Finite Group Databasebeta

← #405468

Subgroups — #405468

Order 512.

1912 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2417C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic, complemented
3C2×#1872C211yesmaximal, characteristic, complemented
4C2×#1869C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic, complemented
6#13872C211yesmaximal, characteristic
7#11293C212yesmaximal, complemented
8#11200C212yesmaximal, complemented
9#11190C212yesmaximal, complemented
10#10970C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2212yes—
22#2417—24no—
23#2417—24no—
24C2×#398—21no—
25C2×#396—21no—
26C2×#391C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#386—21no—
29C2×#386—21no—
30C2×#380—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34#1892—24no—
35#1872—24no—
36#1869—24no—
37C2×#340—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45#1441—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C23×Q8C2311yescharacteristic
50C23×D4D411yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—42no—
58C23×C8—42no—
59C23×C8D414yes—
60C23×C8C2312yes—
61C23×C8C2312yes—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—42no—
71C2×C22⋊Q8D414yes—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4D411yescharacteristic
91C22×C4⋊C4D414yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—22no—
102C22×C22⋊C4D411yescharacteristic
103C22×C22⋊C4D411yescharacteristic
104C22×C42C2311yescharacteristic
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—42no—
109C2×#103—42no—
110C2×#103—42no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—24no—
115C2×C22⋊C8—42no—
116C2×C22⋊C8—42no—
117#398—28no—
118#396—28no—
119#391—22no—
120#391—22no—
121#389—22no—
122#389—22no—
123#386—28no—
124#386—44no—
125#380—28no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380—22no—
130#380—24no—
131#380—42no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95—22no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—42no—
146#336—24no—
147C25C2×D411yescharacteristic
148C25C2×D411yescharacteristic
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8C2×D412yes—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4Q1614yes—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—82no—
195C23×C4—82no—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4#4112yes—
204C23×C4#4112yes—
205C23×C4#4112yes—
206C23×C4C2411yescharacteristic
207C22×C8—28no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—48no—
222C22×C8—44no—
223C22×C8—44no—
224C22×C8—48no—
225C22×C8C2×D414yes—
226C22⋊Q8—28no—
227C22⋊Q8—28no—
228C22⋊Q8—48no—
229C22⋊Q8—44no—
230C22⋊Q8—44no—
231C22⋊Q8—44no—
232C22⋊Q8—44no—
233C4×D4—44no—
234C4×D4—44no—
235C4×D4—44no—
236C4×D4—44no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4#4114yes—
282C2×C4⋊C4#4114yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—82no—
301C2×C22⋊C4—82no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308#103—48no—
309#103—48no—
310C22⋊C8—48no—
311C22⋊C8—44no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—82no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24#12012yes—
335C24#12112yes—
336C24#12312yes—
337C24C2×Q1612yes—
338C24C2×Q1612yes—
339C24#13512yes—
340C24AGL(1,ℤ/8)12yes—
341C24#13712yes—
342C24C22×D411yescharacteristic
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—48no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—48no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—48no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—84no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4#12012yes—
463C22×C4#12114yes—
464C22×C4#12114yes—
465C22×C4#12112yes—
466C22×C4#12314yes—
467C22×C4#12312yes—
468C22×C4#12414yes—
469C22×C4C2×Q1612yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4111yescharacteristic
475C22×C4C2×#4111yescharacteristic
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—416no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—84no—
518C4⋊C4#12314yes—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—84no—
524C22⋊C4—88no—
525C22⋊C4—84no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—22no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23#39414yes—
570C23#44914yes—
571C23#45114yes—
572C23#46714yes—
573C23#46814yes—
574C23#48414yes—
575C23#48514yes—
576C23C2×#12011yescharacteristic
577C23C2×#12111yescharacteristic
578C23C2×#12311yescharacteristic
579C23#54612yes—
580C23#54612yes—
581C23C22×Q1611yescharacteristic
582C23C2×#13511yescharacteristic
583C23C2×AGL(1,ℤ/8)11yescharacteristic
584C23C2×#13711yescharacteristic
585C23#57712yes—
586C23#57712yes—
587Q8—44no—
588Q8—44no—
589Q8—81no—
590Q8—81no—
591Q8—81no—
592Q8—81no—
593D4—416no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—816no—
639C2×C4—816no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—82no—
654C2×C4—81no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—82no—
660C2×C4—82no—
661C2×C4—82no—
662C2×C4—82no—
663C2×C4#39414yes—
664C2×C4C2×#12111yescharacteristic
665C2×C4C2×#12111yescharacteristic
666C2×C4C2×#12311yescharacteristic
667C2×C4C2×#12311yescharacteristic
668C2×C4#52511yescharacteristic
669C2×C4C2×#12411yescharacteristic
670C2×C4#52912yes—
671C2×C4#53212yes—
672C2×C4#53411yescharacteristic
673C2×C4#53412yes—
674C2×C4#53511yescharacteristic
675C2×C4#53812yes—
676C2×C4#54612yes—
677C2×C4#54612yes—
678C2×C4#55312yes—
679C2×C4#55312yes—
680C2×C4#55312yes—
681C2×C4#55312yes—
682C8—44no—
683C8—82no—
684C22—24no—
685C22—28no—
686C22—28no—
687C22—28no—
688C22—28no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—44no—
695C22—44no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—81no—
700C22—82no—
701C22—82no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22#203618yes—
706C22C2×#39411yescharacteristic
707C22#240411yescharacteristic
708C22#240511yescharacteristic
709C22#244412yes—
710C22#254312yes—
711C22C2×#45111yescharacteristic
712C22C2×#44911yescharacteristic
713C22#300911yescharacteristic
714C22#301011yescharacteristic
715C22C2×#46711yescharacteristic
716C22C2×#46811yescharacteristic
717C22#306012yes—
718C22C2×#48511yescharacteristic
719C22C2×#48411yescharacteristic
720C22#309912yes—
721C22#329612yes—
722C22#329612yes—
723C22#329912yes—
724C22#329912yes—
725C22#330612yes—
726C22#331012yes—
727C22#331612yes—
728C22#332012yes—
729C4—24no—
730C4—42no—
731C4—44no—
732C4—81no—
733C4—82no—
734C4—84no—
735C4—81no—
736C4—82no—
737C4—82no—
738C4—162no—
739C4#244412yes—
740C4#250411yescharacteristic
741C4#254312yes—
742C4#259111yescharacteristic
743C4#261711yescharacteristic
744C4#285411yescharacteristic
745C2—28no—
746C2—82no—
747C2#1491311yescharacteristic
748C2C2×#203611yescharacteristic
749C2#2348911yescharacteristic
750C2#2396611yescharacteristic
751C2#2397011yescharacteristic
752C2#2403811yescharacteristic
753C2#2404211yescharacteristic
754C2#2620412yes—
755C2#2620612yes—
756C2#2855312yes—
757C2#2856512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×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×#1892C2semidirect
3. C2×#1872C2semidirect
5. C22×#340C2semidirect
7. #11293C2semidirect
8. #11200C2semidirect
9. #11190C2semidirect
10. #10970C2semidirect
11. #10730C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24172
C2×#18922
C2×#18722
C2×#18692
C22×#3402
#138722
#112932
#112002
#111902
#109702
#107302