Finite Group Databasebeta

← C2×#24585 #6453053

Subgroups — C2×#24585 #6453053

Order 512.

2310 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#482C211yesmaximal, characteristic
2C22×#427C211yesmaximal, characteristic, complemented
3C2×#2500C211yesmaximal, characteristic
4C2×#2347C211yesmaximal, characteristic, complemented
5#24585C2116yesmaximal, direct product factor
6C2×#2084C211yesmaximal, characteristic, complemented
7C2×#2085C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic, complemented
9C2×#2065C211yesmaximal, characteristic, complemented
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1947C211yesmaximal, characteristic, complemented
12C2×#1926C211yesmaximal, characteristic, complemented
13C2×#1925C211yesmaximal, characteristic, complemented
14C2×#1893C211yesmaximal, characteristic
15C2×#1850C211yesmaximal, characteristic, complemented
16C2×#1825C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C2×#482—28no—
22C2×#482C2218yes—
23C2×#427—28no—
24C2×#427C2218yescomplemented
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38#2500—24no—
39#2500C2218yes—
40C2×#394—21no—
41C2×#394—21no—
42#2347—24no—
43#2347C2218yescomplemented
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#388—21no—
49C2×#388—21no—
50C2×#388—21no—
51C2×#388—21no—
52C2×#384C2211yescharacteristic
53C2×#383—21no—
54C2×#382C2211yescharacteristic
55C2×#380C2211yescharacteristic
56C2×C4×C4⋊C4—21no—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2085C2218yescomplemented
61#2084C2218yescomplemented
62#2067C2218yescomplemented
63#2065C2218yescomplemented
64#1949C2218yescomplemented
65#1947C2218yescomplemented
66#1926C2218yescomplemented
67#1925C2218yescomplemented
68#1893C2218yes—
69#1850C2218yescomplemented
70#1825C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C23×C8—24no—
83C23×C8—24no—
84C23×C8—24no—
85C23×C8C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×#121—22no—
91C2×#121—24no—
92C2×#121—24no—
93C2×#121C2314yes—
94C2×#117—22no—
95C2×#117—24no—
96C2×#117—22no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117D418yes—
100C2×#117C2314yes—
101C2×#117C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4D411yescharacteristic
122C22×C22⋊C4D411yescharacteristic
123C22×C42—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#482—232no—
130#427—232no—
131C2×#107—24no—
132C2×#107—24no—
133C2×#107—24no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106—24no—
138C2×#106—24no—
139C2×#106—24no—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8—24no—
144C2×C4⋊C8—24no—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102—24no—
150C2×#102—24no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8—24no—
156C2×C22⋊C8C2314yes—
157#394—28no—
158#394—28no—
159#390—22no—
160#390—22no—
161#390C2314yes—
162#390C2314yes—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#389C2314yes—
167#388—28no—
168#388—28no—
169#388—28no—
170#388—28no—
171#384—22no—
172#384C2314yes—
173#383—28no—
174#382—22no—
175#382C2314yes—
176#380—22no—
177#380C2314yes—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—28no—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C4×C22⋊C4D414yes—
184C4×C22⋊C4C2314yes—
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#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202C25—21no—
203C25C2×D411yescharacteristic
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—44no—
213C22×D4—82no—
214C22×D4—82no—
215C22×D4C2412yes—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D414yes—
233C23×C4C2×D411yescharacteristic
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C22×C8—232no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—28no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—28no—
244C22×C8—28no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8#4114yes—
252C22×C8#4114yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257#121—28no—
258#121—28no—
259#121—48no—
260#117—216no—
261#117—28no—
262#117—28no—
263#117—28no—
264#117—48no—
265#117C2×D4116yes—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
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—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—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—28no—
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—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—82no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4—82no—
320C2×C22⋊C4C2×D414yes—
321C2×C22⋊C4C2×D414yes—
322C2×C42—28no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—82no—
333C2×C42—82no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338#107—28no—
339#107—28no—
340#107—28no—
341#106—28no—
342#106—28no—
343#106—28no—
344C4⋊C8—28no—
345C4⋊C8—28no—
346#102—28no—
347#102—28no—
348#102—28no—
349#102—28no—
350C22⋊C8—28no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95C2×D414yes—
359#95C2×D414yes—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24#12114yes—
375C24C22×D412yes—
376C24C22×D411yescharacteristic
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—48no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—816no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
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—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—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4#12114yes—
446C22×C4#12118yes—
447C22×C4#13514yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C42−1+412yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463C2×C8—232no—
464C2×C8—28no—
465C2×C8—28no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—44no—
477C2×C8—44no—
478C2×C8C2×#4114yes—
479C2×C8C2×#4114yes—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
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—
502C4⋊C4—84no—
503C4⋊C4C22×D414yes—
504C4⋊C4C22×D414yes—
505C22⋊C4—216no—
506C22⋊C4—28no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C42—216no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—48no—
523C42—84no—
524C42—84no—
525C23—28no—
526C23—22no—
527C23—22no—
528C23—28no—
529C23—28no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23C2×#12114yes—
553C23C2×#12112yes—
554C23#54712yes—
555C23#54712yes—
556C23C2×#13512yes—
557C23#57512yes—
558C23#57612yes—
559C23#57712yes—
560C23C23×D411yescharacteristic
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—816no—
566D4—84no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—216no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—48no—
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—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4C2×#12114yes—
619C2×C4C2×#12112yes—
620C2×C4#52514yes—
621C2×C4#53614yes—
622C2×C4#54712yes—
623C2×C4#54712yes—
624C2×C4#55214yes—
625C2×C4#55214yes—
626C2×C4C2×#13514yes—
627C2×C4C23×D411yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C8—24no—
634C8—24no—
635C8—42no—
636C8—42no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—416no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—168no—
653C22#241314yes—
654C22#303514yes—
655C22#304514yes—
656C22#333914yes—
657C22#334014yes—
658C22#334714yes—
659C22#335014yes—
660C22C22×#12111yescharacteristic
661C22C2×#54711yescharacteristic
662C22C2×#54711yescharacteristic
663C22C22×#13511yescharacteristic
664C22C2×#57511yescharacteristic
665C22C2×#57611yescharacteristic
666C22C2×#57711yescharacteristic
667C4—24no—
668C4—24no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#241314yes—
680C4C2×#52511yescharacteristic
681C4C2×#53611yescharacteristic
682C4C2×#55211yescharacteristic
683C4C2×#55211yescharacteristic
684C2—44no—
685C2—162no—
686C2#2458518yesdirect product factor
687C2C2×#241311yescharacteristic
688C2C2×#303511yescharacteristic
689C2C2×#304511yescharacteristic
690C2C2×#333911yescharacteristic
691C2C2×#334011yescharacteristic
692C2C2×#334711yescharacteristic
693C2C2×#335011yescharacteristic
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×#427C2semidirect
4. C2×#2347C2semidirect
5. #24585C2direct (N × H)
6. C2×#2084C2semidirect
7. C2×#2085C2semidirect
8. C2×#2067C2semidirect
9. C2×#2065C2semidirect
10. C2×#1949C2semidirect
11. C2×#1947C2semidirect
12. C2×#1926C2semidirect
13. C2×#1925C2semidirect
15. C2×#1850C2semidirect
16. C2×#1825C2semidirect
24. C2×#427C22semidirect
43. #2347C22semidirect
60. #2085C22semidirect
61. #2084C22semidirect
62. #2067C22semidirect
63. #2065C22semidirect
64. #1949C22semidirect
65. #1947C22semidirect
66. #1926C22semidirect
67. #1925C22semidirect
69. #1850C22semidirect
70. #1825C22semidirect
686. C2#24585direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4822
C22×#4272
C2×#25002
C2×#23472
#245852
C2×#20842
C2×#20852
C2×#20672
C2×#20652
C2×#19492
C2×#19472
C2×#19262
C2×#19252
C2×#18932
C2×#18502
C2×#18252