Finite Group Databasebeta

← #204980

Subgroups — #204980

Order 512.

1704 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3C2×#1520C212yesmaximal, complemented
4C2×#1519C212yesmaximal
5C4×#380C211yesmaximal, characteristic, complemented
6#9716C214yesmaximal, complemented
7#9713C214yesmaximal, complemented
8C22×#124C411yescharacteristic, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C22×C4⋊C8—21no—
13C22×C22⋊C8C2212yes—
14#2850—21no—
15#2850—21no—
16#2850—21no—
17#2850—21no—
18#2850C412yescomplemented
19#2850C412yescomplemented
20#2850C2212yes—
21#2850C2212yes—
22C2×#391C411yescharacteristic, complemented
23C2×#390C411yescharacteristic, complemented
24C2×#388—21no—
25C2×#388—21no—
26C2×#386—21no—
27C2×#386—21no—
28C2×#382C411yescharacteristic, complemented
29C2×#380C2212yes—
30C2×C4×C4⋊C4—21no—
31C2×C4×C22⋊C4—21no—
32#1797—21no—
33#1797—21no—
34#1797C2212yescomplemented
35#1797C2212yescomplemented
36C2×#336—21no—
37#1520—28no—
38#1519—28no—
39C2×#328C2218yescomplemented
40C2×#327C2218yes—
41C4⋊#95—21no—
42#1434C2212yescomplemented
43C22⋊#95—21no—
44C4×#95C2212yescomplemented
45C23×Q8C2×C411yescharacteristic
46C23×D4—21no—
47C23×D4C2×C411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C2×#124—21no—
53C2×#124—21no—
54C2×#124—44no—
55C2×#124C2×C412yes—
56C2×#124C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8C2×C412yes—
62C2×C22⋊Q8C2312yes—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—24no—
66C2×#121—24no—
67C2×#121C2×C412yes—
68C2×#121C2312yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C42—22no—
81C22×C42—44no—
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—44no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8C2314yes—
101#391—21no—
102#391—21no—
103#391C2×C412yes—
104#391C2×C412yes—
105#390—21no—
106#390—21no—
107#390C2×C412yes—
108#390C2×C412yes—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#382—21no—
126#382—21no—
127#382C2×C412yes—
128#382C2×C412yes—
129#380—24no—
130#380D418yes—
131C4×C4⋊C4—44no—
132C4×C22⋊C4—44no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95C2314yes—
141#336—22no—
142#336—22no—
143#328—216no—
144#327—216no—
145C25—21no—
146C25C2×D411yescharacteristic
147C22×Q8—21no—
148C22×Q8—24no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8C22×C412yes—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—24no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4C22×C412yes—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—24no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D414yes—
194C23×C4#4114yes—
195C23×C4C2411yescharacteristic
196C22×C8—24no—
197C22×C8—44no—
198C22×C8—44no—
199C22×C8—44no—
200#124—44no—
201#124—44no—
202#124—44no—
203#124—44no—
204C22⋊Q8—28no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—44no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210#121—28no—
211#121—24no—
212#121—24no—
213#121—44no—
214#121—42no—
215#121—42no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C4⋊C8—44no—
283C4⋊C8—44no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286C22⋊C8—44no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95C2×D418yes—
300C24—22no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24#12114yes—
315C24#13514yes—
316C24AGL(1,ℤ/8)12yes—
317C24#13712yes—
318C24C22×D411yescharacteristic
319C2×Q8—24no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—88no—
326C2×D4—24no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—88no—
340C2×D4—88no—
341C2×D4—88no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
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—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
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—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—48no—
399C22×C4—416no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
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×C4C2×C22⋊C411yescharacteristic
426C22×C4C4×D414yes—
427C22×C4C4×D414yes—
428C22×C4#12114yes—
429C22×C4#12412yes—
430C22×C4#12412yes—
431C22×C4#12712yes—
432C22×C4#12712yes—
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C2×#4112yes—
437C2×C8—44no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8—88no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—416no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—416no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—88no—
479C42#9914yes—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23#39012yes—
515C23#39012yes—
516C23#41612yes—
517C23#41612yes—
518C23#41712yes—
519C23#41712yes—
520C23#46318yes—
521C23#46418yes—
522C23C22×C22⋊C411yescharacteristic
523C23C2×#12112yes—
524C23C2×#13512yes—
525C23C2×AGL(1,ℤ/8)11yescharacteristic
526C23C2×#13711yescharacteristic
527Q8—84no—
528Q8—84no—
529D4—84no—
530D4—88no—
531D4—84no—
532D4—84no—
533D4—84no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
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—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4#39012yes—
603C2×C4#39012yes—
604C2×C4C2×#9912yes—
605C2×C4C2×#9912yes—
606C2×C4C2×#9912yes—
607C2×C4#41012yes—
608C2×C4#42112yes—
609C2×C4#42112yes—
610C2×C4C2×C4×D412yes—
611C2×C4C2×#12411yescharacteristic
612C2×C4C2×#12711yescharacteristic
613C8—84no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—21no—
621C22—22no—
622C22—21no—
623C22—22no—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—48no—
635C22—48no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—168no—
644C22#198314yes—
645C22#198314yes—
646C22C2×#39011yescharacteristic
647C22C22×#9911yescharacteristic
648C22C2×#41011yescharacteristic
649C22C2×#41611yescharacteristic
650C22C2×#41711yescharacteristic
651C22C2×#42111yescharacteristic
652C22C2×#42111yescharacteristic
653C22C2×#46312yes—
654C22C2×#46412yes—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—21no—
660C4—22no—
661C4—21no—
662C4—22no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—88no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4#196414yes—
672C2—22no—
673C2—24no—
674C2—24no—
675C2—22no—
676C2—162no—
677C2#1756711yescharacteristic
678C2#1757211yescharacteristic
679C2#1758111yescharacteristic
680C2#1759211yescharacteristic
681C2#2120911yescharacteristic
682C2C2×#196411yescharacteristic
683C2C2×#198311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1797C2semidirect
3. C2×#1520C2semidirect
5. C4×#380C2semidirect
6. #9716C2semidirect
7. #9713C2semidirect
8. C22×#124C4semidirect
18. #2850C4semidirect
19. #2850C4semidirect
22. C2×#391C4semidirect
23. C2×#390C4semidirect
28. C2×#382C4semidirect
34. #1797C22semidirect
35. #1797C22semidirect
39. C2×#328C22semidirect
42. #1434C22semidirect
44. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
C2×#17972
C2×#15202
C2×#15192
C4×#3802
#97162
#97132