Finite Group Databasebeta

← #416183

Subgroups — #416183

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2C2×#1954C211yesmaximal, characteristic
3C2×#1835C211yesmaximal, characteristic, complemented
4C2×#1805C211yesmaximal, characteristic, complemented
5C2×C4×#103C211yesmaximal, characteristic, complemented
6#13905C211yesmaximal, characteristic
7#11200C212yesmaximal, complemented
8#11199C212yesmaximal, complemented
9#10970C212yesmaximal, complemented
10#10970C212yesmaximal, complemented
11#10850C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2536—24no—
23#2536—24no—
24C2×#398—21no—
25C2×#394—21no—
26C2×#388—22no—
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1954—24no—
34#1835—24no—
35#1805—24no—
36C4×#103—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2212yes—
44#1445—21no—
45#1438—21no—
46#1437—22no—
47C23×Q8C2311yescharacteristic
48C23×D4D411yescharacteristic
49C24×C4C2312yes—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8—42no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8D414yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D412yes—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8—42no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8D414yes—
108#398—28no—
109#394—28no—
110#388—216no—
111#382—22no—
112#382—22no—
113#382—22no—
114#382D414yes—
115#380—24no—
116#380—24no—
117#380—24no—
118#380—24no—
119#380—42no—
120#380—42no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—21no—
131#340—22no—
132#340—22no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—42no—
137#340—42no—
138C25C2×D412yes—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8C2×D412yes—
151C22×Q8#4112yes—
152C22×Q8#4112yes—
153C22×D4—24no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—42no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205C22⋊Q8—48no—
206C22⋊Q8—48no—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—44no—
210C4×Q8—44no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—44no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4#4112yes—
253C2×C4⋊C4#4112yes—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—82no—
268C2×C22⋊C4—82no—
269C2×C42—24no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—82no—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279C2×C42#4112yes—
280C4⋊C8—44no—
281#103—28no—
282#103—28no—
283#103—48no—
284C22⋊C8—48no—
285C22⋊C8—48no—
286C4×C8—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—44no—
291#95—44no—
292#95—42no—
293#95—42no—
294#95—42no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24#13514yes—
310C24C22×D411yescharacteristic
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8#12414yes—
339C2×D4—28no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—44no—
343C2×D4—44no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
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—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4#12012yes—
420C22×C4#12012yes—
421C22×C4#12112yes—
422C22×C4#12112yes—
423C22×C4#12312yes—
424C22×C4#12312yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4H(5,𝔽2)11yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—416no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—42no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—42no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C42—48no—
489C42—84no—
490C42#12114yes—
491C42#12114yes—
492C42#12314yes—
493C42#12414yes—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23C2×#12011yescharacteristic
530C23C2×#12111yescharacteristic
531C23C2×#12311yescharacteristic
532C23#54611yescharacteristic
533C23#54611yescharacteristic
534C23#54611yescharacteristic
535C23#54611yescharacteristic
536C23C2×#13512yes—
537C23#57512yes—
538C23#57712yes—
539C23#57812yes—
540Q8—24no—
541Q8—24no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548D4—416no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4#39414yes—
609C2×C4#39414yes—
610C2×C4C2×#12111yescharacteristic
611C2×C4C2×#12111yescharacteristic
612C2×C4C2×#12311yescharacteristic
613C2×C4#52511yescharacteristic
614C2×C4#52511yescharacteristic
615C2×C4C2×#12411yescharacteristic
616C2×C4C2×#12411yescharacteristic
617C2×C4#52911yescharacteristic
618C2×C4#52911yescharacteristic
619C2×C4#53211yescharacteristic
620C2×C4#53511yescharacteristic
621C2×C4#53511yescharacteristic
622C2×C4#53511yescharacteristic
623C2×C4#53511yescharacteristic
624C2×C4#54011yescharacteristic
625C2×C4#54011yescharacteristic
626C2×C4#54611yescharacteristic
627C2×C4#54611yescharacteristic
628C2×C4#55011yescharacteristic
629C2×C4#55011yescharacteristic
630C2×C4#55311yescharacteristic
631C2×C4#55311yescharacteristic
632C2×C4#55311yescharacteristic
633C2×C4#55311yescharacteristic
634C2×C4#55311yescharacteristic
635C2×C4#55311yescharacteristic
636C2×C4#55511yescharacteristic
637C2×C4#55511yescharacteristic
638C8—42no—
639C8—42no—
640C8—82no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22C2×#39411yescharacteristic
657C22#242011yescharacteristic
658C22#242011yescharacteristic
659C22#243111yescharacteristic
660C22#243511yescharacteristic
661C22#253511yescharacteristic
662C22#253511yescharacteristic
663C22#300212yes—
664C22#301312yes—
665C22#305011yescharacteristic
666C22#305411yescharacteristic
667C22#305711yescharacteristic
668C22#305911yescharacteristic
669C22#309112yes—
670C22#309712yes—
671C22#329612yes—
672C22#330012yes—
673C22#330012yes—
674C22#330112yes—
675C22#332312yes—
676C22#332612yes—
677C22#332712yes—
678C22#332712yes—
679C4—24no—
680C4—24no—
681C4—44no—
682C4—84no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—162no—
688C4#241411yescharacteristic
689C4#241411yescharacteristic
690C4#249011yescharacteristic
691C4#249011yescharacteristic
692C4#250011yescharacteristic
693C4#250411yescharacteristic
694C4#261911yescharacteristic
695C4#285011yescharacteristic
696C2—84no—
697C2#1521911yescharacteristic
698C2#2347911yescharacteristic
699C2#2348711yescharacteristic
700C2#2497512yes—
701C2#2497612yes—
702C2#2557612yes—
703C2#2588912yes—
704C2#2814411yescharacteristic
705C2#2814511yescharacteristic
706C2#2814711yescharacteristic
707C2#2814811yescharacteristic
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
3. C2×#1835C2semidirect
4. C2×#1805C2semidirect
5. C2×C4×#103C2semidirect
7. #11200C2semidirect
8. #11199C2semidirect
9. #10970C2semidirect
10. #10970C2semidirect
11. #10850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
C2×#19542
C2×#18352
C2×#18052
C2×C4×#1032
#139052
#112002
#111992
#109702
#108502