Finite Group Databasebeta

← #398039

Subgroups — #398039

Order 512.

1876 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic, complemented
3C2×#1892C212yesmaximal
4C22×#336C211yesmaximal, characteristic
5#14702C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11293C212yesmaximal, complemented
8#11267C211yesmaximal, characteristic, complemented
9#11067C212yesmaximal, complemented
10#11050C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C24×C8C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C22×C22⋊Q8C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19#2857—28no—
20C2×#391—21no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29#2024—24no—
30#1892—28no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336—24no—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1436—21no—
42#1436—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C22⋊#95—21no—
46C22⋊#95—21no—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8—24no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—24no—
56C23×C8—24no—
57C23×C8—42no—
58C23×C8—42no—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—42no—
65C2×C22⋊Q8—28no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—44no—
70C2×C22⋊Q8—44no—
71C2×C22⋊Q8Q814yescomplemented
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—22no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C42—24no—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—42no—
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99#391—28no—
100#391—22no—
101#391—22no—
102#391—42no—
103#389—28no—
104#389—24no—
105#389—24no—
106#384—24no—
107#380—28no—
108#380—24no—
109#380—24no—
110#380D418yes—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—24no—
129#336—22no—
130#336D414yes—
131C25C2×D411yescharacteristic
132C22×Q8—24no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8C2×Q812yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4SD1614yes—
174C23×C4Q1614yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D414yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×Q812yes—
182C23×C4#4114yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C22×C8—216no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—48no—
199C22×C8—48no—
200C22×C8—48no—
201C22×C8—48no—
202C22×C8—48no—
203C22×C8—48no—
204C22×C8C2×D414yes—
205C22×C8#4114yes—
206C4⋊Q8—48no—
207C4⋊Q8—48no—
208C22⋊Q8—216no—
209C22⋊Q8—432no—
210C22⋊Q8—48no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×Q812yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—48no—
270C2×C42—48no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42C2×D412yes—
274C2×C42#4112yes—
275#103—48no—
276#103—48no—
277#95—28no—
278#95—28no—
279#95—42no—
280#95—42no—
281#95—42no—
282#95—42no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—42no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24#12012yes—
297C24C22⋊Q812yes—
298C24#12412yes—
299C24C2×SD1612yes—
300C24C2×Q1612yes—
301C24AGL(1,ℤ/8)12yes—
302C24#13712yes—
303C24C22×D411yescharacteristic
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—48no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—42no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8—84no—
327C22×C4—21no—
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—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
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—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
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—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4#12012yes—
422C22×C4#12112yes—
423C22×C4#12112yes—
424C22×C4C22⋊Q818yes—
425C22×C4C22⋊Q812yes—
426C22×C4#12418yes—
427C22×C4#12412yes—
428C22×C4C2×SD1612yes—
429C22×C4C2×SD1612yes—
430C22×C4C2×Q1612yes—
431C22×C4C2×Q1612yes—
432C22×C4#13512yes—
433C22×C4#13512yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×Q811yescharacteristic
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+411yescharacteristic
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—432no—
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—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—416no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—84no—
481C42—24no—
482C42—24no—
483C42—44no—
484C42—44no—
485C42—42no—
486C42—42no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
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#39314yes—
516C23#45014yes—
517C23#45114yes—
518C23#47514yes—
519C23#47714yes—
520C23#48714yes—
521C23#48914yes—
522C23C2×#12011yescharacteristic
523C23C2×#12111yescharacteristic
524C23C2×C22⋊Q811yescharacteristic
525C23C2×#12411yescharacteristic
526C23#53211yescharacteristic
527C23#53411yescharacteristic
528C23#53711yescharacteristic
529C23C22×SD1611yescharacteristic
530C23C22×Q1611yescharacteristic
531C23C2×#13511yescharacteristic
532C23C2×#13511yescharacteristic
533C23C2×AGL(1,ℤ/8)11yescharacteristic
534C23C2×#13711yescharacteristic
535C23#57511yescharacteristic
536C23#57511yescharacteristic
537Q8—44no—
538Q8—44no—
539Q8—816no—
540Q8—82no—
541Q8—82no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4#39314yes—
623C2×C4#46514yes—
624C2×C4#46714yes—
625C2×C4C2×C22⋊Q812yes—
626C2×C4#52512yes—
627C2×C4C2×#12412yes—
628C2×C4#52912yes—
629C2×C4#53812yes—
630C2×C4#53912yes—
631C2×C4D4212yes—
632C2×C4#54612yes—
633C2×C4#54612yes—
634C2×C4#54611yescharacteristic
635C2×C4#54712yes—
636C2×C4D4×Q811yescharacteristic
637C2×C4#55112yes—
638C2×C4#55211yescharacteristic
639C2×C4#55412yes—
640C2×C4#55411yescharacteristic
641C8—44no—
642C8—44no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22#202418yescomplemented
658C22C2×#39311yescharacteristic
659C22#241011yescharacteristic
660C22#241711yescharacteristic
661C22#243011yescharacteristic
662C22#256411yescharacteristic
663C22#263911yescharacteristic
664C22#285011yescharacteristic
665C22C2×#45011yescharacteristic
666C22C2×#45111yescharacteristic
667C22#300211yescharacteristic
668C22#300211yescharacteristic
669C22C2×#46511yescharacteristic
670C22C2×#46711yescharacteristic
671C22#305511yescharacteristic
672C22#305611yescharacteristic
673C22C2×#47511yescharacteristic
674C22C2×#47711yescharacteristic
675C22#307611yescharacteristic
676C22#307611yescharacteristic
677C22C2×#48711yescharacteristic
678C22#313111yescharacteristic
679C22C2×#48911yescharacteristic
680C22#313511yescharacteristic
681C22#316711yescharacteristic
682C22#317011yescharacteristic
683C22#318011yescharacteristic
684C22#318111yescharacteristic
685C22#318711yescharacteristic
686C22#318811yescharacteristic
687C22#318911yescharacteristic
688C22#319011yescharacteristic
689C22#324611yescharacteristic
690C22#324711yescharacteristic
691C22#324811yescharacteristic
692C22#324911yescharacteristic
693C4—24no—
694C4—24no—
695C4—24no—
696C4—42no—
697C4—42no—
698C4—84no—
699C4—84no—
700C4—82no—
701C4—82no—
702C4—161no—
703C4—161no—
704C4#244912yes—
705C4#245012yes—
706C4#247212yes—
707C4#248812yes—
708C2—28no—
709C2#1475911yescharacteristic
710C2C2×#202411yescharacteristic
711C2#2345111yescharacteristic
712C2#2437911yescharacteristic
713C2#2438011yescharacteristic
714C2#2483711yescharacteristic
715C2#2483811yescharacteristic
716C2#2549511yescharacteristic
717C2#2549811yescharacteristic
718C2#2915911yescharacteristic
719C2#2916011yescharacteristic
720C2#3018711yescharacteristic
721C2#3018811yescharacteristic
722C2#3304311yescharacteristic
723C2#3305111yescharacteristic
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
2. C2×#2024C2semidirect
6. #11384C2semidirect
7. #11293C2semidirect
8. #11267C2semidirect
9. #11067C2semidirect
10. #11050C2semidirect
71. C2×C22⋊Q8Q8semidirect
657. C22#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#20242
C2×#18922
C22×#3362
#147022
#113842
#112932
#112672
#110672
#110502
#107212