Finite Group Databasebeta

← C2×#28896 #6460086

Subgroups — C2×#28896 #6460086

Order 512.

1946 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic
2C2×#2446C211yesmaximal, characteristic, complemented
3#28896C2116yesmaximal, direct product factor
4C2×#2085C211yesmaximal, characteristic, complemented
5C2×#2066C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic
7C2×#2013C211yesmaximal, characteristic
8C2×#1973C211yesmaximal, characteristic
9C2×#1968C211yesmaximal, characteristic
10C2×#1960C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1928C211yesmaximal, characteristic, complemented
14C2×#1880C211yesmaximal, characteristic
15C2×#1771C211yesmaximal, characteristic, complemented
16C2×C4×#103C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2513—24no—
36#2513C2218yes—
37#2446—24no—
38#2446C2218yescomplemented
39C2×#400—21no—
40C2×#398—21no—
41C2×#397—21no—
42C2×#397—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#388—21no—
52C2×#388—21no—
53C2×#387—21no—
54C2×#387—21no—
55C2×#382C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2085C2218yescomplemented
59#2066C2218yescomplemented
60#2055C2218yes—
61#2013C2218yes—
62#1973C2218yes—
63#1968C2218yes—
64#1960C2218yescomplemented
65#1936C2218yes—
66#1935C2218yescomplemented
67#1928C2218yescomplemented
68#1880C2218yes—
69#1771C2218yescomplemented
70C4×#103C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C22×SD16—24no—
81C22×SD16—24no—
82C22×SD16—24no—
83C22×SD16—42no—
84C22×SD16—42no—
85C22×SD16—42no—
86C22×SD16C2314yes—
87C22×SD16C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—42no—
95C2×C4⋊Q8C2314yes—
96C2×C4×Q8—22no—
97C2×C4×Q8—24no—
98C2×C4×Q8—42no—
99C2×C4×Q8C2314yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
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×C42—21no—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C2×#107—42no—
124C2×#107C2314yes—
125C2×#106—42no—
126C2×#106C2314yes—
127C2×C4⋊C8—24no—
128C2×C4⋊C8C2314yes—
129C2×#103—24no—
130C2×#103—24no—
131C2×#103—24no—
132C2×#103—24no—
133C2×#103—42no—
134C2×#103—42no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102—24no—
142C2×#102—42no—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#97—24no—
146C2×#97—42no—
147C2×#97C2314yes—
148C2×#97C2314yes—
149C2×C4×C8—42no—
150C2×C4×C8C2314yes—
151#400—28no—
152#398—28no—
153#397—28no—
154#397—28no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2314yes—
159#391C2314yes—
160#391C2314yes—
161#390—22no—
162#390—22no—
163#390—22no—
164#390C2314yes—
165#390C2314yes—
166#390C2314yes—
167#389—22no—
168#389—22no—
169#389C2314yes—
170#389C2314yes—
171#388—28no—
172#388—28no—
173#387—28no—
174#387—28no—
175#382—22no—
176#382C2314yes—
177C4×C4⋊C4—22no—
178C4×C4⋊C4D414yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2314yes—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95D411yescharacteristic
193C2×#95D411yescharacteristic
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8#4114yes—
215C22×Q8C2412yes—
216C22×Q8C2412yes—
217C22×D4—21no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×SD16—28no—
240C2×SD16—28no—
241C2×SD16—28no—
242C2×SD16—44no—
243C2×SD16—44no—
244C2×SD16—44no—
245C2×SD16—416no—
246C2×SD16—416no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8#4114yes—
257C22×C8#4114yes—
258C22×C8#4114yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—28no—
264C4⋊Q8—44no—
265C4⋊Q8—48no—
266C4×Q8—28no—
267C4×Q8—44no—
268C4×Q8—48no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—82no—
312C2×C22⋊C4—82no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4—82no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42—82no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331#107—44no—
332#106—44no—
333C4⋊C8—28no—
334#103—28no—
335#103—28no—
336#103—28no—
337#103—28no—
338#103—44no—
339#103—44no—
340#102—28no—
341#102—44no—
342#97—28no—
343#97—44no—
344C4×C8—44no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95C2×D414yes—
357#95C2×D414yes—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24C22×D411yescharacteristic
367C2×Q8—24no—
368C2×Q8—22no—
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8C2×#4114yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
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—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4#13714yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4C2511yescharacteristic
466SD16—416no—
467SD16—416no—
468C2×C8—28no—
469C2×C8—28no—
470C2×C8—28no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—44no—
481C2×C8C2×#4114yes—
482C2×C8C2×#4114yes—
483C2×C8C2×#4114yes—
484C4⋊C4—28no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—84no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C42—24no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—48no—
522C42—84no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23#53212yes—
538C23#54612yes—
539C23#55012yes—
540C23C2×#13712yes—
541C23#57512yes—
542C23#57712yes—
543C23#57712yes—
544C23C23×D411yescharacteristic
545Q8—24no—
546Q8—24no—
547Q8—42no—
548Q8—42no—
549Q8—48no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—84no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—84no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
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—42no—
604C2×C4—42no—
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#53814yes—
614C2×C4#54612yes—
615C2×C4#55012yes—
616C2×C4#55114yes—
617C2×C4#55314yes—
618C2×C4C2×#13714yes—
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C8—24no—
626C8—24no—
627C8—24no—
628C8—42no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—168no—
637C22#255414yes—
638C22#317214yes—
639C22#317414yes—
640C22#332014yes—
641C22#332114yes—
642C22#340014yes—
643C22#340114yes—
644C22C2×#53211yescharacteristic
645C22C2×#54611yescharacteristic
646C22C2×#55011yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57511yescharacteristic
649C22C2×#57711yescharacteristic
650C22C2×#57711yescharacteristic
651C4—24no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#255414yes—
665C4C2×#52911yescharacteristic
666C4C2×#53811yescharacteristic
667C4C2×#55111yescharacteristic
668C4C2×#55311yescharacteristic
669C2—162no—
670C2#2889618yesdirect product factor
671C2C2×#255411yescharacteristic
672C2C2×#317211yescharacteristic
673C2C2×#317411yescharacteristic
674C2C2×#332011yescharacteristic
675C2C2×#332111yescharacteristic
676C2C2×#340011yescharacteristic
677C2C2×#340111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2446C2semidirect
3. #28896C2direct (N × H)
4. C2×#2085C2semidirect
5. C2×#2066C2semidirect
10. C2×#1960C2semidirect
12. C2×#1935C2semidirect
13. C2×#1928C2semidirect
15. C2×#1771C2semidirect
16. C2×C4×#103C2semidirect
38. #2446C22semidirect
58. #2085C22semidirect
59. #2066C22semidirect
64. #1960C22semidirect
66. #1935C22semidirect
67. #1928C22semidirect
69. #1771C22semidirect
70. C4×#103C22semidirect
670. C2#28896direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25132
C2×#24462
#288962
C2×#20852
C2×#20662
C2×#20552
C2×#20132
C2×#19732
C2×#19682
C2×#19602
C2×#19362
C2×#19352
C2×#19282
C2×#18802
C2×#17712
C2×C4×#1032