Finite Group Databasebeta

← C2×#24651 #6448938

Subgroups — C2×#24651 #6448938

Order 512.

2918 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#461C211yesmaximal, characteristic, complemented
2C22×#431C211yesmaximal, characteristic, complemented
3C2×#2857C211yesmaximal, characteristic, complemented
4C2×C4×#121C211yesmaximal, characteristic
5#24651C2116yesmaximal, direct product factor
6C2×#2035C212yesmaximal, complemented
7C2×#2000C212yesmaximal
8C2×#1981C211yesmaximal, characteristic, complemented
9C2×#1968C212yesmaximal, complemented
10C2×#1948C211yesmaximal, characteristic, complemented
11C2×#1933C212yesmaximal, complemented
12C2×#1802C211yesmaximal, characteristic, complemented
13C23×SD16C2212yes—
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×#121C2211yescharacteristic
17C22×C4×D4—21no—
18C22×C4×D4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#461—28no—
21C2×#461C2218yescomplemented
22C2×#431—28no—
23C2×#431C2218yescomplemented
24C22×#106C2212yes—
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103C2212yes—
28C22×#103C2212yes—
29C22×#103C2212yes—
30C22×#102C2212yes—
31C22×C22⋊C8C2212yes—
32C22×C4×C8C2212yes—
33#2857—24no—
34#2857C2218yescomplemented
35C4×#121—24no—
36C4×#121C2218yes—
37C2×#391C2212yes—
38C2×#390C2211yescharacteristic
39C2×#389—21no—
40C2×#389—21no—
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#386—21no—
44C2×#384C2211yescharacteristic
45C2×#380—21no—
46C2×#380—21no—
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50#2035C22116yescomplemented
51#2000C22116yes—
52#1981C2218yescomplemented
53#1968C22116yescomplemented
54#1948C2218yescomplemented
55#1933C22116yescomplemented
56#1802C2218yescomplemented
57C2×#340C2212yes—
58C2×#340C2212yes—
59C23×Q8C2312yes—
60C23×D4—21no—
61C23×D4C2311yescharacteristic
62C24×C4—21no—
63C24×C4D411yescharacteristic
64C24×C4D411yescharacteristic
65C24×C4C2311yescharacteristic
66C22×SD16—28no—
67C22×SD16—28no—
68C22×SD16—44no—
69C22×SD16C2318yes—
70C23×C8C2312yes—
71C23×C8C2312yes—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8C2314yes—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—44no—
78C2×C22⋊Q8—44no—
79C2×C22⋊Q8C2318yes—
80C2×#121—24no—
81C2×#121—22no—
82C2×#121—24no—
83C2×#121C2314yes—
84C2×C4×D4—28no—
85C2×C4×D4—22no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—48no—
89C2×C4×D4C2314yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—22no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C42—24no—
107C22×C42—24no—
108C22×C42—21no—
109C22×C42D414yes—
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42C2314yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#461—232no—
116#431—232no—
117C2×#106—44no—
118C2×#106C2318yes—
119C2×C4⋊C8—24no—
120C2×C4⋊C8D418yescomplemented
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2314yes—
123C2×#103—28no—
124C2×#103—28no—
125C2×#103—44no—
126C2×#103C2318yes—
127C2×#103C2318yes—
128C2×#103C2318yes—
129C2×#102—44no—
130C2×#102C2318yes—
131C2×C22⋊C8—28no—
132C2×C22⋊C8C2318yes—
133C2×C4×C8—28no—
134C2×C4×C8C2318yes—
135#391—24no—
136#391C2318yes—
137#390—22no—
138#390C2314yes—
139#389—28no—
140#389—28no—
141#389—22no—
142#389C2314yes—
143#388—28no—
144#386—28no—
145#384—22no—
146#384C2314yes—
147#380—28no—
148#380—28no—
149#380—22no—
150#380C2314yes—
151C4×C4⋊C4—22no—
152C4×C4⋊C4C2314yes—
153C4×C22⋊C4—28no—
154C2×#95—21no—
155C2×#95—21no—
156C2×#95—21no—
157C2×#95—22no—
158C2×#95—22no—
159#340C2318yes—
160#340C2318yes—
161C25—21no—
162C25C2×D411yescharacteristic
163C22×Q8—22no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8C2414yes—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—22no—
174C22×D4—21no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4#4114yes—
182C22×D4C2412yes—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—21no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4SD1618yescomplemented
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D414yes—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D414yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C2×SD16—216no—
215C2×SD16—216no—
216C2×SD16—48no—
217C2×SD16—432no—
218C22×C8—24no—
219C22×C8—24no—
220C22×C8—24no—
221C22×C8—24no—
222C22×C8—24no—
223C22×C8C2×D418yes—
224C22×C8C2414yes—
225C22×C8C2414yes—
226C4⋊Q8—28no—
227C4⋊Q8—28no—
228C4⋊Q8—48no—
229C22⋊Q8—48no—
230C22⋊Q8—48no—
231C22⋊Q8—416no—
232#121—216no—
233#121—28no—
234#121—28no—
235C4×D4—28no—
236C4×D4—28no—
237C4×D4—432no—
238C4×D4—48no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4#4114yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—216no—
284C2×C42—28no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—44no—
288C2×C42C2×D414yes—
289C2×C42C2×D414yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D414yes—
293C2×C42#4114yes—
294C2×C42C2412yes—
295C2×C42C2412yes—
296#106—48no—
297C4⋊C8—28no—
298C4⋊C8C2×D4116yescomplemented
299#103—216no—
300#103—216no—
301#103—48no—
302#102—48no—
303C22⋊C8—216no—
304C4×C8—216no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24—22no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24C2×SD1614yes—
325C24C22×D412yes—
326C24C22×D411yescharacteristic
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×D4—216no—
341C2×D4—24no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—432no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4C2×#4114yes—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—232no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—44no—
387C22×C4—416no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—816no—
410C22×C4#12114yes—
411C22×C4#12114yes—
412C22×C4C2×SD1614yes—
413C22×C4C2×SD1614yes—
414C22×C4C2×SD16116yescomplemented
415C22×C4#13514yes—
416C22×C4#13714yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C4C2511yescharacteristic
430SD16—432no—
431C2×C8—216no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—48no—
438C2×C8C22×D418yes—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—48no—
448C4⋊C4—416no—
449C4⋊C4—42no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4C2×#4114yes—
457C22⋊C4—216no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C42—24no—
464C42—28no—
465C42—28no—
466C42—22no—
467C42—22no—
468C42—48no—
469C42#12118yes—
470C42C22×D414yes—
471C42C2×#4114yes—
472C23—28no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—28no—
477C23—28no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—48no—
487C23—48no—
488C23—42no—
489C23—48no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23C2×#12112yes—
500C23#52512yes—
501C23#53512yes—
502C23C22×SD1614yes—
503C23C22×SD1612yes—
504C23C2×#13512yes—
505C23C2×#13712yes—
506C23#57512yes—
507C23C23×D411yescharacteristic
508Q8—44no—
509Q8—44no—
510Q8—44no—
511Q8—44no—
512Q8—88no—
513D4—24no—
514D4—24no—
515D4—432no—
516D4—48no—
517D4—42no—
518D4—42no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—232no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—416no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—88no—
556C2×C4—84no—
557C2×C4—88no—
558C2×C4—84no—
559C2×C4—816no—
560C2×C4#465116yes—
561C2×C4C2×#12114yes—
562C2×C4C2×#12112yes—
563C2×C4C2×#12114yes—
564C2×C4#52512yes—
565C2×C4#53512yes—
566C2×C4#54614yes—
567C2×C4#55014yes—
568C2×C4#55314yes—
569C2×C4#55514yes—
570C2×C4C22×SD1614yes—
571C2×C4C22×SD1614yes—
572C2×C4C2×#13514yes—
573C2×C4C2×#13714yes—
574C2×C4C23×D411yescharacteristic
575C2×C4C22×#4111yescharacteristic
576C2×C4C22×#4111yescharacteristic
577C2×C4C22×#4111yescharacteristic
578C2×C4C2×H(5,𝔽2)11yescharacteristic
579C2×C4C2×H(5,𝔽2)11yescharacteristic
580C8—28no—
581C8—44no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—42no—
586C22—416no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—82no—
594C22—82no—
595C22—816no—
596C22—82no—
597C22—82no—
598C22#241414yes—
599C22C2×#46514yes—
600C22#305614yes—
601C22#310114yes—
602C22#310714yes—
603C22#320614yes—
604C22#321214yes—
605C22C22×#12111yescharacteristic
606C22C2×#52511yescharacteristic
607C22C2×#53511yescharacteristic
608C22C23×SD1611yescharacteristic
609C22C22×#13511yescharacteristic
610C22C22×#13711yescharacteristic
611C22C2×#57511yescharacteristic
612C4—24no—
613C4—24no—
614C4—24no—
615C4—44no—
616C4—42no—
617C4—82no—
618C4—82no—
619C4—84no—
620C4—84no—
621C4#241414yes—
622C4C2×#46518yes—
623C4C2×#54611yescharacteristic
624C4C2×#55011yescharacteristic
625C4C2×#55311yescharacteristic
626C4C2×#55511yescharacteristic
627C2—44no—
628C2—84no—
629C2#2465118yesdirect product factor
630C2C2×#241411yescharacteristic
631C2C22×#46511yescharacteristic
632C2C2×#305611yescharacteristic
633C2C2×#310111yescharacteristic
634C2C2×#310711yescharacteristic
635C2C2×#320611yescharacteristic
636C2C2×#321211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#461C2semidirect
2. C22×#431C2semidirect
3. C2×#2857C2semidirect
5. #24651C2direct (N × H)
6. C2×#2035C2semidirect
8. C2×#1981C2semidirect
9. C2×#1968C2semidirect
10. C2×#1948C2semidirect
11. C2×#1933C2semidirect
12. C2×#1802C2semidirect
21. C2×#461C22semidirect
23. C2×#431C22semidirect
34. #2857C22semidirect
50. #2035C22semidirect
52. #1981C22semidirect
53. #1968C22semidirect
54. #1948C22semidirect
55. #1933C22semidirect
56. #1802C22semidirect
120. C2×C4⋊C8D4semidirect
203. C23×C4SD16semidirect
298. C4⋊C8C2×D4semidirect
414. C22×C4C2×SD16semidirect
629. C2#24651direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4612
C22×#4312
C2×#28572
C2×C4×#1212
#246512
C2×#20352
C2×#20002
C2×#19812
C2×#19682
C2×#19482
C2×#19332
C2×#18022