Finite Group Databasebeta

← C2×#28850 #6459879

Subgroups — C2×#28850 #6459879

Order 512.

2120 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic, complemented
2C2×#2420C211yesmaximal, characteristic
3#28850C2116yesmaximal, direct product factor
4C2×#2094C211yesmaximal, characteristic, complemented
5C2×#2085C211yesmaximal, characteristic, complemented
6C2×#2077C211yesmaximal, characteristic, complemented
7C2×#2033C211yesmaximal, characteristic
8C2×#1944C211yesmaximal, characteristic, complemented
9C2×#1936C211yesmaximal, characteristic, complemented
10C2×#1919C211yesmaximal, characteristic, complemented
11C2×#1893C211yesmaximal, characteristic
12C2×#1864C211yesmaximal, characteristic, complemented
13C2×#1836C211yesmaximal, characteristic, complemented
14C2×#1807C211yesmaximal, characteristic, complemented
15C2×#1762C211yesmaximal, characteristic, complemented
16C2×#1752C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×#126—21no—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×#97C2211yescharacteristic
32#2496—24no—
33#2496C2218yescomplemented
34#2420—24no—
35#2420C2218yes—
36C2×#400—21no—
37C2×#397—21no—
38C2×#397—21no—
39C2×#394—21no—
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#386—21no—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#381—21no—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2094C2218yescomplemented
55#2085C2218yescomplemented
56#2077C2218yescomplemented
57#2033C2218yes—
58#1944C2218yescomplemented
59#1936C2218yescomplemented
60#1919C2218yescomplemented
61#1893C2218yes—
62#1864C2218yescomplemented
63#1836C2218yescomplemented
64#1807C2218yescomplemented
65#1762C2218yescomplemented
66#1752C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C23×C8—24no—
82C23×C8—24no—
83C23×C8—42no—
84C23×C8C2311yescharacteristic
85C23×C8C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×#126—28no—
90C2×#126—48no—
91C2×#121—22no—
92C2×#121—24no—
93C2×#121—42no—
94C2×#121C2314yes—
95C2×C4×D4—24no—
96C2×C4×D4—22no—
97C2×C4×D4—42no—
98C2×C4×D4C2314yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4D411yescharacteristic
119C22×C22⋊C4D411yescharacteristic
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C2×#107—42no—
124C2×#107C2314yes—
125C2×#106—24no—
126C2×#106—24no—
127C2×#106—42no—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#102—24no—
132C2×#102—24no—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8—24no—
140C2×C22⋊C8C2314yes—
141C2×#97—24no—
142C2×#97—42no—
143C2×#97C2314yes—
144C2×#97C2314yes—
145#400—28no—
146#397—28no—
147#397—28no—
148#394—28no—
149#390—22no—
150#390C2314yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#387—28no—
158#386—28no—
159#384—22no—
160#384C2314yes—
161#382—22no—
162#382C2314yes—
163#381—28no—
164#380—22no—
165#380—22no—
166#380—22no—
167#380C2314yes—
168#380C2314yes—
169#380C2314yes—
170C4×C4⋊C4—22no—
171C4×C4⋊C4C2314yes—
172C4×C22⋊C4D414yes—
173C4×C22⋊C4C2314yes—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95D411yescharacteristic
184C2×#95D411yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25C2×D411yescharacteristic
197C22×D4—24no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—82no—
207C22×D4#4114yes—
208C22×D4C2412yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—82no—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—28no—
244C22×C8—28no—
245C22×C8—44no—
246C22×C8—416no—
247C22×C8#4114yes—
248C22×C8#4114yes—
249C22×C8#4114yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254#126—432no—
255#121—28no—
256#121—44no—
257#121—48no—
258C4×D4—28no—
259C4×D4—44no—
260C4×D4—48no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—82no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4C2×D414yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327#107—44no—
328#106—28no—
329#106—28no—
330#106—44no—
331#102—28no—
332#102—28no—
333#102—28no—
334#102—28no—
335C22⋊C8—28no—
336#97—28no—
337#97—44no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95C2×D414yes—
348#95C2×D414yes—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24AGL(1,ℤ/8)14yes—
364C24C22×D412yes—
365C24C22×D411yescharacteristic
366C2×D4—28no—
367C2×D4—24no—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—84no—
381C2×D4—816no—
382C2×D4C2×#4114yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
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—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
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—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—84no—
442C22×C4—816no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456C2×C8—28no—
457C2×C8—28no—
458C2×C8—28no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—416no—
469C2×C8—44no—
470C2×C8C2×#4114yes—
471C2×C8C2×#4114yes—
472C2×C8C2×#4114yes—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—84no—
499C22⋊C4—28no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—48no—
515C23—28no—
516C23—28no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23#53212yes—
543C23#54612yes—
544C23#54712yes—
545C23C2×AGL(1,ℤ/8)14yes—
546C23C2×AGL(1,ℤ/8)12yes—
547C23#57512yes—
548C23#57712yes—
549C23#57712yes—
550C23C23×D411yescharacteristic
551D4—24no—
552D4—24no—
553D4—42no—
554D4—42no—
555D4—48no—
556D4—816no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—816no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4#52914yes—
612C2×C4#53212yes—
613C2×C4#53914yes—
614C2×C4#54612yes—
615C2×C4#54712yes—
616C2×C4#55114yes—
617C2×C4#55214yes—
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C8—24no—
625C8—24no—
626C8—24no—
627C8—42no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—416no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—816no—
644C22#255314yes—
645C22#317414yes—
646C22#318014yes—
647C22#330614yes—
648C22#331114yes—
649C22#334514yes—
650C22#334814yes—
651C22C2×#53211yescharacteristic
652C22C2×#54611yescharacteristic
653C22C2×#54711yescharacteristic
654C22C22×AGL(1,ℤ/8)11yescharacteristic
655C22C2×#57511yescharacteristic
656C22C2×#57711yescharacteristic
657C22C2×#57711yescharacteristic
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#255314yes—
671C4C2×#52911yescharacteristic
672C4C2×#53911yescharacteristic
673C4C2×#55111yescharacteristic
674C4C2×#55211yescharacteristic
675C2—44no—
676C2—84no—
677C2#2885018yesdirect product factor
678C2C2×#255311yescharacteristic
679C2C2×#317411yescharacteristic
680C2C2×#318011yescharacteristic
681C2C2×#330611yescharacteristic
682C2C2×#331111yescharacteristic
683C2C2×#334511yescharacteristic
684C2C2×#334811yescharacteristic
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
1. C2×#2496C2semidirect
3. #28850C2direct (N × H)
4. C2×#2094C2semidirect
5. C2×#2085C2semidirect
6. C2×#2077C2semidirect
8. C2×#1944C2semidirect
9. C2×#1936C2semidirect
10. C2×#1919C2semidirect
12. C2×#1864C2semidirect
13. C2×#1836C2semidirect
14. C2×#1807C2semidirect
15. C2×#1762C2semidirect
16. C2×#1752C2semidirect
33. #2496C22semidirect
54. #2094C22semidirect
55. #2085C22semidirect
56. #2077C22semidirect
58. #1944C22semidirect
59. #1936C22semidirect
60. #1919C22semidirect
62. #1864C22semidirect
63. #1836C22semidirect
64. #1807C22semidirect
65. #1762C22semidirect
66. #1752C22semidirect
677. C2#28850direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#24202
#288502
C2×#20942
C2×#20852
C2×#20772
C2×#20332
C2×#19442
C2×#19362
C2×#19192
C2×#18932
C2×#18642
C2×#18362
C2×#18072
C2×#17622
C2×#17522