Finite Group Databasebeta

← #361683

Subgroups — #361683

Order 512.

4190 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#550C211yesmaximal, characteristic
2C2×#1982C211yesmaximal, characteristic, complemented
3C2×#1957C211yesmaximal, characteristic, complemented
4C2×#1954C211yesmaximal, characteristic, complemented
5C2×#1803C211yesmaximal, characteristic, complemented
6C2×#1788C214yesmaximal, complemented
7C2×C4×#103C211yesmaximal, characteristic, complemented
8#13906C211yesmaximal, characteristic, complemented
9#10969C214yesmaximal, complemented
10C23×#41C2212yes—
11C2×#550—24no—
12C2×#550—24no—
13C2×#550—24no—
14C2×#550—24no—
15C2×#550—24no—
16C2×#550C418yescomplemented
17C22×#127—21no—
18C22×#127C2211yescharacteristic
19C22×#121—22no—
20C22×#121C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4—21no—
23C22×C4×D4C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×#102C2214yes—
28C22×#102C2214yes—
29C22×C22⋊C8C2214yes—
30C22×C4×C8C2211yescharacteristic
31C2×#390—21no—
32C2×#384C2211yescharacteristic
33C2×#380C2214yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1982—24no—
36#1957—24no—
37#1954—24no—
38#1803—24no—
39#1788—216no—
40C4×#103—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1439—22no—
44C4⋊#95—21no—
45C22×#41—24no—
46C22×#41—24no—
47C22×#41—24no—
48C22×#41—24no—
49C22×#41—24no—
50C22×#41—28no—
51C22×#41—28no—
52C22×#41—44no—
53C22×#41—44no—
54C22×#41C2×C418yes—
55C23×Q8C2311yescharacteristic
56C23×D4—22no—
57C23×D4—22no—
58C23×D4—21no—
59C23×D4—21no—
60C23×D4D411yescharacteristic
61C23×D4C2312yes—
62C24×C4—22no—
63C24×C4C2312yes—
64C23×C8C2312yes—
65C23×C8C2312yes—
66#550—232no—
67#550—232no—
68#550—232no—
69#550—232no—
70C2×#127—28no—
71C2×#127—28no—
72C2×#127—28no—
73C2×#127—22no—
74C2×#127—22no—
75C2×#127—22no—
76C2×#127C2×C414yes—
77C2×#127D414yes—
78C2×#127D414yes—
79C2×#121—216no—
80C2×#121—216no—
81C2×#121—24no—
82C2×#121—24no—
83C2×#121—24no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121—216no—
87C2×#121C2×C418yes—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8C2×C414yes—
92C2×C4×Q8D414yes—
93C2×C4×Q8D414yes—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4C2×C414yes—
102C2×C4×D4D414yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—22no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4D412yes—
113C22×C42—21no—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×#103—24no—
120C2×#103—24no—
121C2×#103—24no—
122C2×#103—28no—
123C2×#103D418yescomplemented
124C2×#102—28no—
125C2×#102—28no—
126C2×#102—28no—
127C2×#102—28no—
128C2×#102—44no—
129C2×#102—44no—
130C2×C22⋊C8—28no—
131C2×C22⋊C8—28no—
132C2×C22⋊C8—44no—
133C2×C4×C8—22no—
134C2×C4×C8—22no—
135C2×C4×C8—22no—
136#390—44no—
137#384—22no—
138#384—22no—
139#380—28no—
140#380—44no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C2×#95—24no—
144C2×#95—21no—
145#340—24no—
146#340—24no—
147C25—22no—
148C25C2×D412yes—
149C2×#41—216no—
150C2×#41—216no—
151C2×#41—216no—
152C2×#41—216no—
153C2×#41—232no—
154C2×#41—416no—
155C2×#41—416no—
156C2×#41—416no—
157C2×#41—416no—
158C2×#41—416no—
159C2×#41—416no—
160C2×#41—416no—
161C2×#41—416no—
162C2×#41—48no—
163C2×#41—48no—
164C2×#41—416no—
165C2×#41—48no—
166C2×#41—48no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8C22×C412yes—
173C22×Q8C2×D412yes—
174C22×Q8C2×D412yes—
175C22×Q8C2×D414yes—
176C22×Q8#4114yes—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—48no—
203C22×D4—48no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—82no—
209C22×D4—82no—
210C22×D4—84no—
211C22×D4—84no—
212C22×D4C22⋊C414yes—
213C22×D4C22×C414yes—
214C22×D4C2×D414yes—
215C22×D4C2×D414yes—
216C23×C4—21no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—21no—
220C23×C4—28no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—48no—
232C23×C4—48no—
233C23×C4C22×C414yes—
234C23×C4C2×D412yes—
235C23×C4C2×D414yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—24no—
248C22×C8—22no—
249C22×C8—24no—
250C22×C8C2×D414yes—
251#127—264no—
252#127—28no—
253#127—28no—
254#127—28no—
255#127C22⋊C4116yescomplemented
256#121—216no—
257#121—216no—
258#121—2128no—
259#121—216no—
260#121—216no—
261C4×Q8—28no—
262C4×Q8—28no—
263C4×Q8—28no—
264C4×Q8C22⋊C4116yescomplemented
265C4×D4—264no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4C22⋊C414yes—
293C2×C4⋊C4C22×C412yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—84no—
305C2×C22⋊C4C22⋊C418yes—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42C22×C412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C4⋊C8—44no—
319#103—216no—
320#103—48no—
321#102—416no—
322#102—416no—
323C22⋊C8—416no—
324C4×C8—44no—
325#95—48no—
326#95—42no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—44no—
341C24—44no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—44no—
347C24—44no—
348C24C2×C22⋊C414yes—
349C24C22×D411yescharacteristic
350#41—464no—
351#41—432no—
352#41—464no—
353#41—432no—
354#41—432no—
355#41—432no—
356C2×Q8—22no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—24no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8C2×C22⋊C414yes—
370C2×Q8C4×D418yes—
371C2×Q8#12118yes—
372C2×Q8#12414yes—
373C2×Q8#12714yes—
374C2×D4—232no—
375C2×D4—232no—
376C2×D4—24no—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—216no—
387C2×D4—216no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—432no—
391C2×D4—432no—
392C2×D4—416no—
393C2×D4—416no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—88no—
413C2×D4—84no—
414C2×D4—84no—
415C2×D4—88no—
416C2×D4—816no—
417C2×D4—88no—
418C2×D4—88no—
419C2×D4—816no—
420C2×D4C2×C22⋊C418yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—232no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—432no—
446C22×C4—432no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4C2×C22⋊C414yes—
473C22×C4C2×C22⋊C412yes—
474C22×C4C2×C22⋊C412yes—
475C22×C4#12012yes—
476C22×C4#12012yes—
477C22×C4#12012yes—
478C22×C4#12012yes—
479C22×C4#13512yes—
480C22×C4#13512yes—
481C22×C4AGL(1,ℤ/8)12yes—
482C22×C4#13712yes—
483C22×C4C23×C411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8—42no—
501C2×C8—42no—
502C2×C8—42no—
503C2×C8—44no—
504C2×C8—44no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—216no—
509C4⋊C4—28no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4C2×C22⋊C414yes—
520C22⋊C4—232no—
521C22⋊C4—216no—
522C22⋊C4—816no—
523C42—216no—
524C42—84no—
525C42C2×C22⋊C414yes—
526C42C2×C22⋊C414yes—
527C42#12014yes—
528C42#12014yes—
529C23—28no—
530C23—24no—
531C23—24no—
532C23—28no—
533C23—24no—
534C23—28no—
535C23—44no—
536C23—44no—
537C23—48no—
538C23—44no—
539C23—44no—
540C23—48no—
541C23—48no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—48no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23C22×C22⋊C411yescharacteristic
576C23C2×#12011yescharacteristic
577C23C2×#12011yescharacteristic
578C23D4212yes—
579C23#54612yes—
580C23C2×#13511yescharacteristic
581C23C2×#13511yescharacteristic
582C23C2×AGL(1,ℤ/8)11yescharacteristic
583C23C2×#13711yescharacteristic
584C23#57612yes—
585C23#57712yes—
586Q8—24no—
587Q8—28no—
588Q8—28no—
589Q8—24no—
590Q8—24no—
591Q8#39018yescomplemented
592D4—216no—
593D4—464no—
594D4—44no—
595D4—48no—
596D4—44no—
597D4—44no—
598D4—44no—
599D4—416no—
600D4—432no—
601D4—816no—
602D4—832no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—464no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4#37914yes—
642C2×C4#37914yes—
643C2×C4#41614yes—
644C2×C4#41714yes—
645C2×C4#44914yes—
646C2×C4#44914yes—
647C2×C4#45214yes—
648C2×C4#45214yes—
649C2×C4C22×C22⋊C411yescharacteristic
650C2×C4C22×C22⋊C411yescharacteristic
651C2×C4C2×C4×D412yes—
652C2×C4#51812yes—
653C2×C4C2×#12011yescharacteristic
654C2×C4C2×#12011yescharacteristic
655C2×C4C2×#12112yes—
656C2×C4C2×#12411yescharacteristic
657C2×C4C2×#12711yescharacteristic
658C2×C4#53512yes—
659C2×C4#53511yescharacteristic
660C2×C4#53511yescharacteristic
661C2×C4D4212yes—
662C2×C4D4212yes—
663C2×C4D4212yes—
664C2×C4#54612yes—
665C2×C4#54612yes—
666C2×C4#54612yes—
667C8—44no—
668C8—82no—
669C22—48no—
670C22—44no—
671C22—44no—
672C22—48no—
673C22—44no—
674C22—48no—
675C22—48no—
676C22—48no—
677C22—82no—
678C22—84no—
679C22—82no—
680C22—84no—
681C22—88no—
682C22—82no—
683C22—84no—
684C22—82no—
685C22—84no—
686C22—88no—
687C22—84no—
688C22—88no—
689C22—84no—
690C22—88no—
691C22C2×#37911yescharacteristic
692C22D4×C22⋊C412yes—
693C22#240812yes—
694C22#243311yescharacteristic
695C22#243511yescharacteristic
696C22C2×#41611yescharacteristic
697C22C2×#41711yescharacteristic
698C22#289712yes—
699C22C2×#44911yescharacteristic
700C22C2×#44911yescharacteristic
701C22C2×#45211yescharacteristic
702C22C2×#45211yescharacteristic
703C22#300811yescharacteristic
704C22#300911yescharacteristic
705C22#301212yes—
706C22#327412yes—
707C22#327812yes—
708C22#328412yes—
709C22#328712yes—
710C22#330412yes—
711C22#330512yes—
712C22#333212yes—
713C22#333312yes—
714C4—28no—
715C4—24no—
716C4—44no—
717C4—44no—
718C4—84no—
719C4—162no—
720C4#178818yescomplemented
721C4C2×#39011yescharacteristic
722C4#230911yescharacteristic
723C4D4×C22⋊C412yes—
724C4#240812yes—
725C4#243311yescharacteristic
726C4#243511yescharacteristic
727C2—84no—
728C2—88no—
729C2#1390211yescharacteristic
730C2C2×#178811yescharacteristic
731C2#2060211yescharacteristic
732C2#2241812yes—
733C2#2243212yes—
734C2#2423812yes—
735C2#2424212yes—
736C2#2575511yescharacteristic
737C2#2575911yescharacteristic
738C2#2588311yescharacteristic
739C2#2588711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1982C2semidirect
3. C2×#1957C2semidirect
4. C2×#1954C2semidirect
5. C2×#1803C2semidirect
6. C2×#1788C2semidirect
7. C2×C4×#103C2semidirect
8. #13906C2semidirect
9. #10969C2semidirect
16. C2×#550C4semidirect
123. C2×#103D4semidirect
255. #127C22⋊C4semidirect
264. C4×Q8C22⋊C4semidirect
591. Q8#390semidirect
720. C4#1788semidirect

Maximal subgroups

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

subgroupindex
C22×#5502
C2×#19822
C2×#19572
C2×#19542
C2×#18032
C2×#17882
C2×C4×#1032
#139062
#109692