Finite Group Databasebeta

← #414952

Subgroups — #414952

Order 512.

1556 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1805C211yesmaximal, characteristic, complemented
3C2×#1803C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5C2×#1802C211yesmaximal, characteristic
6#14277C211yesmaximal, characteristic, complemented
7#11267C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#11066C212yesmaximal
10#11009C212yesmaximal
11#10833C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21#2326—24no—
22#2326—24no—
23C2×#391C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2212yes—
26C2×#388—21no—
27C2×#388—21no—
28C2×#385—21no—
29C2×#385—21no—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1805—24no—
35#1804—24no—
36#1803—24no—
37#1802—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1444—21no—
46#1442—21no—
47#1442—21no—
48C4⋊#95C2212yes—
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124Q814yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8D414yes—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—42no—
107C2×#102—42no—
108C2×#102—42no—
109#391—22no—
110#391—22no—
111#390—22no—
112#390—22no—
113#389—42no—
114#389—42no—
115#388—28no—
116#388—28no—
117#385—28no—
118#385—28no—
119#384—22no—
120#384—22no—
121#384—22no—
122#384—22no—
123#382—22no—
124#382D414yes—
125#376—22no—
126#376—22no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D411yescharacteristic
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137C2×#95D411yescharacteristic
138#340—24no—
139#340—24no—
140#340—24no—
141#340—24no—
142#336—24no—
143#336—42no—
144C25—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8C2×D412yes—
157C22×Q8C2×Q812yes—
158C22×Q8#4112yes—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C2×D412yes—
171C22×D4C2×Q812yes—
172C22×D4#4112yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—82no—
182C23×C4—82no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2411yescharacteristic
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—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204#124—28no—
205#124—28no—
206#124—28no—
207#124—28no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—44no—
211C4×Q8—44no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4—44no—
215C4×D4—44no—
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—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4—82no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—82no—
266C2×C22⋊C4—82no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—24no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×Q812yes—
284C2×C42#4112yes—
285C2×C42#4112yes—
286C2×C42#4112yes—
287C2×C42#4112yes—
288C4⋊C8—44no—
289C4⋊C8—44no—
290#103—48no—
291#103—48no—
292#102—48no—
293#102—48no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—44no—
303#95—82no—
304#95—82no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24C22×D411yescharacteristic
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
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—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8C22⋊Q814yes—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4C22⋊Q814yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—41no—
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—42no—
419C22×C4—48no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4#12012yes—
432C22×C4#12012yes—
433C22×C4#12112yes—
434C22×C4#12112yes—
435C22×C4C22⋊Q812yes—
436C22×C4C22⋊Q812yes—
437C22×C4#12412yes—
438C22×C4#12412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×Q811yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C4H(5,𝔽2)11yescharacteristic
451C22×C4H(5,𝔽2)11yescharacteristic
452C22×C42−1+411yescharacteristic
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—88no—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—48no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—48no—
503C42#12114yes—
504C42#12114yes—
505C42#12314yes—
506C42#12314yes—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23C2×#12011yescharacteristic
531C23C2×#12111yescharacteristic
532C23C2×C22⋊Q811yescharacteristic
533C23#52511yescharacteristic
534C23C2×#12411yescharacteristic
535C23#53511yescharacteristic
536C23#53711yescharacteristic
537C23#57612yes—
538C23#57712yes—
539C23#57712yes—
540C23#57812yes—
541Q8—24no—
542Q8—24no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549D4—24no—
550D4—24no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—816no—
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—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4#39414yes—
618C2×C4#39414yes—
619C2×C4C2×#12111yescharacteristic
620C2×C4C2×#12111yescharacteristic
621C2×C4C2×C22⋊Q811yescharacteristic
622C2×C4C2×C22⋊Q811yescharacteristic
623C2×C4C2×#12311yescharacteristic
624C2×C4C2×#12311yescharacteristic
625C2×C4#52511yescharacteristic
626C2×C4#52511yescharacteristic
627C2×C4#53511yescharacteristic
628C2×C4#53511yescharacteristic
629C2×C4#53711yescharacteristic
630C2×C4#53711yescharacteristic
631C2×C4#54611yescharacteristic
632C2×C4#54611yescharacteristic
633C2×C4D4×Q811yescharacteristic
634C2×C4#55011yescharacteristic
635C2×C4#55211yescharacteristic
636C2×C4#55211yescharacteristic
637C2×C4#55311yescharacteristic
638C2×C4#55311yescharacteristic
639C2×C4#55311yescharacteristic
640C2×C4#55311yescharacteristic
641C2×C4#55411yescharacteristic
642C2×C4#55411yescharacteristic
643C2×C4#55411yescharacteristic
644C2×C4#55511yescharacteristic
645C2×C4#55511yescharacteristic
646C2×C4#55611yescharacteristic
647C8—82no—
648C8—82no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—82no—
656C22—82no—
657C22—84no—
658C22—82no—
659C22—82no—
660C22C2×#39411yescharacteristic
661C22#253511yescharacteristic
662C22#253811yescharacteristic
663C22#256211yescharacteristic
664C22#256311yescharacteristic
665C22#261511yescharacteristic
666C22#284811yescharacteristic
667C22#300811yescharacteristic
668C22#300911yescharacteristic
669C22#301011yescharacteristic
670C22#301111yescharacteristic
671C22#304512yes—
672C22#304612yes—
673C22#308312yes—
674C22#308412yes—
675C22#312412yes—
676C22#312512yes—
677C22#313612yes—
678C22#313712yes—
679C22#322412yes—
680C22#322512yes—
681C22#325712yes—
682C22#325812yes—
683C4—24no—
684C4—24no—
685C4—44no—
686C4—42no—
687C4—42no—
688C4—84no—
689C4—82no—
690C4—82no—
691C4—81no—
692C4—81no—
693C4—162no—
694C4#241411yescharacteristic
695C4#241411yescharacteristic
696C4#241711yescharacteristic
697C4#241711yescharacteristic
698C4#248511yescharacteristic
699C4#248511yescharacteristic
700C4#248711yescharacteristic
701C4#248711yescharacteristic
702C2—84no—
703C2#1518111yescharacteristic
704C2#2350911yescharacteristic
705C2#2351011yescharacteristic
706C2#2817212yes—
707C2#2829312yes—
708C2#2913712yes—
709C2#2915112yes—
710C2#2966911yescharacteristic
711C2#2967211yescharacteristic
712C2#3296311yescharacteristic
713C2#3296511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1805C2semidirect
3. C2×#1803C2semidirect
6. #14277C2semidirect
7. #11267C2semidirect
8. #11067C2semidirect
11. #10833C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#18052
C2×#18032
C2×#18042
C2×#18022
#142772
#112672
#110672
#110662
#110092
#108332