Finite Group Databasebeta

← C2×#28880 #6460079

Subgroups — C2×#28880 #6460079

Order 512.

1956 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2514C211yesmaximal, characteristic, complemented
2C2×#2448C211yesmaximal, characteristic
3#28880C2116yesmaximal, direct product factor
4C2×#2085C211yesmaximal, characteristic, complemented
5C2×#2066C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic
7C2×#2015C211yesmaximal, characteristic
8C2×#1969C211yesmaximal, characteristic, complemented
9C2×#1968C211yesmaximal, characteristic
10C2×#1958C211yesmaximal, characteristic
11C2×#1934C211yesmaximal, characteristic
12C2×#1933C211yesmaximal, characteristic, complemented
13C2×#1925C211yesmaximal, characteristic
14C2×#1878C211yesmaximal, characteristic, complemented
15C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
16C2×C4×#103C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2514—24no—
36#2514C2218yescomplemented
37#2448—24no—
38#2448C2218yes—
39C2×#401—21no—
40C2×#400—21no—
41C2×#399—21no—
42C2×#397—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#388—21no—
52C2×#388—21no—
53C2×#387—21no—
54C2×#383—21no—
55C2×#382C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2085C2218yescomplemented
59#2066C2218yescomplemented
60#2055C2218yes—
61#2015C2218yes—
62#1969C2218yescomplemented
63#1968C2218yes—
64#1958C2218yes—
65#1934C2218yes—
66#1933C2218yescomplemented
67#1925C2218yes—
68#1878C2218yescomplemented
69C8×C4⋊C4C2218yescomplemented
70C4×#103C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C22×Q16—24no—
81C22×Q16—24no—
82C22×Q16—42no—
83C22×Q16C2314yes—
84C22×SD16—24no—
85C22×SD16—42no—
86C22×SD16—42no—
87C22×SD16C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×#124—24no—
93C2×#124—22no—
94C2×#124—42no—
95C2×#124C2314yes—
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⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C2×#107—42no—
122C2×#107C2314yes—
123C2×#106—42no—
124C2×#106C2314yes—
125C2×C4⋊C8—24no—
126C2×C4⋊C8C2314yes—
127C2×#103—24no—
128C2×#103—24no—
129C2×#103—24no—
130C2×#103—42no—
131C2×#103—42no—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102—42no—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C4×C8—24no—
144C2×C4×C8—42no—
145C2×C4×C8—42no—
146C2×C4×C8C2314yes—
147C2×C4×C8C2314yes—
148C2×C4×C8C2314yes—
149#401—28no—
150#400—28no—
151#399—28no—
152#397—28no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391C2314yes—
158#391C2314yes—
159#391C2314yes—
160#391C2314yes—
161#390—22no—
162#390—22no—
163#390C2314yes—
164#390C2314yes—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#388—28no—
170#388—28no—
171#387—28no—
172#383—28no—
173#382—22no—
174#382C2314yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4D414yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C2×#95—21no—
180C2×#95—21no—
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×Q16—28no—
240C2×Q16—28no—
241C2×Q16—44no—
242C2×Q16—416no—
243C2×SD16—28no—
244C2×SD16—44no—
245C2×SD16—44no—
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—
263#124—28no—
264#124—44no—
265#124—48no—
266C4×Q8—28no—
267C4×Q8—44no—
268C4×Q8—48no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
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—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
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—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—82no—
310C2×C22⋊C4—82no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2412yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327#107—44no—
328#106—44no—
329C4⋊C8—28no—
330#103—28no—
331#103—28no—
332#103—28no—
333#103—44no—
334#103—44no—
335#102—28no—
336#102—28no—
337#102—44no—
338C4×C8—28no—
339C4×C8—44no—
340C4×C8—44no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
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#95C2×D414yes—
355#95C2×D414yes—
356C24—22no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24C22×D411yescharacteristic
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—22no—
370C2×Q8—22no—
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—42no—
380C2×Q8—42no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8C2×#4114yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—44no—
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—24no—
413C22×C4—22no—
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—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—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4#13514yes—
449C22×C4#13514yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4C2511yescharacteristic
464Q16—416no—
465SD16—416no—
466C2×C8—28no—
467C2×C8—28no—
468C2×C8—28no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—44no—
479C2×C8C2×#4114yes—
480C2×C8C2×#4114yes—
481C2×C8C2×#4114yes—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—22no—
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—48no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—48no—
501C4⋊C4—84no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—48no—
518C42—48no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23#53212yes—
534C23#54612yes—
535C23#55012yes—
536C23C2×#13512yes—
537C23C2×#13512yes—
538C23#57712yes—
539C23#57812yes—
540C23C23×D411yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—42no—
544Q8—42no—
545Q8—48no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—84no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—84no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
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—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4#52914yes—
607C2×C4#53212yes—
608C2×C4#53814yes—
609C2×C4#54612yes—
610C2×C4#55012yes—
611C2×C4#55114yes—
612C2×C4#55314yes—
613C2×C4C2×#13514yes—
614C2×C4C2×#13514yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C8—24no—
622C8—24no—
623C8—24no—
624C8—42no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—168no—
633C22#255414yes—
634C22#316814yes—
635C22#317814yes—
636C22#330014yes—
637C22#330414yes—
638C22#339314yes—
639C22#339414yes—
640C22C2×#53211yescharacteristic
641C22C2×#54611yescharacteristic
642C22C2×#55011yescharacteristic
643C22C22×#13511yescharacteristic
644C22C22×#13511yescharacteristic
645C22C2×#57711yescharacteristic
646C22C2×#57811yescharacteristic
647C4—24no—
648C4—24no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#255414yes—
661C4C2×#52911yescharacteristic
662C4C2×#53811yescharacteristic
663C4C2×#55111yescharacteristic
664C4C2×#55311yescharacteristic
665C2—162no—
666C2#2888018yesdirect product factor
667C2C2×#255411yescharacteristic
668C2C2×#316811yescharacteristic
669C2C2×#317811yescharacteristic
670C2C2×#330011yescharacteristic
671C2C2×#330411yescharacteristic
672C2C2×#339311yescharacteristic
673C2C2×#339411yescharacteristic
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
1. C2×#2514C2semidirect
3. #28880C2direct (N × H)
4. C2×#2085C2semidirect
5. C2×#2066C2semidirect
8. C2×#1969C2semidirect
12. C2×#1933C2semidirect
14. C2×#1878C2semidirect
15. C2×C8×C4⋊C4C2semidirect
16. C2×C4×#103C2semidirect
36. #2514C22semidirect
58. #2085C22semidirect
59. #2066C22semidirect
62. #1969C22semidirect
66. #1933C22semidirect
68. #1878C22semidirect
69. C8×C4⋊C4C22semidirect
70. C4×#103C22semidirect
666. C2#28880direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25142
C2×#24482
#288802
C2×#20852
C2×#20662
C2×#20552
C2×#20152
C2×#19692
C2×#19682
C2×#19582
C2×#19342
C2×#19332
C2×#19252
C2×#18782
C2×C8×C4⋊C42
C2×C4×#1032