Finite Group Databasebeta

← C2×#25036 #6444050

Subgroups — C2×#25036 #6444050

Order 512.

2510 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#468C211yesmaximal, characteristic
2C22×C4×SD16C211yesmaximal, characteristic, complemented
3C2×#2421C211yesmaximal, characteristic, complemented
4C2×#2422C211yesmaximal, characteristic
5#25036C2116yesmaximal, direct product factor
6C2×#2072C211yesmaximal, characteristic, complemented
7C2×#2073C211yesmaximal, characteristic, complemented
8C2×#2036C211yesmaximal, characteristic, complemented
9C2×#2034C211yesmaximal, characteristic
10C2×#1973C211yesmaximal, characteristic, complemented
11C2×#1942C211yesmaximal, characteristic, complemented
12C2×#1865C211yesmaximal, characteristic, complemented
13C2×#1862C211yesmaximal, characteristic, complemented
14C2×#1847C211yesmaximal, characteristic, complemented
15C2×#1809C211yesmaximal, characteristic, complemented
16C2×#1753C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×M16C2211yescharacteristic
19C22×#123—21no—
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C2×#468—28no—
25C2×#468C2218yes—
26C2×C4×SD16—28no—
27C2×C4×SD16C2218yescomplemented
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#107C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×#106C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40C22×#97C2211yescharacteristic
41C22×C4×C8C2211yescharacteristic
42#2422—24no—
43#2422C2218yes—
44#2421—24no—
45#2421C2218yescomplemented
46C2×#400—21no—
47C2×#397—21no—
48C2×#397—21no—
49C2×#394—21no—
50C2×#391C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#388—21no—
55C2×#387—21no—
56C2×#385—21no—
57C2×#384C2211yescharacteristic
58C2×#382C2211yescharacteristic
59C2×#380C2211yescharacteristic
60C2×#380C2211yescharacteristic
61C2×C4×C22⋊C4C2211yescharacteristic
62#2073C2218yescomplemented
63#2072C2218yescomplemented
64#2036C2218yescomplemented
65#2034C2218yes—
66#1973C2218yescomplemented
67#1942C2218yescomplemented
68#1865C2218yescomplemented
69#1862C2218yescomplemented
70#1847C2218yescomplemented
71#1809C2218yescomplemented
72#1753C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16D418yes—
87C22×SD16C2314yes—
88C22×M16—24no—
89C22×M16—24no—
90C22×M16—24no—
91C22×M16C2314yes—
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×#123—28no—
97C2×#123—48no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—42no—
101C2×C22⋊Q8C2314yes—
102C2×#121—22no—
103C2×#121—24no—
104C2×#121—42no—
105C2×#121C2314yes—
106C2×C4×Q8—22no—
107C2×C4×Q8—24no—
108C2×C4×Q8—42no—
109C2×C4×Q8C2314yes—
110C2×C4×D4—24no—
111C2×C4×D4—22no—
112C2×C4×D4—42no—
113C2×C4×D4C2314yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#468—232no—
139C4×SD16—232no—
140C2×#107—24no—
141C2×#107—24no—
142C2×#107—42no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106—24no—
147C2×#106—42no—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102—24no—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8—24no—
163C2×C22⋊C8C2314yes—
164C2×#97—24no—
165C2×#97C2314yes—
166C2×C4×C8—24no—
167C2×C4×C8C2314yes—
168#400—28no—
169#397—28no—
170#397—28no—
171#394—28no—
172#391—22no—
173#391C2314yes—
174#390—22no—
175#390C2314yes—
176#389—22no—
177#389—22no—
178#389C2314yes—
179#389C2314yes—
180#388—28no—
181#387—28no—
182#385—28no—
183#384—22no—
184#384C2314yes—
185#382—22no—
186#382C2314yes—
187#380—22no—
188#380—22no—
189#380C2314yes—
190#380C2314yes—
191C4×C22⋊C4D414yes—
192C4×C22⋊C4C2314yes—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95D411yescharacteristic
202C2×#95D411yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2314yes—
209C25—21no—
210C25C2×D411yescharacteristic
211C22×Q8—21no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8C2×D414yes—
218C22×Q8C2412yes—
219C22×D4—24no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—82no—
229C22×D4C2×D414yes—
230C22×D4C2412yes—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C2×SD16—232no—
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16C2×D4116yes—
260C2×M16—28no—
261C2×M16—28no—
262C2×M16—28no—
263C2×M16—416no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8C2×D414yes—
273C22×C8#4114yes—
274C22×C8#4114yes—
275C22×C8#4114yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280#123—432no—
281C22⋊Q8—28no—
282C22⋊Q8—44no—
283C22⋊Q8—48no—
284#121—28no—
285#121—44no—
286#121—48no—
287C4×Q8—28no—
288C4×Q8—44no—
289C4×Q8—48no—
290C4×D4—28no—
291C4×D4—44no—
292C4×D4—48no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—42no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4C2×D414yes—
346C2×C22⋊C4C2×D414yes—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42—44no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2412yes—
357#107—28no—
358#107—28no—
359#107—44no—
360#106—28no—
361#106—44no—
362#103—28no—
363#103—28no—
364#103—28no—
365#102—28no—
366#102—28no—
367#102—28no—
368C22⋊C8—28no—
369#97—28no—
370C4×C8—28no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95C2×D414yes—
380#95C2×D414yes—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—28no—
401C2×Q8—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8C22×D414yes—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—84no—
426C2×D4—816no—
427C2×D4C22×D414yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—48no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—48no—
485C22×C4—84no—
486C22×C4—816no—
487C22×C4AGL(1,ℤ/8)14yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C42−1+412yes—
500C22×C4C2511yescharacteristic
501SD16—232no—
502M16—416no—
503C2×C8—28no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—44no—
513C2×C8—44no—
514C2×C8#123116yescomplemented
515C2×C8C22×D414yes—
516C2×C8C2×#4114yes—
517C2×C8C2×#4114yes—
518C2×C8C2×#4114yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—48no—
541C4⋊C4—48no—
542C4⋊C4—84no—
543C22⋊C4—28no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—48no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—44no—
573C23—48no—
574C23—42no—
575C23—48no—
576C23—44no—
577C23—44no—
578C23—48no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23#53212yes—
588C23D4212yes—
589C23#54712yes—
590C23C2×AGL(1,ℤ/8)12yes—
591C23#57512yes—
592C23#57712yes—
593C23#57812yes—
594C23C23×D411yescharacteristic
595Q8—24no—
596Q8—24no—
597Q8—48no—
598Q8—42no—
599Q8—42no—
600D4—24no—
601D4—24no—
602D4—48no—
603D4—42no—
604D4—42no—
605D4—816no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—48no—
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—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—816no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4C2×#12314yes—
660C2×C4#53212yes—
661C2×C4#54214yes—
662C2×C4D4212yes—
663C2×C4#54712yes—
664C2×C4#54714yes—
665C2×C4#55314yes—
666C2×C4C2×AGL(1,ℤ/8)14yes—
667C2×C4C23×D411yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×2−1+411yescharacteristic
673C8—24no—
674C8—42no—
675C8—42no—
676C8C2×#12318yescomplemented
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—42no—
686C22—44no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—88no—
692C22—816no—
693C22#242214yes—
694C22#317114yes—
695C22#317814yes—
696C22#326514yes—
697C22#326914yes—
698C22#335114yes—
699C22#335314yes—
700C22C2×#53211yescharacteristic
701C22C2×D4211yescharacteristic
702C22C2×#54711yescharacteristic
703C22C22×AGL(1,ℤ/8)11yescharacteristic
704C22C2×#57511yescharacteristic
705C22C2×#57711yescharacteristic
706C22C2×#57811yescharacteristic
707C4—24no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—44no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4#242214yes—
720C4C22×#12311yescharacteristic
721C4C2×#54211yescharacteristic
722C4C2×#54711yescharacteristic
723C4C2×#55311yescharacteristic
724C2—82no—
725C2—84no—
726C2#2503618yesdirect product factor
727C2C2×#242211yescharacteristic
728C2C2×#317111yescharacteristic
729C2C2×#317811yescharacteristic
730C2C2×#326511yescharacteristic
731C2C2×#326911yescharacteristic
732C2C2×#335111yescharacteristic
733C2C2×#335311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×SD16C2semidirect
3. C2×#2421C2semidirect
5. #25036C2direct (N × H)
6. C2×#2072C2semidirect
7. C2×#2073C2semidirect
8. C2×#2036C2semidirect
10. C2×#1973C2semidirect
11. C2×#1942C2semidirect
12. C2×#1865C2semidirect
13. C2×#1862C2semidirect
14. C2×#1847C2semidirect
15. C2×#1809C2semidirect
16. C2×#1753C2semidirect
27. C2×C4×SD16C22semidirect
45. #2421C22semidirect
62. #2073C22semidirect
63. #2072C22semidirect
64. #2036C22semidirect
66. #1973C22semidirect
67. #1942C22semidirect
68. #1865C22semidirect
69. #1862C22semidirect
70. #1847C22semidirect
71. #1809C22semidirect
72. #1753C22semidirect
514. C2×C8#123semidirect
676. C8C2×#123semidirect
726. C2#25036direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4682
C22×C4×SD162
C2×#24212
C2×#24222
#250362
C2×#20722
C2×#20732
C2×#20362
C2×#20342
C2×#19732
C2×#19422
C2×#18652
C2×#18622
C2×#18472
C2×#18092
C2×#17532