Finite Group Databasebeta

← C2×C4×AΣL(1,𝔽16) #492985

Subgroups — C2×C4×AΣL(1,𝔽16) #492985

Order 512.

2716 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#120C211yesmaximal, characteristic
2C22×AΣL(1,𝔽16)C212yesmaximal
3C2×C4×#100C211yesmaximal, characteristic, complemented
4C2×C4×#99C211yesmaximal, characteristic, complemented
5C4×AΣL(1,𝔽16)C218yesmaximal, direct product factor
6C2×#1390C212yesmaximal, complemented
7C25×C4C411yescharacteristic, complemented
8C22×#120C2211yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C2211yescharacteristic
11C22×#100C2212yes—
12C22×#99C2212yes—
13C2×C4×M16—21no—
14C4×#120—22no—
15C4×#120—22no—
16C4×#120C414yescomplemented
17C4×#120C2214yes—
18C2×#386—21no—
19C2×#386C2211yescharacteristic
20C2×#385—21no—
21C2×#385C411yescharacteristic, complemented
22C2×#379C411yescharacteristic, complemented
23C2×C4×C22⋊C4—21no—
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4C411yescharacteristic, complemented
26C2×AΣL(1,𝔽16)C4116yesdirect product factor
27C2×AΣL(1,𝔽16)C2218yes—
28C4×#100—22no—
29C4×#100C2214yescomplemented
30C4×#99—22no—
31C4×#99C2214yescomplemented
32C2×#343—21no—
33C2×#342—21no—
34#1390C2218yescomplemented
35C26C2×C411yescharacteristic
36C23×D4—21no—
37C23×D4C2311yescharacteristic
38C24×C4—21no—
39C24×C4—21no—
40C24×C4—22no—
41C24×C4—48no—
42C24×C4—41no—
43C24×C4—42no—
44C24×C4—41no—
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C414yescomplemented
48C24×C4D411yescharacteristic
49C22×M16—22no—
50C2×#120—21no—
51C2×#120—21no—
52C2×#120—24no—
53C2×#120C2×C418yes—
54C2×#120C2×C412yes—
55C2×#120C2312yes—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—41no—
59C2×C4×D4—42no—
60C2×C4×D4—41no—
61C2×C4×D4—48no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4D412yes—
64C2×C4×D4D412yes—
65C2×C4×D4C2312yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42—41no—
79C22×C42D411yescharacteristic
80C2×#100—22no—
81C2×#100—24no—
82C2×#100C2×C418yescomplemented
83C2×#100C2314yes—
84C2×#99—22no—
85C2×#99—24no—
86C2×#99C2×C418yescomplemented
87C2×#99C2314yes—
88C4×M16—24no—
89C4×M16—42no—
90C2×#97—41no—
91C2×C4×C8—41no—
92#386—21no—
93#386—21no—
94#386—24no—
95#386—24no—
96#386C2×C412yes—
97#386C2312yes—
98#385—24no—
99#385—24no—
100#385—22no—
101#385C2×C414yescomplemented
102#379—22no—
103#379C2×C414yescomplemented
104C4×C22⋊C4—22no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—24no—
108C4×C22⋊C4—42no—
109C4×C22⋊C4C2×C414yescomplemented
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—41no—
114AΣL(1,𝔽16)C2×C4132yesdirect product factor
115#343—24no—
116#342—24no—
117C25—21no—
118C25—21no—
119C25—22no—
120C25—44no—
121C25—41no—
122C25—41no—
123C25—44no—
124C25—42no—
125C25C4214yescomplemented
126C25C22×C411yescharacteristic
127C25C22×C412yes—
128C25C2×D411yescharacteristic
129C22×D4—22no—
130C22×D4—28no—
131C22×D4—22no—
132C22×D4—41no—
133C22×D4—41no—
134C22×D4—42no—
135C22×D4—44no—
136C22×D4—44no—
137C22×D4C22×C411yescharacteristic
138C22×D4C22×C414yes—
139C22×D4C2×D411yescharacteristic
140C22×D4C2×D411yescharacteristic
141C22×D4#4114yes—
142C22×D4C2411yescharacteristic
143C23×C4—28no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—416no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—416no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—48no—
171C23×C4—48no—
172C23×C4—432no—
173C23×C4—48no—
174C23×C4—41no—
175C23×C4—42no—
176C23×C4—41no—
177C23×C4—41no—
178C23×C4—41no—
179C23×C4—48no—
180C23×C4—42no—
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C412yes—
185C23×C4C22⋊C412yes—
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22×C412yes—
188C23×C4C22×C412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C2×M16—28no—
192C2×M16—24no—
193C2×M16—44no—
194C2×M16—42no—
195C22×C8—42no—
196#120—24no—
197#120—28no—
198#120—28no—
199#120—24no—
200#120C4218yescomplemented
201#120C22×C418yes—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—416no—
211C4×D4—416no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4C4214yescomplemented
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22×C412yes—
244C2×C22⋊C4C22×C412yes—
245C2×C22⋊C4C22×C412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—24no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42—84no—
254C2×C42C22⋊C412yes—
255C2×C42C2×D412yes—
256#100—28no—
257#100—24no—
258#100C22×C418yescomplemented
259#99—28no—
260#99—24no—
261#99C22×C418yescomplemented
262#97—44no—
263C4×C8—44no—
264#95—22no—
265#95—24no—
266#95—24no—
267#95—22no—
268#95—44no—
269C24—24no—
270C24—24no—
271C24—22no—
272C24—24no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—21no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—42no—
282C24—44no—
283C24—42no—
284C24—416no—
285C24—44no—
286C24—44no—
287C24—42no—
288C24—44no—
289C24—42no—
290C24—42no—
291C24—44no—
292C24—42no—
293C24—42no—
294C24—44no—
295C24—416no—
296C24—42no—
297C24—44no—
298C24—44no—
299C24—42no—
300C24—42no—
301C24—44no—
302C24—44no—
303C24—48no—
304C24—44no—
305C24—48no—
306C24—432no—
307C24—44no—
308C24—41no—
309C24—41no—
310C24—42no—
311C24C2×C4212yes—
312C24C2×C4214yescomplemented
313C24C2×C22⋊C411yescharacteristic
314C24C2×C22⋊C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C24#11712yes—
317C24C4×D414yes—
318C24C23×C411yescharacteristic
319C24C22×D411yescharacteristic
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—41no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—432no—
339C2×D4C2×C4212yes—
340C2×D4#11712yes—
341C2×D4C4×D414yes—
342C2×D4C4×D414yes—
343C2×D4C23×C412yes—
344C2×D4C2×#4112yes—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—216no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—48no—
370C22×C4—44no—
371C22×C4—48no—
372C22×C4—416no—
373C22×C4—48no—
374C22×C4—44no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—48no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—416no—
388C22×C4—432no—
389C22×C4—416no—
390C22×C4—416no—
391C22×C4—48no—
392C22×C4—416no—
393C22×C4—44no—
394C22×C4—432no—
395C22×C4—416no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—416no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—48no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—84no—
415C22×C4—82no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4#9911yescharacteristic
419C22×C4#9912yes—
420C22×C4#9911yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C412yes—
423C22×C4C2×C22⋊C412yes—
424C22×C4C2×C22⋊C412yes—
425C22×C4C2×C22⋊C412yes—
426C22×C4C4×D414yes—
427C22×C4C22×D411yescharacteristic
428M16—28no—
429M16—44no—
430M16—48no—
431C2×C8—44no—
432C2×C8—48no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—28no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—48no—
448C22⋊C4—44no—
449C22⋊C4—48no—
450C22⋊C4—42no—
451C22⋊C4—48no—
452C22⋊C4—42no—
453C22⋊C4C2×C4214yescomplemented
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—48no—
459C42—88no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—216no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—21no—
478C23—21no—
479C23—24no—
480C23—41no—
481C23—41no—
482C23—44no—
483C23—48no—
484C23—44no—
485C23—41no—
486C23—44no—
487C23—41no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—48no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—416no—
500C23—44no—
501C23—48no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—48no—
506C23—41no—
507C23—44no—
508C23—44no—
509C23—416no—
510C23—44no—
511C23—48no—
512C23—416no—
513C23—432no—
514C23—416no—
515C23—432no—
516C23—44no—
517C23—48no—
518C23—416no—
519C23—416no—
520C23—416no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—81no—
532C23—81no—
533C23—84no—
534C23C4×C22⋊C412yes—
535C23C4×C22⋊C412yes—
536C23C4×C22⋊C412yes—
537C23C2×#9911yescharacteristic
538C23C2×#9911yescharacteristic
539C23#41012yes—
540C23C22×C4211yescharacteristic
541C23C22×C22⋊C411yescharacteristic
542C23C2×#11711yescharacteristic
543C23C2×C4×D412yes—
544D4—42no—
545D4—42no—
546D4—44no—
547D4—44no—
548D4—42no—
549D4—42no—
550D4—416no—
551D4—416no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—416no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—432no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—416no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—816no—
604C2×C4AΣL(1,𝔽16)14yesdirect product factor
605C2×C4C4×C22⋊C412yes—
606C2×C4C2×#9912yes—
607C2×C4C2×#9912yes—
608C2×C4C2×C4×D412yes—
609C8—48no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—24no—
615C22—24no—
616C22—28no—
617C22—22no—
618C22—21no—
619C22—22no—
620C22—21no—
621C22—22no—
622C22—42no—
623C22—42no—
624C22—41no—
625C22—42no—
626C22—41no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—416no—
633C22—44no—
634C22—44no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—432no—
640C22—44no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—44no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—48no—
651C22—416no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—81no—
656C22—81no—
657C22—84no—
658C22C4×#9912yes—
659C22C4×#9912yes—
660C22C2×AΣL(1,𝔽16)12yes—
661C22#212012yes—
662C22C2×C4×C22⋊C411yescharacteristic
663C22C22×#9911yescharacteristic
664C22C2×#41011yescharacteristic
665C4—22no—
666C4—22no—
667C4—22no—
668C4—44no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—82no—
674C4—88no—
675C4C2×AΣL(1,𝔽16)14yesdirect product factor
676C2—21no—
677C2—21no—
678C2—22no—
679C2—42no—
680C2—48no—
681C2—41no—
682C2—42no—
683C2—41no—
684C2—42no—
685C2C4×AΣL(1,𝔽16)14yesdirect product factor
686C2C2×C4×#9911yescharacteristic
687C2C22×AΣL(1,𝔽16)11yescharacteristic
688C2C2×#212011yescharacteristic
subgroupG/H
centerC22×C4#99
commutatorC23C22×C42
Frattini, cosocleC22×D4C24
socleC23C2×#99
prime radicalC22×#120C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×#100C2semidirect
4. C2×C4×#99C2semidirect
5. C4×AΣL(1,𝔽16)C2direct (N × H)
6. C2×#1390C2semidirect
7. C25×C4C4semidirect
16. C4×#120C4semidirect
21. C2×#385C4semidirect
22. C2×#379C4semidirect
25. C2×C4×C22⋊C4C4semidirect
26. C2×AΣL(1,𝔽16)C4direct (N × H)
29. C4×#100C22semidirect
31. C4×#99C22semidirect
34. #1390C22semidirect
47. C24×C4C2×C4semidirect
82. C2×#100C2×C4semidirect
86. C2×#99C2×C4semidirect
101. #385C2×C4semidirect
103. #379C2×C4semidirect
109. C4×C22⋊C4C2×C4semidirect
114. AΣL(1,𝔽16)C2×C4direct (N × H)
125. C25C42semidirect
200. #120C42semidirect
240. C2×C22⋊C4C42semidirect
258. #100C22×C4semidirect
261. #99C22×C4semidirect
312. C24C2×C42semidirect
453. C22⋊C4C2×C42semidirect
604. C2×C4AΣL(1,𝔽16)direct (N × H)
675. C4C2×AΣL(1,𝔽16)direct (N × H)
685. C2C4×AΣL(1,𝔽16)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1202
C22×AΣL(1,𝔽16)2
C2×C4×#1002
C2×C4×#992
C4×AΣL(1,𝔽16)2
C2×#13902