Finite Group Databasebeta

← #364654

Subgroups — #364654

Order 512.

1900 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic
3C2×#1892C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11190C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#483—24no—
18C2×#483—24no—
19C2×#483C418yescomplemented
20C22×#106C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24#2360—24no—
25#2360—24no—
26C2×#391C2211yescharacteristic
27C2×#389—21no—
28C2×#389C2211yescharacteristic
29C2×#387—21no—
30C2×#385—21no—
31C2×#381—21no—
32C2×#381—21no—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4—21no—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1892—24no—
38C2×#340—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1445—21no—
47#1445—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C23×Q8C2311yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55C23×C8—24no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—42no—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C414yes—
65C2×C22⋊Q8D414yes—
66C2×C22⋊Q8D414yes—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—24no—
70C2×C4×Q8—42no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4C2×C414yes—
85C22×C4⋊C4D414yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4D411yescharacteristic
94C22×C22⋊C4D411yescharacteristic
95C22×C42—21no—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#483—232no—
100C2×#106—24no—
101C2×#106—24no—
102C2×#106C2×C418yescomplemented
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—42no—
108C2×#103C2×C418yescomplemented
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8C2×C414yes—
112#391—22no—
113#391—22no—
114#389—28no—
115#389—22no—
116#389—22no—
117#387—28no—
118#385—28no—
119#381—44no—
120#381—44no—
121#380—22no—
122#380—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C22⋊C4—28no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95D411yescharacteristic
138C2×#95D411yescharacteristic
139#340—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—42no—
144#336—24no—
145#336—42no—
146C25C2×D411yescharacteristic
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8C22×C412yes—
155C22×Q8C2×D412yes—
156C22×Q8C2×D412yes—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—44no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—82no—
179C23×C4C22×C412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4#4112yes—
189C23×C4#4112yes—
190C23×C4C2411yescharacteristic
191C22×C8—28no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—48no—
205C22×C8—48no—
206C22×C8—44no—
207C22×C8—44no—
208C22×C8C22×C414yes—
209C22×C8#4114yes—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C22⋊Q8—28no—
213C22⋊Q8C22⋊C4116yes—
214C4×Q8—48no—
215C4×Q8—44no—
216C4×Q8—44no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4C22×C414yes—
256C2×C4⋊C4C22×C412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4#4114yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4SD1614yes—
274C2×C42—24no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—44no—
285C2×C42—82no—
286#106—216no—
287#103—216no—
288#103—48no—
289C22⋊C8—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—44no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24C4×D412yes—
306C24#12012yes—
307C24#12112yes—
308C24C2×SD1612yes—
309C24#13512yes—
310C24AGL(1,ℤ/8)12yes—
311C24#13712yes—
312C24C22×D411yescharacteristic
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—42no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8C2×C22⋊C414yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—48no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—84no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4C2×C22⋊C414yes—
412C22×C4C4×D414yes—
413C22×C4C4×D412yes—
414C22×C4#12012yes—
415C22×C4#12114yes—
416C22×C4#12112yes—
417C22×C4#12314yes—
418C22×C4C2×SD1612yes—
419C22×C4C2×SD1612yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4111yescharacteristic
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+412yes—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—416no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—216no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—84no—
466C4⋊C4C2×C22⋊C414yes—
467C4⋊C4#12314yes—
468C22⋊C4—216no—
469C22⋊C4—28no—
470C22⋊C4—44no—
471C22⋊C4—84no—
472C42—48no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—42no—
478C42—42no—
479C42—84no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23#38614yes—
497C23C4×SD1614yes—
498C23#44014yes—
499C23#44814yes—
500C23#44914yes—
501C23#46014yes—
502C23#46414yes—
503C23C2×C4×D411yescharacteristic
504C23C2×#12011yescharacteristic
505C23C2×#12111yescharacteristic
506C23#54612yes—
507C23#54712yes—
508C23C22×SD1611yescharacteristic
509C23C2×#13511yescharacteristic
510C23C2×AGL(1,ℤ/8)11yescharacteristic
511C23C2×#13711yescharacteristic
512C23#57712yes—
513C23#57812yes—
514Q8—28no—
515Q8—42no—
516Q8—44no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—816no—
572C2×C4—816no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4#38614yes—
587C2×C4C22×C22⋊C411yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C4×#4112yes—
590C2×C4#52012yes—
591C2×C4C2×#12111yescharacteristic
592C2×C4C2×#12311yescharacteristic
593C2×C4C2×#12311yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4#52912yes—
596C2×C4#53411yescharacteristic
597C2×C4#53611yescharacteristic
598C2×C4#53912yes—
599C2×C4#54612yes—
600C2×C4#54612yes—
601C2×C4#54712yes—
602C2×C4#54712yes—
603C2×C4#55112yes—
604C2×C4#55212yes—
605C8—44no—
606C8—44no—
607C22—24no—
608C22—24no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—24no—
613C22—28no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22#188018yes—
618C22C2×#38611yescharacteristic
619C22#235912yes—
620C22#240511yescharacteristic
621C22#240611yescharacteristic
622C22#242512yes—
623C22C2×C4×SD1611yescharacteristic
624C22C2×#44011yescharacteristic
625C22#294712yes—
626C22C2×#44811yescharacteristic
627C22C2×#44911yescharacteristic
628C22#300911yescharacteristic
629C22#301111yescharacteristic
630C22C2×#46011yescharacteristic
631C22C2×#46411yescharacteristic
632C22#304612yes—
633C22#329612yes—
634C22#329812yes—
635C22#330612yes—
636C22#331212yes—
637C22#333512yes—
638C22#333612yes—
639C22#334312yes—
640C22#334412yes—
641C4—24no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—42no—
647C4—81no—
648C4—84no—
649C4—81no—
650C4—82no—
651C4—162no—
652C4C4×#12311yescharacteristic
653C4#232211yescharacteristic
654C4#233912yes—
655C4#241912yes—
656C4#248611yescharacteristic
657C4#253411yescharacteristic
658C2—28no—
659C2#1396811yescharacteristic
660C2C2×#188011yescharacteristic
661C2#2093111yescharacteristic
662C2#2292012yes—
663C2#2293412yes—
664C2#2403811yescharacteristic
665C2#2404111yescharacteristic
666C2#2409411yescharacteristic
667C2#2409711yescharacteristic
668C2#2520412yes—
669C2#2520812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11220C2semidirect
7. #11190C2semidirect
8. #10908C2semidirect
9. #10789C2semidirect
19. C2×#483C4semidirect
102. C2×#106C2×C4semidirect
108. C2×#103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C2×#23602
C2×#18922
C22×#3402
#139042
#112202
#111902
#109082
#107892
#107302
#107142