Finite Group Databasebeta

← #4417008

Subgroups — #4417008

Order 512.

3082 conjugacy classes of subgroups.

The subgroup lattice has 775 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16904C212yesmaximal, complemented
2#16902C212yesmaximal, complemented
3#16825C212yesmaximal, complemented
4#16824C212yesmaximal, complemented
5#16298C214yesmaximal
6#15223C212yesmaximal, complemented
7#15220C212yesmaximal, complemented
8#15220C212yesmaximal, complemented
9#15135C212yesmaximal, complemented
10#15131C212yesmaximal, complemented
11#15131C212yesmaximal, complemented
12#14680C211yesmaximal, characteristic
13#14680C211yesmaximal, characteristic
14#14637C214yesmaximal, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C2×#399C2214yes—
18C2×#396C2214yes—
19C2×#396C2214yes—
20C2×#389C2212yescomplemented
21C2×#389C2212yescomplemented
22C2×#389C2212yescomplemented
23C2×#389C2212yescomplemented
24C2×#384C2212yescomplemented
25C2×#382C2212yescomplemented
26C2×#380C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#379C2212yes—
35C2×#379C2212yes—
36C2×#379C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41#1445C2214yescomplemented
42#1445C2214yescomplemented
43#1445C2214yescomplemented
44#1444C2214yescomplemented
45#1443C2214yescomplemented
46#1443C2214yescomplemented
47#1441C2214yescomplemented
48#1441C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62C4⋊#95C2212yescomplemented
63C4⋊#95C2212yescomplemented
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C26C2311yescharacteristic
76C24×C4—42no—
77C24×C4—42no—
78C24×C4—44no—
79C24×C4—44no—
80C24×C4Q818yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113#399—48no—
114#396—216no—
115#396—48no—
116#389—44no—
117#389—44no—
118#389—44no—
119#389—44no—
120#384—44no—
121#382—44no—
122#380—44no—
123#380—44no—
124#380—42no—
125#380—44no—
126#380—42no—
127#380—44no—
128#380—42no—
129#380—42no—
130#379—28no—
131#379—44no—
132#379—44no—
133C4×C22⋊C4—44no—
134C4×C22⋊C4—42no—
135C4×C22⋊C4—44no—
136C4×C22⋊C4—42no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25—24no—
164C25—22no—
165C25—22no—
166C25—22no—
167C25—22no—
168C25—41no—
169C25—41no—
170C25—42no—
171C25—42no—
172C25—42no—
173C25C2×Q814yes—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—416no—
190C23×C4—416no—
191C23×C4—416no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4—41no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—432no—
205C23×C4—432no—
206C23×C4C2×Q814yes—
207C23×C4C2×Q814yes—
208C23×C4C2×Q814yes—
209C23×C4C2×Q814yes—
210C23×C4C2×Q814yes—
211C23×C4C2×Q814yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
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—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4#4118yes—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302#95—28no—
303#95—44no—
304#95—41no—
305#95—42no—
306#95—41no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—41no—
311#95—41no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—44no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328C24—22no—
329C24—22no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—28no—
354C24—28no—
355C24—28no—
356C24—44no—
357C24—44no—
358C24—44no—
359C24—44no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—44no—
365C24—44no—
366C24—44no—
367C24—44no—
368C24—48no—
369C24—48no—
370C24—48no—
371C24—48no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24C22×Q812yes—
377C24C22×Q812yes—
378C24C22×Q812yes—
379C24C2×#4114yes—
380C24H(5,𝔽2)12yes—
381C24H(5,𝔽2)12yes—
382C242−1+412yes—
383C242−1+412yes—
384C24C2511yescharacteristic
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
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—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—4128no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4C22×Q812yes—
483C22×C4C22×Q812yes—
484C22×C4C22×Q812yes—
485C22×C4C22×Q812yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)14yes—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C4⋊C4—88no—
501C4⋊C4—88no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C42—48no—
517C42—48no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—416no—
559C23—416no—
560C23—416no—
561C23—416no—
562C23—416no—
563C23—416no—
564C23—416no—
565C23—432no—
566C23—432no—
567C23—432no—
568C23—432no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23#53314yes—
596C23#54414yes—
597C23#54414yes—
598C23C23×Q811yescharacteristic
599C23C22×#4112yes—
600C23C2×H(5,𝔽2)11yescharacteristic
601C23C2×H(5,𝔽2)11yescharacteristic
602C23C2×H(5,𝔽2)12yes—
603C23C2×H(5,𝔽2)12yes—
604C23C2×H(5,𝔽2)11yescharacteristic
605C23C2×H(5,𝔽2)11yescharacteristic
606C23C2×2−1+411yescharacteristic
607C23C2×2−1+411yescharacteristic
608C23#58512yes—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—416no—
625C2×C4—416no—
626C2×C4—416no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—416no—
642C2×C4—416no—
643C2×C4—416no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—42no—
686C2×C4#54314yes—
687C2×C4#54314yes—
688C2×C4#54314yes—
689C2×C4#56018yes—
690C2×C4#56018yes—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—24no—
697C22—24no—
698C22—24no—
699C22—24no—
700C22—464no—
701C22—42no—
702C22—42no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—42no—
723C22—42no—
724C22—42no—
725C22C2×#53311yescharacteristic
726C22C2×#54311yescharacteristic
727C22C2×#54311yescharacteristic
728C22C2×#54311yescharacteristic
729C22C2×#54411yescharacteristic
730C22C2×#54411yescharacteristic
731C22#345811yescharacteristic
732C22#350512yes—
733C22#350612yes—
734C22#351912yes—
735C22#351912yes—
736C22#352012yes—
737C22#352012yes—
738C22C2×#56012yes—
739C22C2×#56012yes—
740C22#353611yescharacteristic
741C22#353611yescharacteristic
742C22#353711yescharacteristic
743C22#353811yescharacteristic
744C22#354112yes—
745C22#354112yes—
746C22#354212yes—
747C22#354212yes—
748C4—44no—
749C4—44no—
750C4—42no—
751C4—42no—
752C4—42no—
753C4—84no—
754C4—82no—
755C4—84no—
756C4—84no—
757C4—84no—
758C4—82no—
759C4#284218yes—
760C2—44no—
761C2—44no—
762C2—44no—
763C2#3503311yescharacteristic
764C2#3507911yescharacteristic
765C2#3507911yescharacteristic
766C2C2×#284211yescharacteristic
767C2#3747611yescharacteristic
768C2#3747611yescharacteristic
769C2#3747711yescharacteristic
770C2#4559812yes—
771C2#4560012yes—
772C2#4562312yes—
773C2#4562412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16904C2semidirect
2. #16902C2semidirect
3. #16825C2semidirect
4. #16824C2semidirect
6. #15223C2semidirect
7. #15220C2semidirect
8. #15220C2semidirect
9. #15135C2semidirect
10. #15131C2semidirect
11. #15131C2semidirect
14. #14637C2semidirect
20. C2×#389C22semidirect
21. C2×#389C22semidirect
22. C2×#389C22semidirect
23. C2×#389C22semidirect
24. C2×#384C22semidirect
25. C2×#382C22semidirect
41. #1445C22semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1444C22semidirect
45. #1443C22semidirect
46. #1443C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
#169042
#169022
#168252
#168242
#162982
#152232
#152202
#151352
#151312
#146802
#146372
C4×#3792