Finite Group Databasebeta

← #405297

Subgroups — #405297

Order 512.

1776 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic, complemented
4C2×#1824C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#13872C211yesmaximal, characteristic
7#11293C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#11200C212yesmaximal, complemented
10#11096C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19#2487—24no—
20#2487—24no—
21C2×#400—21no—
22C2×#399—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#386—21no—
26C2×#386—21no—
27C2×#385—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33#1869—24no—
34#1852—24no—
35#1824—24no—
36C2×#340—24no—
37C2×#340C2212yes—
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×D4D411yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55C23×C8—24no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—42no—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D414yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4D411yescharacteristic
94C22×C22⋊C4D411yescharacteristic
95C22×C42C2311yescharacteristic
96C2×#107—24no—
97C2×#107—24no—
98C2×#107—42no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8D414yes—
105#400—28no—
106#399—28no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#386—28no—
112#386—44no—
113#385—28no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380—22no—
119#380—22no—
120#380—22no—
121#380—24no—
122#380—42no—
123#380D414yes—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95—22no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—42no—
140#336—24no—
141C25C2×D411yescharacteristic
142C25C2×D411yescharacteristic
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4Q1614yes—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—44no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—82no—
178C23×C4—82no—
179C23×C4—82no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4#4112yes—
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×C8C2×D414yes—
209C22×C8#4114yes—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—44no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
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—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4—82no—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4#4114yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—82no—
282C2×C22⋊C4—82no—
283C2×C22⋊C4—82no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290#107—48no—
291C22⋊C8—28no—
292C22⋊C8—48no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—44no—
299#95—44no—
300#95—42no—
301#95—82no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24#12012yes—
317C24#12112yes—
318C24#12312yes—
319C24C2×Q1612yes—
320C24C2×Q1612yes—
321C24#13512yes—
322C24AGL(1,ℤ/8)12yes—
323C24#13712yes—
324C24C22×D411yescharacteristic
325C2×D4—28no—
326C2×D4—28no—
327C2×D4—28no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
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—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—48no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
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—48no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
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—84no—
419C22×C4—84no—
420C22×C4—88no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4—84no—
426C22×C4#12012yes—
427C22×C4#12114yes—
428C22×C4#12114yes—
429C22×C4#12112yes—
430C22×C4#12314yes—
431C22×C4#12312yes—
432C22×C4#12414yes—
433C22×C4C2×Q1612yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+412yes—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—416no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—48no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—41no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—88no—
492C22⋊C4—84no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23#39414yes—
537C23#44914yes—
538C23#45114yes—
539C23#46714yes—
540C23#46814yes—
541C23#48414yes—
542C23#48514yes—
543C23C2×#12011yescharacteristic
544C23C2×#12111yescharacteristic
545C23C2×#12311yescharacteristic
546C23#54612yes—
547C23#54712yes—
548C23C22×Q1611yescharacteristic
549C23C2×#13511yescharacteristic
550C23C2×AGL(1,ℤ/8)11yescharacteristic
551C23C2×#13711yescharacteristic
552C23#57612yes—
553C23#57712yes—
554D4—416no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—816no—
598C2×C4—816no—
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—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4#39414yes—
625C2×C4C2×#12111yescharacteristic
626C2×C4C2×#12111yescharacteristic
627C2×C4C2×#12311yescharacteristic
628C2×C4#52511yescharacteristic
629C2×C4#52511yescharacteristic
630C2×C4C2×#12411yescharacteristic
631C2×C4#52912yes—
632C2×C4#52912yes—
633C2×C4#53511yescharacteristic
634C2×C4#53611yescharacteristic
635C2×C4#53912yes—
636C2×C4#53912yes—
637C2×C4#54612yes—
638C2×C4#54612yes—
639C2×C4#54712yes—
640C2×C4#54712yes—
641C2×C4#55112yes—
642C2×C4#55212yes—
643C8—44no—
644C8—44no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—28no—
649C22—28no—
650C22—28no—
651C22—24no—
652C22—24no—
653C22—28no—
654C22—24no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—42no—
659C22—42no—
660C22—81no—
661C22—81no—
662C22—82no—
663C22—81no—
664C22—82no—
665C22—81no—
666C22#203618yes—
667C22C2×#39411yescharacteristic
668C22#240411yescharacteristic
669C22#240611yescharacteristic
670C22#241912yes—
671C22#242512yes—
672C22C2×#45111yescharacteristic
673C22C2×#44911yescharacteristic
674C22#300811yescharacteristic
675C22#300911yescharacteristic
676C22C2×#46711yescharacteristic
677C22C2×#46811yescharacteristic
678C22#306212yes—
679C22C2×#48511yescharacteristic
680C22C2×#48411yescharacteristic
681C22#309612yes—
682C22#329712yes—
683C22#329912yes—
684C22#331612yes—
685C22#331812yes—
686C22#333712yes—
687C22#333812yes—
688C22#334512yes—
689C22#334612yes—
690C4—24no—
691C4—42no—
692C4—82no—
693C4—81no—
694C4—82no—
695C4—84no—
696C4—81no—
697C4—82no—
698C4—82no—
699C4—162no—
700C4#241111yescharacteristic
701C4#241311yescharacteristic
702C4#242512yes—
703C4#248812yes—
704C4#262311yescharacteristic
705C4#285311yescharacteristic
706C2—28no—
707C2—82no—
708C2#1491211yescharacteristic
709C2C2×#203611yescharacteristic
710C2#2348311yescharacteristic
711C2#2396311yescharacteristic
712C2#2397011yescharacteristic
713C2#2409111yescharacteristic
714C2#2409811yescharacteristic
715C2#2493912yes—
716C2#2494312yes—
717C2#2523412yes—
718C2#2525212yes—
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×#1852C2semidirect
4. C2×#1824C2semidirect
5. C22×#340C2semidirect
7. #11293C2semidirect
8. #11219C2semidirect
9. #11200C2semidirect
10. #11096C2semidirect
11. #10721C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#18692
C2×#18522
C2×#18242
C22×#3402
#138722
#112932
#112192
#112002
#110962
#107212