Finite Group Databasebeta

← #396545

Subgroups — #396545

Order 512.

2118 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊C4C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#1865C212yesmaximal, complemented
4C2×#1803C211yesmaximal, characteristic, complemented
5C2×#1787C212yesmaximal
6#15135C211yesmaximal, characteristic
7#11095C212yesmaximal
8#11077C211yesmaximal, characteristic, complemented
9#11067C212yesmaximal, complemented
10#11042C211yesmaximal, characteristic
11#10789C211yesmaximal, characteristic, complemented
12C24×Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2212yes—
22C2×#399—21no—
23Q8×C22⋊C4—24no—
24Q8×C22⋊C4—24no—
25C2×#391—21no—
26C2×#391—21no—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#389C2212yes—
30C2×#384C2211yescharacteristic
31C2×#380—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4C2211yescharacteristic
36#2096—24no—
37#1865—28no—
38#1803—24no—
39#1787—28no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2212yes—
45#1445—21no—
46#1444—21no—
47#1439—21no—
48C4⋊#95C2211yescharacteristic
49C23×Q8—22no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—21no—
54C23×Q8—22no—
55C23×Q8—24no—
56C23×Q8—24no—
57C23×Q8—42no—
58C23×Q8—42no—
59C23×Q8Q814yes—
60C23×Q8C2311yescharacteristic
61C23×Q8C2311yescharacteristic
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C23×C8C2312yes—
65C23×C8C2312yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—24no—
70C2×C4×Q8—24no—
71C2×C4×Q8—42no—
72C2×C4×Q8—42no—
73C2×C4×Q8D414yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—42no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—42no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—42no—
106C2×#103—42no—
107C2×#103D418yes—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—42no—
111#399—44no—
112#391—28no—
113#391—28no—
114#391—22no—
115#391—22no—
116#391—22no—
117#391D414yes—
118#389—24no—
119#389—42no—
120#384—22no—
121#384—22no—
122#380—28no—
123#380—22no—
124#380—22no—
125#380—24no—
126#380—42no—
127C4×C22⋊C4—28no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95D411yescharacteristic
140C2×#95D411yescharacteristic
141#340—24no—
142#340—24no—
143#340—42no—
144#336—24no—
145#336—22no—
146C25C2×D411yescharacteristic
147C22×Q8—24no—
148C22×Q8—28no—
149C22×Q8—28no—
150C22×Q8—24no—
151C22×Q8—28no—
152C22×Q8—28no—
153C22×Q8—24no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—216no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8—48no—
170C22×Q8—48no—
171C22×Q8—48no—
172C22×Q8—48no—
173C22×Q8—48no—
174C22×Q8—48no—
175C22×Q8—48no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—48no—
179C22×Q8—48no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8C2×D412yes—
183C22×Q8C2×D412yes—
184C22×Q8C2×D414yes—
185C22×Q8C2×Q812yes—
186C22×Q8C2×Q812yes—
187C22×Q8#4112yes—
188C22×Q8#4114yes—
189C22×Q8#4112yes—
190C23×C4—21no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×Q812yes—
215C23×C4#4112yes—
216C23×C4C2411yescharacteristic
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8C2×D414yes—
231C4×Q8—28no—
232C4×Q8—44no—
233C4×Q8—48no—
234C4×Q8—48no—
235C4×Q8—44no—
236C4×Q8—44no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—82no—
265C2×C4⋊C4—82no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D414yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—82no—
280C2×C22⋊C4SD1614yes—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293#106—48no—
294C4⋊C8—44no—
295#103—216no—
296#103—48no—
297#103—48no—
298#103—48no—
299C22⋊C8—48no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—42no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24C22⋊Q812yes—
320C24C2×SD1612yes—
321C24#13712yes—
322C24C22×D411yescharacteristic
323C2×Q8—216no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—416no—
347C2×Q8—48no—
348C2×Q8—416no—
349C2×Q8—48no—
350C2×Q8—432no—
351C2×Q8—416no—
352C2×Q8—432no—
353C2×Q8—416no—
354C2×Q8—416no—
355C2×Q8—416no—
356C2×Q8—41no—
357C2×Q8—42no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8C22⋊Q814yes—
365C2×Q8C22⋊Q814yes—
366C2×Q8C22⋊Q818yes—
367C2×Q8#12318yes—
368C2×Q8#12314yes—
369C2×Q8#12714yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—416no—
410C22×C4—48no—
411C22×C4—416no—
412C22×C4—48no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4#12012yes—
447C22×C4#12012yes—
448C22×C4C22⋊Q812yes—
449C22×C4C2×SD1612yes—
450C22×C4C2×SD1612yes—
451C22×C4#13712yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×Q811yescharacteristic
456C22×C4C2×#4111yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4C2×#4111yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—88no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C22⋊C4—28no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—88no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—48no—
503C42—48no—
504C42—41no—
505C42—41no—
506C42—42no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23#47514yes—
530C23C2×#12011yescharacteristic
531C23C2×C22⋊Q811yescharacteristic
532C23#52511yescharacteristic
533C23D4212yes—
534C23#54612yes—
535C23C22×SD1611yescharacteristic
536C23C2×#13711yescharacteristic
537C23C2×#13711yescharacteristic
538C23#57511yescharacteristic
539C23#57712yes—
540C23#57812yes—
541Q8—24no—
542Q8—28no—
543Q8—28no—
544Q8—24no—
545Q8—24no—
546Q8—28no—
547Q8—432no—
548Q8—42no—
549Q8—42no—
550Q8—44no—
551Q8—42no—
552Q8—42no—
553Q8#39918yescomplemented
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—432no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4#44814yes—
620C2×C4C2×C22⋊Q811yescharacteristic
621C2×C4C2×C22⋊Q812yes—
622C2×C4C2×C22⋊Q811yescharacteristic
623C2×C4C2×#12312yes—
624C2×C4C2×#12311yescharacteristic
625C2×C4#52511yescharacteristic
626C2×C4#52511yescharacteristic
627C2×C4C2×#12711yescharacteristic
628C2×C4#53411yescharacteristic
629C2×C4#53511yescharacteristic
630C2×C4#53812yes—
631C2×C4#54312yes—
632C2×C4D4212yes—
633C2×C4#54612yes—
634C2×C4#54612yes—
635C2×C4#54612yes—
636C2×C4#55112yes—
637C2×C4#55112yes—
638C8—44no—
639C8—82no—
640C22—28no—
641C22—24no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22#240511yescharacteristic
652C22#241812yes—
653C22#243311yescharacteristic
654C22#243712yes—
655C22#284811yescharacteristic
656C22C2×#44811yescharacteristic
657C22#300711yescharacteristic
658C22#301011yescharacteristic
659C22#301111yescharacteristic
660C22C2×#47511yescharacteristic
661C22#307811yescharacteristic
662C22#308212yes—
663C22#310311yescharacteristic
664C22#311311yescharacteristic
665C22#312112yes—
666C22#326912yes—
667C22#327012yes—
668C22#327812yes—
669C22D4×SD1612yes—
670C22#329812yes—
671C22#331212yes—
672C22#331612yes—
673C22#332612yes—
674C4—28no—
675C4—24no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—84no—
681C4—81no—
682C4—82no—
683C4—82no—
684C4—162no—
685C4C2×#39911yescharacteristic
686C4#241812yes—
687C4#243712yes—
688C4#248211yescharacteristic
689C4#253911yescharacteristic
690C4#262611yescharacteristic
691C2—44no—
692C2#1473411yescharacteristic
693C2#2404111yescharacteristic
694C2#2405211yescharacteristic
695C2#2489812yes—
696C2#2491812yes—
697C2#2575611yescharacteristic
698C2#2576711yescharacteristic
699C2#2595512yes—
700C2#2597312yes—
701C2#3295411yescharacteristic
702C2#3296411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
2. C2×#2096C2semidirect
3. C2×#1865C2semidirect
4. C2×#1803C2semidirect
8. #11077C2semidirect
9. #11067C2semidirect
11. #10789C2semidirect
553. Q8#399semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊C42
C2×#20962
C2×#18652
C2×#18032
C2×#17872
#151352
#110952
#110772
#110672
#110422
#107892