
39#1797C2212yescomplemented
40#1797C2212yescomplemented
41#1728—21no—
42#1486—28no—
43C2×D4⋊C8C2218yescomplemented
44#1279C2218yescomplemented
45C23×D4—21no—
46C23×D4C2×C411yescharacteristic
47C24×C4—21no—
48C24×C4C2×C411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C22×M16—22no—
52C23×C8—22no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—28no—
56C2×#121C2×C412yes—
57C2×#121C2×C412yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—44no—
67C2×C4×D4—44no—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2312yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42—24no—
80C22×C42—24no—
81C22×C42—21no—
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#431—28no—
95#431—28no—
96#431—24no—
97#431—28no—
98#431D418yescomplemented
99#423—24no—
100#423—24no—
101C2×C4⋊C8—28no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8C2314yes—
106#407—44no—
107C2×C22⋊C8—28no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—28no—
114C2×C4×C8C2314yes—
115#390—21no—
116#390—21no—
117#390C2×C412yes—
118#390C2×C412yes—
119#389—21no—
120#389—21no—
121#389C2×C412yes—
122#389C2×C412yes—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#380—21no—
132#380—21no—
133#380C2×C412yes—
134#380C2×C412yes—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×C412yes—
138C4×C4⋊C4C2312yes—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146#336—22no—
147#336—22no—
148D4⋊C8—216no—
149C25—21no—
150C25C22⋊C411yescharacteristic
151C22×D4—28no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—24no—
155C22×D4—21no—
156C22×D4—24no—
157C22×D4—42no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4C22×C412yes—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—44no—
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4D814yes—
189C23×C4SD1614yes—
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C2×M16—44no—
200C2×M16—44no—
201C2×M16—44no—
202C22×C8—28no—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8—44no—
206C22×C8—44no—
207#121—28no—
208#121—28no—
209#121—24no—
210#121—24no—
211#121—24no—
212#121M1618yescomplemented
213C4×D4—28no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—48no—
217C4×D4—48no—
218C4×D4—44no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—42no—
223C4×D4—44no—
224C4×D4—42no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D414yes—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42#4114yes—
290C2×C42C2411yescharacteristic
291C4⋊C8—28no—
292C4⋊C8—28no—
293C4⋊C8—24no—
294C4⋊C8—24no—
295C4⋊C8—44no—
296C22⋊C8—28no—
297C22⋊C8—28no—
298C22⋊C8—44no—
299C22⋊C8—44no—
300C4×C8—28no—
301C4×C8—24no—
302C4×C8C2×D418yescomplemented
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—24no—
314C24—22no—
315C24—41no—
316C24—41no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24#10214yes—
324C24C2×C22⋊C411yescharacteristic
325C24C2×C22⋊C411yescharacteristic
326C24C2×C22⋊C411yescharacteristic
327C2×D4—24no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—416no—
334C2×D4—42no—
335C2×D4—432no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—42no—
341C2×D4C2×M1614yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—216no—
361C22×C4—216no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
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—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—416no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4#10214yes—
414C22×C4#10214yes—
415C22×C4#10214yes—
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C4×D414yes—
426C22×C4#12012yes—
427C22×C4#12114yes—
428C22×C4#12312yes—
429C22×C4C2×M1614yes—
430C22×C4C2×D812yes—
431C22×C4C2×D812yes—
432C22×C4C2×D812yes—
433C22×C4C2×SD1612yes—
434C22×C4C2×SD1612yes—
435C22×C4C2×SD1612yes—
436C22×C4#13514yes—
437C22×C4C23×C411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440M16—88no—
441C2×C8—28no—
442C2×C8—416no—
443C2×C8—44no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4C2×M1614yes—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—22no—
473C42—22no—
474C42—21no—
475C42—21no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—42no—
483C42—42no—
484C42—44no—
485C42C4×D414yes—
486C42#12012yes—
487C42#12114yes—
488C42#12312yes—
489C42C22×D412yes—
490C42C2×#4112yes—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—28no—
505C23—28no—
506C23—21no—
507C23—24no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23#38612yes—
527C23#40712yes—
528C23C2×#10212yes—
529C23C2×#10212yes—
530C23C2×#10212yes—
531C23#41612yes—
532C23#42112yes—
533C23C22×C22⋊C411yescharacteristic
534D4—28no—
535D4—48no—
536D4—416no—
537D4—416no—
538D4—44no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—88no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4#38612yes—
597C2×C4#38612yes—
598C2×C4#38612yes—
599C2×C4#40712yes—
600C2×C4C2×#10212yes—
601C2×C4C2×#10212yes—
602C2×C4C2×#10212yes—
603C2×C4#41612yes—
604C2×C4#42112yes—
605C2×C4#46518yes—
606C2×C4#46618yes—
607C2×C4C2×C4×D412yes—
608C2×C4C2×#12011yescharacteristic
609C2×C4C2×#12112yes—
610C2×C4C2×#12311yescharacteristic
611C2×C4C22×M1612yes—
612C2×C4#56812yes—
613C2×C4C22×D811yescharacteristic
614C2×C4C22×SD1611yescharacteristic
615C2×C4C2×#13512yes—
616C8—48no—
617C8—84no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—416no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—816no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22#148614yescomplemented
647C22#189314yes—
648C22C2×#38611yescharacteristic
649C22C2×#40711yescharacteristic
650C22#287911yescharacteristic
651C22C22×#10211yescharacteristic
652C22C2×#41611yescharacteristic
653C22C2×#42111yescharacteristic
654C22C2×#42111yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—21no—
660C4—21no—
661C4—22no—
662C4—22no—
663C4—44no—
664C4—41no—
665C4—41no—
666C4—82no—
667C4—82no—
668C4#189314yes—
669C4D4×M1614yes—
670C4C2×#46512yes—
671C4C2×#46612yes—
672C2—24no—
673C2—24no—
674C2—42no—
675C2—84no—
676C2C2×#148611yescharacteristic
677C2#1744311yescharacteristic
678C2#1748511yescharacteristic
679C2#1748711yescharacteristic
680C2#2089811yescharacteristic
681C2C2×#189311yescharacteristic
682C2#2097911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#431C2semidirect
3. #20626C2semidirect
4. C2×#1486C2semidirect
5. #9076C2semidirect
6. #9072C2semidirect
7. #7901C2semidirect
16. C2×#431C22semidirect
21. C22×C4×C8C22semidirect
39. #1797C22semidirect
40. #1797C22semidirect
43. C2×D4⋊C8C22semidirect
44. #1279C22semidirect
98. #431D4semidirect
212. #121M16semidirect
302. C4×C8C2×D4semidirect
646. C22#1486semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
C2×C4×#1212
#206262
C2×#14862
#90762
#90722
#79012