Finite Group Databasebeta

← #405395

Subgroups — #405395

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1850C211yesmaximal, characteristic, complemented
4C22×#340C211yesmaximal, characteristic, complemented
5#14137C211yesmaximal, characteristic
6#13904C211yesmaximal, characteristic, complemented
7#11256C212yesmaximal, complemented
8#11232C212yesmaximal, complemented
9#11189C212yesmaximal, complemented
10#11076C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#480—24no—
17C2×#480—24no—
18C2×#480—24no—
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×C4⋊C8C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C2×#390C2212yes—
27C2×#386—21no—
28C2×#386—21no—
29C2×#381—21no—
30C2×#381—21no—
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1893—24no—
34#1850—24no—
35C2×#340—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42#1445—22no—
43#1439—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—42no—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2312yes—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121D414yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4D414yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4D411yescharacteristic
88C22×C22⋊C4D411yescharacteristic
89C22×C42C2312yes—
90#480—232no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—42no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—42no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8D414yes—
112#390—24no—
113#390—42no—
114#386—44no—
115#386—44no—
116#381—44no—
117#381—44no—
118#380—22no—
119#380—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95D412yes—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#336—24no—
136C25—21no—
137C25C2×D411yescharacteristic
138C22×D4—24no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4C2×D412yes—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—82no—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186C22×C8—28no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
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×C8—48no—
202C22×C8—48no—
203C22×C8—44no—
204C22×C8—44no—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8#4114yes—
208#121—28no—
209#121—28no—
210#121—44no—
211#121—44no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
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—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4#4114yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4—82no—
271C2×C22⋊C4D814yes—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278#107—28no—
279#107—28no—
280C4⋊C8—48no—
281#102—28no—
282#102—28no—
283#102—48no—
284C22⋊C8—28no—
285#95—24no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—42no—
290#95—44no—
291#95—44no—
292#95—44no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—41no—
299C24—41no—
300C24—42no—
301C24—42no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—42no—
307C24—42no—
308C24#12112yes—
309C24#12112yes—
310C24#12112yes—
311C24C2×D812yes—
312C24#13512yes—
313C24AGL(1,ℤ/8)12yes—
314C24AGL(1,ℤ/8)12yes—
315C24C22×D411yescharacteristic
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4—84no—
340C2×D4—84no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—48no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4#12014yes—
418C22×C4#12112yes—
419C22×C4#12112yes—
420C22×C4#12112yes—
421C22×C4#12314yes—
422C22×C4#12314yes—
423C22×C4#12414yes—
424C22×C4C2×D814yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C42−1+412yes—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—416no—
443C2×C8—42no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—42no—
448C2×C8—41no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—48no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—48no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C22⋊C4—28no—
476C22⋊C4—44no—
477C22⋊C4—48no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—84no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—42no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—81no—
511C23—81no—
512C23—82no—
513C23—81no—
514C23—81no—
515C23—82no—
516C23—81no—
517C23—81no—
518C23—82no—
519C23—81no—
520C23—81no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23#39414yes—
525C23#45914yes—
526C23#45914yes—
527C23#46314yes—
528C23#46314yes—
529C23#46614yes—
530C23#46914yes—
531C23C2×#12111yescharacteristic
532C23C2×#12111yescharacteristic
533C23C2×#12111yescharacteristic
534C23#54612yes—
535C23#54712yes—
536C23C22×D811yescharacteristic
537C23C2×#13511yescharacteristic
538C23C2×AGL(1,ℤ/8)11yescharacteristic
539C23C2×AGL(1,ℤ/8)11yescharacteristic
540C23#57712yes—
541C23#57812yes—
542D4—42no—
543D4—42no—
544D4—44no—
545D4—816no—
546D4—81no—
547D4—81no—
548D4—82no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—816no—
586C2×C4—816no—
587C2×C4—82no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
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#39414yes—
611C2×C4C2×#12011yescharacteristic
612C2×C4C2×#12311yescharacteristic
613C2×C4C2×#12311yescharacteristic
614C2×C4#52511yescharacteristic
615C2×C4#52511yescharacteristic
616C2×C4C2×#12411yescharacteristic
617C2×C4#52912yes—
618C2×C4#52912yes—
619C2×C4#53511yescharacteristic
620C2×C4#53611yescharacteristic
621C2×C4#53912yes—
622C2×C4#53912yes—
623C2×C4#54612yes—
624C2×C4#54612yes—
625C2×C4#54712yes—
626C2×C4#54712yes—
627C2×C4#55112yes—
628C2×C4#55212yes—
629C8—42no—
630C8—44no—
631C8—42no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—28no—
636C22—28no—
637C22—28no—
638C22—24no—
639C22—24no—
640C22—28no—
641C22—24no—
642C22—82no—
643C22—84no—
644C22—84no—
645C22—82no—
646C22—82no—
647C22—161no—
648C22—161no—
649C22—162no—
650C22—161no—
651C22—161no—
652C22—162no—
653C22#205418yes—
654C22C2×#39411yescharacteristic
655C22#241111yescharacteristic
656C22#241311yescharacteristic
657C22#242512yes—
658C22#242512yes—
659C22C2×#45911yescharacteristic
660C22C2×#45911yescharacteristic
661C22C2×#46311yescharacteristic
662C22C2×#46311yescharacteristic
663C22#304311yescharacteristic
664C22#304411yescharacteristic
665C22#304612yes—
666C22C2×#46611yescharacteristic
667C22C2×#46911yescharacteristic
668C22#306312yes—
669C22#329412yes—
670C22#330012yes—
671C22#330812yes—
672C22#331012yes—
673C22#333412yes—
674C22#333712yes—
675C22#334312yes—
676C22#334512yes—
677C4—24no—
678C4—44no—
679C4—84no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—162no—
685C4#240411yescharacteristic
686C4#240611yescharacteristic
687C4#241912yes—
688C4#248812yes—
689C4#262311yescharacteristic
690C4#285311yescharacteristic
691C2—28no—
692C2—162no—
693C2#1491211yescharacteristic
694C2C2×#205411yescharacteristic
695C2#2354511yescharacteristic
696C2#2447611yescharacteristic
697C2#2448111yescharacteristic
698C2#2457611yescharacteristic
699C2#2458111yescharacteristic
700C2#2520012yes—
701C2#2521212yes—
702C2#2523212yes—
703C2#2525012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1850C2semidirect
4. C22×#340C2semidirect
6. #13904C2semidirect
7. #11256C2semidirect
8. #11232C2semidirect
9. #11189C2semidirect
10. #11076C2semidirect
11. #10730C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
C2×#18932
C2×#18502
C22×#3402
#141372
#139042
#112562
#112322
#111892
#110762
#107302