Finite Group Databasebeta

← #406330

Subgroups — #406330

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1797C211yesmaximal, characteristic
5#14702C211yesmaximal, characteristic
6C2×C4⋊#95C211yesmaximal, characteristic
7#11256C212yesmaximal, complemented
8#11232C212yesmaximal, complemented
9#11190C212yesmaximal, complemented
10#11077C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#484—24no—
17C2×#484—24no—
18C2×#484—24no—
19C22×#107C2212yes—
20C22×C4⋊C8C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C22×#95—21no—
29C22×#95—21no—
30#1872—24no—
31#1824—24no—
32#1797—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1436—21no—
40#1436—21no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C22⋊#95—21no—
48C22⋊#95—21no—
49C23×Q8C2311yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8D414yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4D414yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42—24no—
86C22×C42—42no—
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#484—232no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—42no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—42no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8D414yes—
110#391—24no—
111#391—42no—
112#380—22no—
113#380—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—44no—
128C2×#95—44no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—42no—
134#336—24no—
135C25C2×D411yescharacteristic
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8C2×D412yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4Q1614yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4#4112yes—
195C23×C4C2411yescharacteristic
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—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8C2×D414yes—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C22⋊Q8—44no—
213C22⋊Q8—44no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4#4114yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—48no—
273C2×C42—48no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42#4114yes—
277#107—216no—
278C4⋊C8—48no—
279#103—216no—
280#103—48no—
281C22⋊C8—28no—
282C22⋊C8—44no—
283#95—42no—
284#95—42no—
285#95—44no—
286#95—44no—
287#95—42no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—42no—
293#95—42no—
294#95—44no—
295#95—44no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24#12112yes—
305C24#12112yes—
306C24#12112yes—
307C24C2×Q1612yes—
308C24#13512yes—
309C24#13712yes—
310C24#13712yes—
311C24C22×D411yescharacteristic
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—48no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4#12014yes—
437C22×C4#12112yes—
438C22×C4#12112yes—
439C22×C4#12112yes—
440C22×C4#12312yes—
441C22×C4#12314yes—
442C22×C4#12314yes—
443C22×C4#12312yes—
444C22×C4#12414yes—
445C22×C4C2×Q1612yes—
446C22×C4C2×Q1612yes—
447C22×C4AGL(1,ℤ/8)12yes—
448C22×C4AGL(1,ℤ/8)12yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—41no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C22⋊C4—48no—
493C22⋊C4—84no—
494C42—24no—
495C42—24no—
496C42—44no—
497C42—44no—
498C42—42no—
499C42—42no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23#39414yes—
529C23#46214yes—
530C23#46214yes—
531C23#46414yes—
532C23#46414yes—
533C23#46714yes—
534C23#47014yes—
535C23C2×#12111yescharacteristic
536C23C2×#12111yescharacteristic
537C23C2×#12111yescharacteristic
538C23C2×#12311yescharacteristic
539C23#52511yescharacteristic
540C23#53411yescharacteristic
541C23#53611yescharacteristic
542C23C22×Q1611yescharacteristic
543C23C2×#13511yescharacteristic
544C23C2×AGL(1,ℤ/8)11yescharacteristic
545C23C2×AGL(1,ℤ/8)11yescharacteristic
546C23C2×#13711yescharacteristic
547C23C2×#13711yescharacteristic
548C23#57511yescharacteristic
549C23#57511yescharacteristic
550Q8—44no—
551Q8—44no—
552Q8—82no—
553Q8—82no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—416no—
580C2×C4—416no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—816no—
606C2×C4—816no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4#39414yes—
638C2×C4#48414yescomplemented
639C2×C4C2×#12011yescharacteristic
640C2×C4C2×#12311yescharacteristic
641C2×C4C2×#12311yescharacteristic
642C2×C4C2×#12411yescharacteristic
643C2×C4#52912yes—
644C2×C4#53212yes—
645C2×C4#53712yes—
646C2×C4#53912yes—
647C2×C4#54612yes—
648C2×C4#54612yes—
649C2×C4#54712yes—
650C2×C4#54712yes—
651C2×C4#55112yes—
652C2×C4#55112yes—
653C2×C4#55212yes—
654C2×C4#55212yes—
655C8—44no—
656C8—82no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22#205618yes—
672C22C2×#39411yescharacteristic
673C22#241211yescharacteristic
674C22#241311yescharacteristic
675C22#248611yescharacteristic
676C22#253411yescharacteristic
677C22#261811yescharacteristic
678C22#285411yescharacteristic
679C22C2×#46211yescharacteristic
680C22C2×#46211yescharacteristic
681C22C2×#46411yescharacteristic
682C22C2×#46411yescharacteristic
683C22#303611yescharacteristic
684C22#303611yescharacteristic
685C22#303811yescharacteristic
686C22#304011yescharacteristic
687C22C2×#46711yescharacteristic
688C22C2×#47011yescharacteristic
689C22#305511yescharacteristic
690C22#305511yescharacteristic
691C22C2×#48411yescharacteristic
692C22#309111yescharacteristic
693C22#309211yescharacteristic
694C22#309511yescharacteristic
695C22#310411yescharacteristic
696C22#310511yescharacteristic
697C22#311011yescharacteristic
698C22#311711yescharacteristic
699C22#318811yescharacteristic
700C22#318911yescharacteristic
701C22#319311yescharacteristic
702C22#319611yescharacteristic
703C22#323011yescharacteristic
704C22#323211yescharacteristic
705C22#323611yescharacteristic
706C22#323911yescharacteristic
707C4—24no—
708C4—24no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—82no—
713C4—84no—
714C4—82no—
715C4—84no—
716C4—161no—
717C4—161no—
718C4#241912yes—
719C4#243012yes—
720C4#248812yes—
721C4#249812yes—
722C2—28no—
723C2#1492711yescharacteristic
724C2C2×#205611yescharacteristic
725C2#2354611yescharacteristic
726C2#2455211yescharacteristic
727C2#2455611yescharacteristic
728C2#2460211yescharacteristic
729C2#2460611yescharacteristic
730C2#2764811yescharacteristic
731C2#2765311yescharacteristic
732C2#2812111yescharacteristic
733C2#2812311yescharacteristic
734C2#2975711yescharacteristic
735C2#2976111yescharacteristic
736C2#3310911yescharacteristic
737C2#3311011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11256C2semidirect
8. #11232C2semidirect
9. #11190C2semidirect
10. #11077C2semidirect
638. C2×C4#484semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C2×#18722
C2×#18242
C2×#17972
#147022
C2×C4⋊#952
#112562
#112322
#111902
#110772
#107302