Finite Group Databasebeta

← #641451

Subgroups — #641451

Order 512.

3334 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic, complemented
2C2×#2537C211yesmaximal, characteristic
3C2×C4×#124C211yesmaximal, characteristic, complemented
4C2×#2293C211yesmaximal, characteristic, complemented
5#16817C218yesmaximal, complemented
6#15233C212yesmaximal, complemented
7#15220C214yesmaximal, complemented
8#15219C214yesmaximal, complemented
9#15215C212yesmaximal, complemented
10#14687C211yesmaximal, characteristic, complemented
11#14258C212yesmaximal, complemented
12#14257C212yesmaximal, complemented
13#13926C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic, complemented
15C22×#124C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2212yes—
23#2538—28no—
24#2537—28no—
25C2×#396C2214yes—
26C2×#394C2214yes—
27C4×#124—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30#2293—28no—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#384C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2214yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2214yes—
50C2×C43C2211yescharacteristic
51#1443C2218yes—
52#1441C2214yes—
53#1441C2218yes—
54#1440C2218yes—
55#1440C2218yes—
56#1439C2214yes—
57#1439C2218yes—
58#1439C2218yes—
59#1438C2218yes—
60#1438C2218yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C23×Q8C2311yescharacteristic
68C23×D4C2311yescharacteristic
69C23×D4C2311yescharacteristic
70C23×D4C2314yes—
71C24×C4C2312yes—
72C24×C4C2312yes—
73C2×#124—24no—
74C2×#124—24no—
75C2×#124—24no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—28no—
78C2×C22⋊Q8—44no—
79C2×#121—28no—
80C2×#121—28no—
81C2×#121—28no—
82C2×#121—48no—
83C2×#121—44no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4D418yes—
93C2×#117—28no—
94C2×#117—44no—
95C2×#117—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2318yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#396—216no—
128#394—216no—
129#391—24no—
130#391—44no—
131#390—28no—
132#390—28no—
133#390—24no—
134#390—44no—
135#389—28no—
136#389—28no—
137#388—216no—
138#388—216no—
139#384—24no—
140#384—24no—
141#382—24no—
142#382—48no—
143#380—28no—
144#380—28no—
145#380—28no—
146#380—48no—
147C4×C4⋊C4—24no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—216no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C43D418yescomplemented
162C25C2412yes—
163C25C2412yes—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8#4114yes—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—44no—
184C22×D4—48no—
185C22×D4—48no—
186C22×D4C2×D414yes—
187C22×D4#4114yes—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4C2×D418yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212#124—232no—
213C22⋊Q8—432no—
214#121—432no—
215#121—432no—
216C4×Q8—416no—
217C4×D4—232no—
218C4×D4—416no—
219#117—432no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4C2×D418yes—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D418yes—
303C2×C42C2×D418yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42#4114yes—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—48no—
314#95—48no—
315#95—48no—
316#95—48no—
317#95—48no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24C22×D414yes—
339C24C2×#4114yes—
340C24C2511yescharacteristic
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—88no—
363C2×D4—816no—
364C2×D4—816no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+414yes—
449C4⋊C4—28no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C22⋊C4—232no—
463C22⋊C4—216no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—48no—
471C22⋊C4—88no—
472C22⋊C4—816no—
473C42—28no—
474C42—28no—
475C42—216no—
476C42—48no—
477C42—48no—
478C42—48no—
479C42#121116yes—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23C23×D411yescharacteristic
527C23C23×D411yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4111yescharacteristic
531C23C2×H(5,𝔽2)12yes—
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×2−1+412yes—
535C23#58512yes—
536C23#58512yes—
537Q8—416no—
538D4—416no—
539D4—416no—
540D4—832no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
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—42no—
572C2×C4—42no—
573C2×C4—432no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—816no—
602C2×C4C2×#12114yes—
603C2×C4C2×#12114yes—
604C2×C4C2×#12114yes—
605C2×C4#52514yes—
606C2×C4#53514yes—
607C2×C4#53514yes—
608C2×C4#53614yes—
609C2×C4#53718yes—
610C2×C4#54218yes—
611C2×C4#54614yes—
612C2×C4#54614yes—
613C2×C4#55314yes—
614C2×C4#55314yes—
615C22—44no—
616C22—48no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—48no—
627C22—82no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22C22×#12111yescharacteristic
638C22C2×#52511yescharacteristic
639C22C2×#53511yescharacteristic
640C22C2×#53511yescharacteristic
641C22C2×#53611yescharacteristic
642C22C2×#53712yes—
643C22#344912yes—
644C22C2×#54212yes—
645C22C2×#54611yescharacteristic
646C22C2×#54611yescharacteristic
647C22C2×#55311yescharacteristic
648C22C2×#55311yescharacteristic
649C22#350011yescharacteristic
650C22#350011yescharacteristic
651C22#351212yes—
652C22#351412yes—
653C22#351512yes—
654C22#351712yes—
655C22#351812yes—
656C22#354612yes—
657C22#354612yes—
658C22#354712yes—
659C22#354712yes—
660C4—24no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—82no—
666C4—84no—
667C4—84no—
668C4—82no—
669C4—84no—
670C4#253518yes—
671C4#262018yes—
672C2—84no—
673C2—84no—
674C2C2×#262011yescharacteristic
675C2C2×#253511yescharacteristic
676C2#3518912yes—
677C2#3532811yescharacteristic
678C2#3762112yes—
679C2#3895511yescharacteristic
680C2#3895611yescharacteristic
681C2#3895711yescharacteristic
682C2#3895811yescharacteristic
683C2#3919312yes—
684C2#3919412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2538C2semidirect
3. C2×C4×#124C2semidirect
4. C2×#2293C2semidirect
5. #16817C2semidirect
6. #15233C2semidirect
7. #15220C2semidirect
8. #15219C2semidirect
9. #15215C2semidirect
10. #14687C2semidirect
11. #14258C2semidirect
12. #14257C2semidirect
13. #13926C2semidirect
14. #13906C2semidirect
161. C43D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25372
C2×C4×#1242
C2×#22932
#168172
#152332
#152202
#152192
#152152
#146872
#142582
#142572
#139262
#139062