Finite Group Databasebeta

← #412046

Subgroups — #412046

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2042C212yesmaximal
3C2×#1981C211yesmaximal, characteristic, complemented
4C2×#1956C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6C2×C4×#102C211yesmaximal, characteristic
7#14151C211yesmaximal, characteristic, complemented
8#11190C212yesmaximal, complemented
9#11067C212yesmaximal, complemented
10#11009C212yesmaximal
11#10834C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2547—24no—
25#2547—24no—
26C2×#397—22no—
27C2×#391C2212yes—
28C2×#389—21no—
29C2×#389C2212yes—
30C2×#384C2211yescharacteristic
31C2×#380—21no—
32C2×#380—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#2042—28no—
36#1981—24no—
37#1956—24no—
38#1802—24no—
39C4×#102—24no—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1441—22no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C23×Q8C2312yes—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8D414yes—
60C2×#124—24no—
61C2×#124—24no—
62C2×#124—24no—
63C2×#124—24no—
64C2×#124—42no—
65C2×#124—42no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C42—21no—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102—42no—
112C2×#102D414yes—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116#397—216no—
117#391—24no—
118#391—24no—
119#389—24no—
120#389—42no—
121#389—44no—
122#384—22no—
123#384D414yes—
124#380—28no—
125#380—28no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95D412yes—
137#340—24no—
138#340—22no—
139#340—22no—
140#336—24no—
141#336—42no—
142C25—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8C2×D414yes—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4C2×D412yes—
169C22×D4C2×D412yes—
170C22×D4#4112yes—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—24no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8C2×D412yes—
208C22×C8#4112yes—
209C4⋊Q8—28no—
210C4⋊Q8—28no—
211C4⋊Q8—44no—
212C4⋊Q8—44no—
213#124—48no—
214#124—48no—
215#124—48no—
216#124—48no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—44no—
220C4×D4—44no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—81no—
260C2×C4⋊C4—81no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4#4112yes—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42#4112yes—
288C2×C42#4112yes—
289C2×C42#4112yes—
290C4⋊C8—44no—
291#103—48no—
292#103—48no—
293#103—48no—
294#103—48no—
295#102—28no—
296#102—28no—
297#102—48no—
298C4×C8—44no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—82no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24C22×D411yescharacteristic
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4#12114yes—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4#12012yes—
439C22×C4#12012yes—
440C22×C4#12112yes—
441C22×C4#12112yes—
442C22×C4#12412yes—
443C22×C4#12412yes—
444C22×C4#13512yes—
445C22×C4#13512yes—
446C22×C4AGL(1,ℤ/8)12yes—
447C22×C4#13712yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4C2×#4111yescharacteristic
457C22×C4C2×#4111yescharacteristic
458C22×C4H(5,𝔽2)11yescharacteristic
459C22×C4H(5,𝔽2)11yescharacteristic
460C22×C4H(5,𝔽2)11yescharacteristic
461C22×C42−1+411yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—42no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—84no—
505C4⋊C4—88no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—84no—
514C42#12114yes—
515C42#12114yes—
516C42#12314yes—
517C42#12314yes—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—81no—
538C23—81no—
539C23—81no—
540C23—81no—
541C23—81no—
542C23—81no—
543C23—81no—
544C23—81no—
545C23—81no—
546C23—81no—
547C23—81no—
548C23—81no—
549C23—81no—
550C23—81no—
551C23—81no—
552C23—81no—
553C23C2×#12011yescharacteristic
554C23C2×#12111yescharacteristic
555C23C2×#12411yescharacteristic
556C23D4211yescharacteristic
557C23#54611yescharacteristic
558C23#54811yescharacteristic
559C23#55011yescharacteristic
560C23C2×#13511yescharacteristic
561C23C2×#13511yescharacteristic
562C23C2×AGL(1,ℤ/8)11yescharacteristic
563C23C2×#13711yescharacteristic
564C23#57711yescharacteristic
565C23#57711yescharacteristic
566C23#57811yescharacteristic
567C23#57811yescharacteristic
568Q8—44no—
569Q8—44no—
570Q8—44no—
571Q8—44no—
572Q8—82no—
573Q8—82no—
574Q8—82no—
575Q8—82no—
576D4—24no—
577D4—24no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—81no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—816no—
640C2×C4#39414yes—
641C2×C4#39414yes—
642C2×C4#44914yes—
643C2×C4#45214yes—
644C2×C4#46314yes—
645C2×C4#46414yes—
646C2×C4#48814yes—
647C2×C4#48914yes—
648C2×C4C2×#12111yescharacteristic
649C2×C4C2×#12111yescharacteristic
650C2×C4C2×#12111yescharacteristic
651C2×C4C2×#12311yescharacteristic
652C2×C4C2×#12311yescharacteristic
653C2×C4#52511yescharacteristic
654C2×C4#52911yescharacteristic
655C2×C4#52911yescharacteristic
656C2×C4#53211yescharacteristic
657C2×C4#53211yescharacteristic
658C2×C4#53511yescharacteristic
659C2×C4#53711yescharacteristic
660C2×C4#54011yescharacteristic
661C2×C4#54011yescharacteristic
662C2×C4#54211yescharacteristic
663C2×C4#54211yescharacteristic
664C2×C4D4211yescharacteristic
665C2×C4D4211yescharacteristic
666C2×C4#54611yescharacteristic
667C2×C4#54611yescharacteristic
668C2×C4#54611yescharacteristic
669C2×C4#54711yescharacteristic
670C2×C4#54711yescharacteristic
671C2×C4#54811yescharacteristic
672C2×C4#55011yescharacteristic
673C2×C4#55311yescharacteristic
674C2×C4#55311yescharacteristic
675C2×C4#55511yescharacteristic
676C8—42no—
677C8—42no—
678C8—82no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22C2×#39411yescharacteristic
695C22#240711yescharacteristic
696C22#240811yescharacteristic
697C22#242611yescharacteristic
698C22#242811yescharacteristic
699C22#249011yescharacteristic
700C22#249111yescharacteristic
701C22C2×#44911yescharacteristic
702C22C2×#45211yescharacteristic
703C22#301311yescharacteristic
704C22#301311yescharacteristic
705C22C2×#46311yescharacteristic
706C22C2×#46411yescharacteristic
707C22#304211yescharacteristic
708C22#304411yescharacteristic
709C22C2×#48811yescharacteristic
710C22C2×#48911yescharacteristic
711C22#313711yescharacteristic
712C22#313711yescharacteristic
713C22#327211yescharacteristic
714C22#327611yescharacteristic
715C22#328311yescharacteristic
716C22#328411yescharacteristic
717C22#330211yescharacteristic
718C22#330311yescharacteristic
719C22#333011yescharacteristic
720C22#333111yescharacteristic
721C22#336011yescharacteristic
722C22#336411yescharacteristic
723C22#336711yescharacteristic
724C22#336911yescharacteristic
725C22#339311yescharacteristic
726C22#339411yescharacteristic
727C22#339711yescharacteristic
728C22#339811yescharacteristic
729C4—24no—
730C4—84no—
731C4—84no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4#204218yescomplemented
738C4#241411yescharacteristic
739C4#241611yescharacteristic
740C4#242011yescharacteristic
741C4#242211yescharacteristic
742C4#242711yescharacteristic
743C4#242811yescharacteristic
744C4#262411yescharacteristic
745C4#285411yescharacteristic
746C2—84no—
747C2#1512611yescharacteristic
748C2C2×#204211yescharacteristic
749C2#2351211yescharacteristic
750C2#2418511yescharacteristic
751C2#2418911yescharacteristic
752C2#2423711yescharacteristic
753C2#2424111yescharacteristic
754C2#2527311yescharacteristic
755C2#2527511yescharacteristic
756C2#2536511yescharacteristic
757C2#2536711yescharacteristic
758C2#2774811yescharacteristic
759C2#2775611yescharacteristic
760C2#2776811yescharacteristic
761C2#2777611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1981C2semidirect
4. C2×#1956C2semidirect
7. #14151C2semidirect
8. #11190C2semidirect
9. #11067C2semidirect
11. #10834C2semidirect
737. C4#2042semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#20422
C2×#19812
C2×#19562
C2×#18022
C2×C4×#1022
#141512
#111902
#110672
#110092
#108342