Finite Group Databasebeta

← #405502

Subgroups — #405502

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#1854C212yesmaximal
3#13756C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11324C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11220C212yesmaximal, complemented
9#11219C212yesmaximal, complemented
10#10731C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C22⋊C8C2212yes—
20#2857—28no—
21C2×#391—21no—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#380—21no—
26C2×#380C2212yes—
27#1854—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C22⋊#95—21no—
43C4×#95—21no—
44C4×#95—21no—
45C23×Q8—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C22⋊Q8—28no—
56C2×C22⋊Q8—44no—
57C2×C22⋊Q8—44no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—24no—
62C2×#117—42no—
63C2×#117—42no—
64C2×#117—42no—
65C2×#117Q814yescomplemented
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C42—24no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42—22no—
84C22×C42—42no—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#107—24no—
90C2×#107—24no—
91C2×#107—42no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—42no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8Q818yescomplemented
98#391—28no—
99#389—28no—
100#389—24no—
101#389—24no—
102#384—24no—
103#380—28no—
104#380—24no—
105#380D418yes—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116#340—24no—
117#340—24no—
118#340—42no—
119#340—42no—
120#340—42no—
121#340—42no—
122#336—22no—
123#336—22no—
124#336—42no—
125C25C2×D411yescharacteristic
126C22×Q8—24no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D414yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×Q814yes—
166C23×C4C2×Q812yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8C2×Q814yes—
182C22×C8C2×Q814yes—
183C4⋊Q8—48no—
184C4⋊Q8—48no—
185C22⋊Q8—432no—
186#117—48no—
187#117—48no—
188#117—48no—
189#117—48no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×Q812yes—
227C2×C4⋊C4C2×Q812yes—
228C2×C4⋊C4#4112yes—
229C2×C4⋊C4#4112yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—48no—
252C2×C42—48no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42#4112yes—
260C2×C42#4112yes—
261#107—48no—
262#106—48no—
263C22⋊C8—216no—
264#95—28no—
265#95—28no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—42no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24#12012yes—
287C24C22⋊Q812yes—
288C24C4⋊Q812yes—
289C24C22×D411yescharacteristic
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8—84no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—21no—
326C22×C4—21no—
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—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4#12012yes—
392C22×C4C22⋊Q818yes—
393C22×C4C22⋊Q812yes—
394C22×C4#12312yes—
395C22×C4#12312yes—
396C22×C4C4⋊Q818yes—
397C22×C4C4⋊Q812yes—
398C22×C4#13512yes—
399C22×C4#13512yes—
400C22×C4AGL(1,ℤ/8)12yes—
401C22×C4AGL(1,ℤ/8)12yes—
402C22×C4#13712yes—
403C22×C4#13712yes—
404C22×C4C22×D412yes—
405C22×C4C22×Q812yes—
406C22×C4C22×Q811yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4H(5,𝔽2)12yes—
413C22×C4H(5,𝔽2)12yes—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—416no—
451C42—24no—
452C42—24no—
453C42—44no—
454C42—44no—
455C42—48no—
456C42—48no—
457C42—42no—
458C42—42no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23#39514yes—
494C23C2×#12011yescharacteristic
495C23C2×C22⋊Q811yescharacteristic
496C23C2×#12311yescharacteristic
497C23#52511yescharacteristic
498C23C2×C4⋊Q811yescharacteristic
499C23#53411yescharacteristic
500C23#53511yescharacteristic
501C23C2×#13511yescharacteristic
502C23C2×#13511yescharacteristic
503C23C2×AGL(1,ℤ/8)11yescharacteristic
504C23C2×AGL(1,ℤ/8)11yescharacteristic
505C23C2×#13711yescharacteristic
506C23C2×#13711yescharacteristic
507C23#57511yescharacteristic
508C23#57511yescharacteristic
509Q8—816no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—416no—
539C2×C4—416no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4#39514yes—
591C2×C4#48214yes—
592C2×C4#48514yes—
593C2×C4C2×C22⋊Q812yes—
594C2×C4#52911yescharacteristic
595C2×C4#52911yescharacteristic
596C2×C4C2×C4⋊Q812yes—
597C2×C4#53311yescharacteristic
598C2×C4#53311yescharacteristic
599C2×C4#53812yes—
600C2×C4#54312yes—
601C2×C4#54612yes—
602C2×C4#54612yes—
603C2×C4#54612yes—
604C2×C4#54612yes—
605C2×C4#55412yes—
606C2×C4#55412yes—
607C2×C4#55412yes—
608C2×C4#55412yes—
609C8—44no—
610C8—44no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22C2×#39511yescharacteristic
627C22#240411yescharacteristic
628C22#240511yescharacteristic
629C22#250211yescharacteristic
630C22#258311yescharacteristic
631C22#262611yescharacteristic
632C22#285511yescharacteristic
633C22#300211yescharacteristic
634C22#300211yescharacteristic
635C22#300311yescharacteristic
636C22#300411yescharacteristic
637C22#307611yescharacteristic
638C22#307611yescharacteristic
639C22#307911yescharacteristic
640C22#307911yescharacteristic
641C22C2×#48211yescharacteristic
642C22C2×#48511yescharacteristic
643C22#309411yescharacteristic
644C22#309511yescharacteristic
645C22#310611yescharacteristic
646C22#310711yescharacteristic
647C22#311511yescharacteristic
648C22#311611yescharacteristic
649C22#316011yescharacteristic
650C22#316211yescharacteristic
651C22#316511yescharacteristic
652C22#316511yescharacteristic
653C22#318811yescharacteristic
654C22#319011yescharacteristic
655C22#319411yescharacteristic
656C22#319611yescharacteristic
657C22#320711yescharacteristic
658C22#321011yescharacteristic
659C22#321411yescharacteristic
660C22#321611yescharacteristic
661C4—24no—
662C4—24no—
663C4—24no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4—82no—
671C4—84no—
672C4#245012yes—
673C4#245012yes—
674C4#254912yes—
675C4#254912yes—
676C2—44no—
677C2#1491411yescharacteristic
678C2#2355111yescharacteristic
679C2#2355211yescharacteristic
680C2#2393211yescharacteristic
681C2#2393511yescharacteristic
682C2#2400911yescharacteristic
683C2#2401011yescharacteristic
684C2#2794911yescharacteristic
685C2#2795511yescharacteristic
686C2#2929911yescharacteristic
687C2#2930411yescharacteristic
688C2#2995411yescharacteristic
689C2#2995511yescharacteristic
690C2#3315311yescharacteristic
691C2#3315411yescharacteristic
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
4. #11384C2semidirect
5. #11384C2semidirect
6. #11324C2semidirect
7. #11323C2semidirect
8. #11220C2semidirect
9. #11219C2semidirect
65. C2×#117Q8semidirect
97. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#18542
#137562
#113842
#113242
#113232
#112202
#112192
#107312
#107122