Finite Group Databasebeta

← #1013910

Subgroups — #1013910

Order 512.

3266 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17234C212yesmaximal, complemented
2#17024C214yesmaximal, complemented
3#16391C212yesmaximal, complemented
4#16047C212yesmaximal, complemented
5#15284C214yesmaximal, complemented
6#15224C212yesmaximal, complemented
7#15222C212yesmaximal, complemented
8#15135C211yesmaximal, characteristic, complemented
9#14366C212yesmaximal, complemented
10#14141C212yesmaximal, complemented
11#14001C212yesmaximal, complemented
12#13902C212yesmaximal
13#13902C211yesmaximal, characteristic
14#13875C212yesmaximal, complemented
15C4×#397C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C2×#399C2212yes—
18C2×#397C2211yescharacteristic
19C2×#390C2212yes—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#389C2212yescomplemented
26C2×#388C2214yescomplemented
27C2×#388C2214yescomplemented
28C2×#388C2214yescomplemented
29C2×#384C2211yescharacteristic, complemented
30C2×#382C2212yescomplemented
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×#379C2212yes—
34C2×#379C2214yes—
35C2×#379C2214yes—
36C2×#379C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic, complemented
38C2×C4×C4⋊C4C2212yescomplemented
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41#1445C2212yescomplemented
42#1445C2211yescharacteristic, complemented
43#1444C2212yescomplemented
44#1443C2212yescomplemented
45#1443C2214yescomplemented
46#1442C2211yescharacteristic, complemented
47#1440C2214yes—
48#1440C2212yes—
49#1440C2214yescomplemented
50#1440C2214yescomplemented
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1440C2214yescomplemented
54#1439C2212yes—
55#1439C2211yescharacteristic
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1438C2214yes—
64#1438C2212yes—
65#1438C2212yes—
66#1438C2214yes—
67#1437C2214yes—
68#1437C2214yescomplemented
69C4⋊#95C2211yescharacteristic, complemented
70C4⋊#95C2212yescomplemented
71C4⋊#95C2212yescomplemented
72#1434C2214yescomplemented
73C4×#95C2211yescharacteristic, complemented
74C4×#95C2212yescomplemented
75C26C2312yes—
76C23×D4—28no—
77C23×D4—24no—
78C23×D4—42no—
79C23×D4—44no—
80C23×D4—42no—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C24×C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
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⋊C4C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115#399—44no—
116#397—24no—
117#390—28no—
118#390—28no—
119#390—44no—
120#390—44no—
121#390—44no—
122#390—44no—
123#389—44no—
124#388—48no—
125#388—48no—
126#388—48no—
127#384—42no—
128#382—44no—
129#380—44no—
130#380—42no—
131#379—28no—
132#379—44no—
133#379—48no—
134#379—48no—
135C4×C4⋊C4—42no—
136C4×C4⋊C4—44no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—42no—
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2314yes—
163C25—28no—
164C25—24no—
165C25—24no—
166C25—24no—
167C25—42no—
168C25—44no—
169C25—42no—
170C25—44no—
171C25—42no—
172C25—44no—
173C25—42no—
174C25C2414yes—
175C25C2411yescharacteristic
176C22×D4—28no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—432no—
185C22×D4—44no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—416no—
189C22×D4—416no—
190C22×D4—432no—
191C22×D4—416no—
192C22×D4#4118yes—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4#4114yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
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#4114yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—44no—
299C2×C42—44no—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—44no—
308#95—42no—
309#95—41no—
310#95—42no—
311#95—41no—
312#95—41no—
313#95—42no—
314#95—44no—
315#95—42no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—42no—
324C24—22no—
325C24—22no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—416no—
347C24—48no—
348C24—416no—
349C24—416no—
350C24—48no—
351C24—416no—
352C24—48no—
353C24—48no—
354C24—48no—
355C24—88no—
356C24—84no—
357C24—84no—
358C24—88no—
359C24—84no—
360C24—88no—
361C24—84no—
362C24—88no—
363C24—88no—
364C24—84no—
365C24—88no—
366C24—88no—
367C24—84no—
368C24—88no—
369C24C2×#4118yes—
370C24C2×#4112yes—
371C24C2×#4112yes—
372C24H(5,𝔽2)12yes—
373C24C2511yescharacteristic
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—416no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—416no—
381C2×D4—48no—
382C2×D4—4128no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—216no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4114yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)14yes—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—88no—
479C22⋊C4—44no—
480C22⋊C4—416no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—84no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C42—48no—
498C42—44no—
499C42—48no—
500C42—44no—
501C42—88no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—432no—
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—832no—
561C23—816no—
562C23—816no—
563C23—816no—
564C23—832no—
565C23—832no—
566C23—832no—
567C23—816no—
568C23—816no—
569C23—816no—
570C23#55514yes—
571C23C22×#4111yescharacteristic
572C23C22×#4112yes—
573C23C22×#4111yescharacteristic
574C23C2×H(5,𝔽2)11yescharacteristic
575C23C2×H(5,𝔽2)11yescharacteristic
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)12yes—
579C23C2×H(5,𝔽2)11yescharacteristic
580C23#58512yes—
581C23#58511yescharacteristic
582D4—432no—
583D4—432no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
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—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4#53814yes—
660C2×C4#53818yes—
661C2×C4#55118yes—
662C2×C4#55114yes—
663C2×C4#56014yes—
664C22—24no—
665C22—28no—
666C22—42no—
667C22—44no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—44no—
672C22—42no—
673C22—42no—
674C22—44no—
675C22—48no—
676C22—48no—
677C22—48no—
678C22—48no—
679C22—48no—
680C22—48no—
681C22—48no—
682C22—864no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—84no—
690C22—84no—
691C22C2×#53812yes—
692C22C2×#53811yescharacteristic
693C22C2×#55111yescharacteristic
694C22C2×#55112yes—
695C22C2×#55511yescharacteristic
696C22#348112yes—
697C22#349412yes—
698C22#350511yescharacteristic
699C22#350611yescharacteristic
700C22#350811yescharacteristic
701C22#351312yes—
702C22#351312yes—
703C22#351312yes—
704C22#351412yes—
705C22C2×#56011yescharacteristic
706C22#353711yescharacteristic
707C22#354011yescharacteristic
708C22#354312yes—
709C22#354312yes—
710C22#354612yes—
711C22#354911yescharacteristic
712C22#355811yescharacteristic
713C22#356212yes—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—44no—
719C4—84no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—81no—
724C4—84no—
725C4—84no—
726C4#273518yes—
727C2—44no—
728C2—88no—
729C2#3950311yescharacteristic
730C2#4017612yes—
731C2#4254412yes—
732C2#4297511yescharacteristic
733C2#4501912yes—
734C2#4538011yescharacteristic
735C2#4569711yescharacteristic
736C2C2×#273511yescharacteristic
737C2#5296911yescharacteristic
738C2#5922412yes—
739C2#5994511yescharacteristic
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. #17234C2semidirect
2. #17024C2semidirect
3. #16391C2semidirect
4. #16047C2semidirect
5. #15284C2semidirect
6. #15224C2semidirect
7. #15222C2semidirect
8. #15135C2semidirect
9. #14366C2semidirect
10. #14141C2semidirect
11. #14001C2semidirect
14. #13875C2semidirect
15. C4×#397C2semidirect
25. C2×#389C22semidirect
26. C2×#388C22semidirect
27. C2×#388C22semidirect
28. C2×#388C22semidirect
29. C2×#384C22semidirect
30. C2×#382C22semidirect
37. C2×C4×C4⋊C4C22semidirect
38. C2×C4×C4⋊C4C22semidirect
41. #1445C22semidirect
42. #1445C22semidirect
43. #1444C22semidirect
44. #1443C22semidirect
45. #1443C22semidirect
46. #1442C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
68. #1437C22semidirect
69. C4⋊#95C22semidirect
70. C4⋊#95C22semidirect
71. C4⋊#95C22semidirect
72. #1434C22semidirect
73. C4×#95C22semidirect
74. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
#172342
#170242
#163912
#160472
#152842
#152242
#152222
#151352
#143662
#141412
#140012
#139022
#138752
C4×#3972