Finite Group Databasebeta

← C2×#24584 #6453052

Subgroups — C2×#24584 #6453052

Order 512.

2180 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C22×#427C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic
4C2×#2347C211yesmaximal, characteristic
5#24584C2116yesmaximal, direct product factor
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2083C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2066C211yesmaximal, characteristic, complemented
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1948C211yesmaximal, characteristic, complemented
12C2×#1928C211yesmaximal, characteristic, complemented
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1892C211yesmaximal, characteristic
15C2×#1850C211yesmaximal, characteristic
16C2×#1825C211yesmaximal, characteristic
17C24×C8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C2×#485—28no—
22C2×#485C2218yes—
23C2×#427—28no—
24C2×#427C2218yes—
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38#2499—24no—
39#2499C2218yes—
40C2×#397—21no—
41C2×#397—21no—
42#2347—24no—
43#2347C2218yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#388—21no—
49C2×#388—21no—
50C2×#387—21no—
51C2×#387—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#2083C2218yescomplemented
62#2068C2218yescomplemented
63#2066C2218yescomplemented
64#1949C2218yescomplemented
65#1948C2218yescomplemented
66#1928C2218yescomplemented
67#1927C2218yescomplemented
68#1892C2218yes—
69#1850C2218yes—
70#1825C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C23×C8—24no—
82C23×C8—24no—
83C23×C8—24no—
84C23×C8C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8C2314yes—
93C2×#117—22no—
94C2×#117—24no—
95C2×#117—22no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117D418yes—
99C2×#117C2314yes—
100C2×#117C2314yes—
101C22×C4⋊C4—21no—
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⋊C4D411yescharacteristic
118C22×C22⋊C4D411yescharacteristic
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#485—232no—
126#427—232no—
127C2×#107—24no—
128C2×#107—24no—
129C2×#107—24no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106—24no—
134C2×#106—24no—
135C2×#106—24no—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8—24no—
140C2×C4⋊C8—24no—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143C2×#103—24no—
144C2×#103—24no—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×C22⋊C8—24no—
152C2×C22⋊C8C2314yes—
153#397—28no—
154#397—28no—
155#391—22no—
156#391—22no—
157#391C2314yes—
158#391C2314yes—
159#389—22no—
160#389—22no—
161#389C2314yes—
162#389C2314yes—
163#388—28no—
164#388—28no—
165#387—28no—
166#387—28no—
167#384—22no—
168#384C2314yes—
169#383—28no—
170#382—22no—
171#382C2314yes—
172#380—22no—
173#380C2314yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—28no—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4D414yes—
180C4×C22⋊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#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202C25C2×D411yescharacteristic
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8C2412yes—
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—21no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D414yes—
229C23×C4C2×D411yescharacteristic
230C23×C4#4114yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C22×C8—232no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—28no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—28no—
240C22×C8—28no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8#4114yes—
248C22×C8#4114yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22⋊Q8—28no—
254C22⋊Q8—28no—
255C22⋊Q8—48no—
256#117—216no—
257#117—28no—
258#117—28no—
259#117—28no—
260#117—48no—
261#117C2×D4116yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
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—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—82no—
310C2×C22⋊C4C2×D414yes—
311C2×C22⋊C4C2×D414yes—
312C2×C42—28no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—44no—
322C2×C42—82no—
323C2×C42—82no—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2412yes—
327C2×C42C2412yes—
328#107—28no—
329#107—28no—
330#107—28no—
331#106—28no—
332#106—28no—
333#106—28no—
334C4⋊C8—28no—
335C4⋊C8—28no—
336#103—28no—
337#103—28no—
338#103—28no—
339#103—28no—
340C22⋊C8—28no—
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#95C2×D414yes—
353#95C2×D414yes—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24#12114yes—
361C24C22×D412yes—
362C24C22×D411yescharacteristic
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
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—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
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#12114yes—
432C22×C4#12118yes—
433C22×C4#13514yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C42−1+412yes—
447C22×C42−1+412yes—
448C22×C4C2511yescharacteristic
449C2×C8—232no—
450C2×C8—28no—
451C2×C8—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—44no—
463C2×C8—44no—
464C2×C8C2×#4114yes—
465C2×C8C2×#4114yes—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4C22×D414yes—
491C4⋊C4C22×D414yes—
492C22⋊C4—216no—
493C22⋊C4—28no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—84no—
498C42—216no—
499C42—24no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—48no—
506C42—84no—
507C42—84no—
508C23—28no—
509C23—22no—
510C23—22no—
511C23—28no—
512C23—28no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23C2×#12114yes—
522C23C2×#12112yes—
523C23#54712yes—
524C23#54712yes—
525C23C2×#13512yes—
526C23#57512yes—
527C23#57712yes—
528C23#57812yes—
529C23C23×D411yescharacteristic
530Q8—42no—
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—84no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—216no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4C2×#12114yes—
591C2×C4C2×#12112yes—
592C2×C4#52514yes—
593C2×C4#53614yes—
594C2×C4#54712yes—
595C2×C4#54712yes—
596C2×C4#55214yes—
597C2×C4#55214yes—
598C2×C4C2×#13514yes—
599C2×C4C23×D411yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C2×2−1+411yescharacteristic
604C2×C4C2×2−1+411yescharacteristic
605C8—24no—
606C8—24no—
607C8—42no—
608C8—42no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—416no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22#241314yes—
618C22#303514yes—
619C22#304614yes—
620C22#333714yes—
621C22#334214yes—
622C22#334914yes—
623C22#335014yes—
624C22C22×#12111yescharacteristic
625C22C2×#54711yescharacteristic
626C22C2×#54711yescharacteristic
627C22C22×#13511yescharacteristic
628C22C2×#57511yescharacteristic
629C22C2×#57711yescharacteristic
630C22C2×#57811yescharacteristic
631C4—24no—
632C4—24no—
633C4—44no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4#241314yes—
645C4C2×#52511yescharacteristic
646C4C2×#53611yescharacteristic
647C4C2×#55211yescharacteristic
648C4C2×#55211yescharacteristic
649C2—44no—
650C2#2458418yesdirect product factor
651C2C2×#241311yescharacteristic
652C2C2×#303511yescharacteristic
653C2C2×#304611yescharacteristic
654C2C2×#333711yescharacteristic
655C2C2×#334211yescharacteristic
656C2C2×#334911yescharacteristic
657C2C2×#335011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
5. #24584C2direct (N × H)
6. C2×#2085C2semidirect
7. C2×#2083C2semidirect
8. C2×#2068C2semidirect
9. C2×#2066C2semidirect
10. C2×#1949C2semidirect
11. C2×#1948C2semidirect
12. C2×#1928C2semidirect
13. C2×#1927C2semidirect
60. #2085C22semidirect
61. #2083C22semidirect
62. #2068C22semidirect
63. #2066C22semidirect
64. #1949C22semidirect
65. #1948C22semidirect
66. #1928C22semidirect
67. #1927C22semidirect
650. C2#24584direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4852
C22×#4272
C2×#24992
C2×#23472
#245842
C2×#20852
C2×#20832
C2×#20682
C2×#20662
C2×#19492
C2×#19482
C2×#19282
C2×#19272
C2×#18922
C2×#18502
C2×#18252