Finite Group Databasebeta

← #1013911

Subgroups — #1013911

Order 512.

3490 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2735C212yesmaximal, complemented
2#17121C212yesmaximal, complemented
3#16589C214yesmaximal, complemented
4#16298C212yesmaximal, complemented
5#15417C212yesmaximal, complemented
6#15282C214yesmaximal, complemented
7#15136C212yesmaximal, complemented
8#15135C211yesmaximal, characteristic, complemented
9#15134C212yesmaximal, complemented
10#14141C212yesmaximal, complemented
11#14001C212yesmaximal, complemented
12#13902C212yesmaximal
13#13902C211yesmaximal, characteristic
14#13876C212yesmaximal, complemented
15C4×#397C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17#2735—216no—
18C2×#399C2212yes—
19C2×#397C2211yescharacteristic
20C2×#397C2214yescomplemented
21C2×#397C2214yescomplemented
22C2×#396C2214yescomplemented
23C2×#396C2214yescomplemented
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2212yescomplemented
31C2×#388C2214yes—
32C2×#388C2214yescomplemented
33C2×#388C2214yescomplemented
34C2×#388C2214yescomplemented
35C2×#384C2212yescomplemented
36C2×#384C2211yescharacteristic, complemented
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×#379C2212yes—
40C2×#379C2214yes—
41C2×#379C2214yes—
42C2×#379C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic, complemented
44C2×C4×C4⋊C4C2212yescomplemented
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47#1445C2212yescomplemented
48#1445C2211yescharacteristic, complemented
49#1444C2212yescomplemented
50#1443C2212yescomplemented
51#1442C2211yescharacteristic, complemented
52#1442C2214yescomplemented
53#1440C2212yes—
54#1440C2214yescomplemented
55#1439C2212yes—
56#1439C2211yescharacteristic
57#1439C2214yes—
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1438C2214yes—
66#1438C2212yes—
67#1438C2212yes—
68#1438C2214yes—
69#1437C2214yescomplemented
70#1436C2214yescomplemented
71C4⋊#95C2212yescomplemented
72C4⋊#95C2211yescharacteristic, complemented
73C4⋊#95C2212yescomplemented
74C4×#95C2212yescomplemented
75C4×#95C2211yescharacteristic, complemented
76C26C2312yes—
77C23×D4—28no—
78C23×D4—24no—
79C23×D4—42no—
80C23×D4—44no—
81C23×D4—42no—
82C23×D4C2312yes—
83C23×D4C2312yes—
84C24×C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118#399—28no—
119#397—216no—
120#397—48no—
121#397D418yescomplemented
122#396—216no—
123#396—48no—
124#390—28no—
125#390—28no—
126#390—44no—
127#390—44no—
128#390—44no—
129#390—44no—
130#389—44no—
131#388—216no—
132#388—216no—
133#388—216no—
134#388—48no—
135#384—42no—
136#384—44no—
137#380—44no—
138#380—42no—
139#379—28no—
140#379—216no—
141#379—44no—
142#379—48no—
143C4×C4⋊C4—42no—
144C4×C4⋊C4—44no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—42no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C25—28no—
170C25—24no—
171C25—24no—
172C25—24no—
173C25—42no—
174C25—44no—
175C25—42no—
176C25—44no—
177C25—42no—
178C25—44no—
179C25—42no—
180C25C2414yes—
181C25C2411yescharacteristic
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—432no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—416no—
195C22×D4—416no—
196C22×D4—432no—
197C22×D4—416no—
198C22×D4#4118yes—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4#4114yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
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—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C2×D414yescomplemented
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
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×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D418yes—
298C2×C22⋊C4C2×D414yes—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—44no—
311C2×C42—44no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—42no—
322#95—42no—
323#95—41no—
324#95—41no—
325#95—41no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95C2×D414yescomplemented
333#95C2×D418yescomplemented
334C24—22no—
335C24—22no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—21no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—416no—
357C24—48no—
358C24—416no—
359C24—416no—
360C24—48no—
361C24—416no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—88no—
366C24—84no—
367C24—88no—
368C24—84no—
369C24—84no—
370C24—88no—
371C24—84no—
372C24—88no—
373C24—88no—
374C24—84no—
375C24—88no—
376C24—88no—
377C24—84no—
378C24—88no—
379C24C22×D412yes—
380C24C2×#4118yes—
381C24C2×#4112yes—
382C242−1+412yes—
383C24C2511yescharacteristic
384C2×D4—216no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—416no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—416no—
391C2×D4—48no—
392C2×D4—4128no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—28no—
432C22×C4—216no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)14yes—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—84no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C4⋊C4—84no—
490C4⋊C4—88no—
491C22⋊C4—216no—
492C22⋊C4—44no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—44no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—84no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C42—48no—
510C42—44no—
511C42—48no—
512C42—44no—
513C42—88no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—432no—
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—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—832no—
573C23—816no—
574C23—816no—
575C23—816no—
576C23—832no—
577C23—832no—
578C23—832no—
579C23—816no—
580C23—816no—
581C23—816no—
582C23#54714yes—
583C23C23×D411yescharacteristic
584C23C22×#4111yescharacteristic
585C23C22×#4112yes—
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)11yescharacteristic
588C23C2×H(5,𝔽2)12yes—
589C23C2×H(5,𝔽2)12yes—
590C23C2×H(5,𝔽2)11yescharacteristic
591C23C2×2−1+411yescharacteristic
592C23#58512yes—
593C23#58511yescharacteristic
594D4—432no—
595D4—432no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—416no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—44no—
658C2×C4—41no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—41no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4#53814yes—
672C2×C4#53818yes—
673C2×C4#55118yes—
674C2×C4#55114yes—
675C2×C4#56014yes—
676C22—28no—
677C22—24no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—44no—
687C22—48no—
688C22—48no—
689C22—48no—
690C22—48no—
691C22—48no—
692C22—48no—
693C22—48no—
694C22—864no—
695C22—84no—
696C22—84no—
697C22—84no—
698C22—84no—
699C22—84no—
700C22—84no—
701C22—84no—
702C22—84no—
703C22C2×#53812yes—
704C22C2×#53811yescharacteristic
705C22C2×#54711yescharacteristic
706C22D4×#4112yes—
707C22C2×#55111yescharacteristic
708C22C2×#55112yes—
709C22#349412yes—
710C22#349611yescharacteristic
711C22#349812yes—
712C22#349811yescharacteristic
713C22#350611yescharacteristic
714C22#351912yes—
715C22#352012yes—
716C22#352112yes—
717C22C2×#56011yescharacteristic
718C22#353611yescharacteristic
719C22#353711yescharacteristic
720C22#354612yes—
721C22#354612yes—
722C22#355111yescharacteristic
723C22#355812yes—
724C22#356112yes—
725C22#356111yescharacteristic
726C4—42no—
727C4—44no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—44no—
732C4—44no—
733C4—81no—
734C4—82no—
735C4—84no—
736C4—84no—
737C4—84no—
738C4#273518yescomplemented
739C2—44no—
740C2—88no—
741C2#3937511yescharacteristic
742C2#3940512yes—
743C2#4263412yes—
744C2#4298211yescharacteristic
745C2#4490112yes—
746C2#4537711yescharacteristic
747C2#4548711yescharacteristic
748C2C2×#273511yescharacteristic
749C2#5301811yescharacteristic
750C2#5927312yes—
751C2#5984311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2735C2semidirect
2. #17121C2semidirect
3. #16589C2semidirect
4. #16298C2semidirect
5. #15417C2semidirect
6. #15282C2semidirect
7. #15136C2semidirect
8. #15135C2semidirect
9. #15134C2semidirect
10. #14141C2semidirect
11. #14001C2semidirect
14. #13876C2semidirect
15. C4×#397C2semidirect
20. C2×#397C22semidirect
21. C2×#397C22semidirect
22. C2×#396C22semidirect
23. C2×#396C22semidirect
30. C2×#389C22semidirect
32. C2×#388C22semidirect
33. C2×#388C22semidirect
34. C2×#388C22semidirect
35. C2×#384C22semidirect
36. C2×#384C22semidirect
43. C2×C4×C4⋊C4C22semidirect
44. C2×C4×C4⋊C4C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
54. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
69. #1437C22semidirect
70. #1436C22semidirect
71. C4⋊#95C22semidirect
72. C4⋊#95C22semidirect
73. C4⋊#95C22semidirect
74. C4×#95C22semidirect
75. C4×#95C22semidirect
121. #397D4semidirect
244. C2×C4⋊C4C2×D4semidirect
332. #95C2×D4semidirect
333. #95C2×D4semidirect
738. C4#2735semidirect

Maximal subgroups

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

subgroupindex
C2×#27352
#171212
#165892
#162982
#154172
#152822
#151362
#151352
#151342
#141412
#140012
#139022
#138762
C4×#3972