Finite Group Databasebeta

← #951792

Subgroups — #951792

Order 512.

3082 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2490C212yesmaximal, complemented
2#17115C214yesmaximal, complemented
3#16551C214yesmaximal, complemented
4#16495C214yesmaximal, complemented
5#15423C214yesmaximal, complemented
6#15130C212yesmaximal, complemented
7#15124C212yesmaximal, complemented
8#15123C212yesmaximal, complemented
9#14283C211yesmaximal, characteristic, complemented
10#14272C211yesmaximal, characteristic, complemented
11#14022C212yesmaximal, complemented
12#13875C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×#121C2212yes—
17C22×#121C2212yes—
18C22×C4×D4C2211yescharacteristic
19#2490—216no—
20C2×#397C2214yes—
21C2×#394C2214yes—
22C2×#394C2214yes—
23C2×#394C2214yes—
24C2×#392C2214yes—
25C2×#392C2214yes—
26C2×#392C2214yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2214yes—
31C2×#390C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×#376C2211yescharacteristic
43#1445C2212yes—
44#1443C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2212yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2212yes—
60#1438C2214yes—
61#1438C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C23×D4C2314yes—
71C23×D4C2314yes—
72C23×D4C2314yes—
73C24×C4C2312yes—
74C2×#127—24no—
75C2×#127—42no—
76C2×#127—42no—
77C2×#125—42no—
78C2×#125—42no—
79C2×#125D418yescomplemented
80C2×#121—28no—
81C2×#121—44no—
82C2×#121—44no—
83C2×#121—44no—
84C2×#121—44no—
85C2×#121—44no—
86C2×C4×D4—24no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119#397—216no—
120#394—216no—
121#394—48no—
122#394—48no—
123#392—48no—
124#392—48no—
125#392—48no—
126#390—28no—
127#390—216no—
128#390—28no—
129#390—28no—
130#390—44no—
131#388—216no—
132#388—48no—
133#384—42no—
134#382—42no—
135#380—28no—
136#380—44no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—42no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—28no—
141#376—42no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C25C2414yes—
160C25C2412yes—
161C22×D4—24no—
162C22×D4—28no—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—28no—
166C22×D4—22no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—28no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—42no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212#127—416no—
213#125—416no—
214#121—432no—
215#121—432no—
216C4×D4—416no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C2×D418yes—
238C2×C4⋊C4C2×D4116yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
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—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42C2×D414yes—
303C2×C42#4114yes—
304C2×C42#4114yes—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24H(5,𝔽2)14yes—
352C24C2511yescharacteristic
353C2×D4—48no—
354C2×D4—44no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—416no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—88no—
373C2×D4—88no—
374C2×D4—88no—
375C2×D4—88no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4C22×D418yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)12yes—
466C4⋊C4—216no—
467C4⋊C4—416no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—88no—
474C22⋊C4—416no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C42—28no—
492C42—28no—
493C42—28no—
494C42—48no—
495C42—88no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23C23×D411yescharacteristic
567C23C22×#4111yescharacteristic
568C23C22×#4111yescharacteristic
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×H(5,𝔽2)12yes—
574C23C2×H(5,𝔽2)12yes—
575C23#58512yes—
576C23#58512yes—
577D4—416no—
578D4—832no—
579D4—832no—
580D4—832no—
581D4—816no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—816no—
649C2×C4#53414yes—
650C2×C4#53814yes—
651C2×C4#53814yes—
652C2×C4#55014yes—
653C2×C4#55014yes—
654C2×C4#55514yes—
655C2×C4H(3,𝔽4)14yes—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22C2×#53411yescharacteristic
687C22C2×#53811yescharacteristic
688C22C2×#53811yescharacteristic
689C22#346012yes—
690C22C2×#55011yescharacteristic
691C22C2×#55011yescharacteristic
692C22C2×#55511yescharacteristic
693C22#348812yes—
694C22#348812yes—
695C22#349412yes—
696C22#349412yes—
697C22#349412yes—
698C22C2×H(3,𝔽4)11yescharacteristic
699C22#353711yescharacteristic
700C22#353711yescharacteristic
701C22#353911yescharacteristic
702C22#353911yescharacteristic
703C22#354212yes—
704C22#354312yes—
705C22#354312yes—
706C22#354612yes—
707C22#356212yes—
708C22#356212yes—
709C4—44no—
710C4—42no—
711C4—42no—
712C4—44no—
713C4—48no—
714C4—44no—
715C4—82no—
716C4—84no—
717C4—82no—
718C4—84no—
719C4#273318yes—
720C2—84no—
721C2—164no—
722C2#3728311yescharacteristic
723C2#3751511yescharacteristic
724C2#3777512yes—
725C2#3795412yes—
726C2#4295112yes—
727C2#4297412yes—
728C2#4542311yescharacteristic
729C2#4542311yescharacteristic
730C2C2×#273311yescharacteristic
731C2#5298611yescharacteristic
732C2#5307411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2490C2semidirect
2. #17115C2semidirect
3. #16551C2semidirect
4. #16495C2semidirect
5. #15423C2semidirect
6. #15130C2semidirect
7. #15124C2semidirect
8. #15123C2semidirect
9. #14283C2semidirect
10. #14272C2semidirect
11. #14022C2semidirect
12. #13875C2semidirect
13. #13847C2semidirect
79. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24902
#171152
#165512
#164952
#154232
#151302
#151242
#151232
#142832
#142722
#140222
#138752
#138472