Finite Group Databasebeta

← C2×#18750 #6348663

Subgroups — C2×#18750 #6348663

Order 512.

3402 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#559C211yesmaximal, characteristic, complemented
2C2×#3205C211yesmaximal, characteristic
3C2×#3103C211yesmaximal, characteristic, complemented
4C2×#3037C211yesmaximal, characteristic, complemented
5#18750C2116yesmaximal, direct product factor
6C2×#1640C214yesmaximal, complemented
7C2×#1620C214yesmaximal, complemented
8C2×#1545C211yesmaximal, characteristic, complemented
9C2×#1487C211yesmaximal, characteristic, complemented
10C2×#1479C211yesmaximal, characteristic, complemented
11C22×#137—21no—
12C2×#559—22no—
13C2×#559—24no—
14C2×#559—22no—
15C2×#559—22no—
16C2×#559—24no—
17C2×#559—24no—
18C2×#559C2214yes—
19C2×#559C2218yescomplemented
20C2×#532C2211yescharacteristic
21C22×#127C2211yescharacteristic
22C22×#124—21no—
23C22×#124—21no—
24C22×#123—22no—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4—21no—
28C22×C4×D4—21no—
29#3205—24no—
30#3205C2218yes—
31C2×#493C2212yes—
32C2×#486C2212yes—
33#3103—24no—
34#3103C2218yescomplemented
35C2×#481—21no—
36C2×#475C2212yescomplemented
37#3037—24no—
38#3037C2218yescomplemented
39C2×#464—22no—
40C2×#462C2212yescomplemented
41C2×#460C2212yescomplemented
42C2×#460C2212yescomplemented
43C2×#459—22no—
44C2×#448—21no—
45C2×#447—22no—
46C2×C4×SD16—22no—
47C2×#431C2211yescharacteristic
48C2×#431C2211yescharacteristic
49C2×#423C2211yescharacteristic
50C2×#417—21no—
51#1640C22132yescomplemented
52#1620C22132yescomplemented
53#1545C2218yescomplemented
54C2×#331C2214yes—
55#1487C2218yescomplemented
56#1479C2218yescomplemented
57C2×Q8⋊C8C2214yescomplemented
58C2×Q8⋊C8C2214yescomplemented
59C22×#41—21no—
60C23×Q8D411yescharacteristic
61C23×Q8D411yescharacteristic
62C23×D4—21no—
63C23×D4—21no—
64C23×D4D412yes—
65C24×C4—22no—
66C2×#137—28no—
67C2×#137—48no—
68C22×Q16—22no—
69C22×SD16—22no—
70C22×SD16—22no—
71C22×D8—24no—
72C22×M16—21no—
73#559—216no—
74#559—216no—
75#559—216no—
76#559—28no—
77#559—216no—
78#559—28no—
79#559—28no—
80#559—216no—
81#559—216no—
82#559C23116yes—
83#532—22no—
84#532—22no—
85#532—22no—
86#532C2314yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#127—21no—
89C2×#127—21no—
90C2×#127—22no—
91C2×#127—22no—
92C2×#127D414yes—
93C2×#127D412yes—
94C2×#127C2311yescharacteristic
95C2×#127C2312yes—
96C2×#127C2314yes—
97C2×#124—28no—
98C2×#124—28no—
99C2×#124—24no—
100C2×#124—24no—
101C2×#124—24no—
102C2×#124—24no—
103C2×#124—28no—
104C2×#124—28no—
105C2×#124—21no—
106C2×#123—216no—
107C2×#123—28no—
108C2×#123—28no—
109C2×#123—216no—
110C2×#121—21no—
111C2×#121—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8D414yes—
121C2×C4×Q8D412yes—
122C2×C4×Q8D412yes—
123C2×C4×Q8D414yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2314yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2314yes—
128C2×C4×D4—28no—
129C2×C4×D4—28no—
130C2×C4×D4—24no—
131C2×C4×D4—24no—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4—28no—
135C2×C4×D4—28no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4D411yescharacteristic
141C22×C4⋊C4D411yescharacteristic
142C22×C22⋊C4—22no—
143C22×C22⋊C4—22no—
144C22×C22⋊C4—22no—
145C22×C42—21no—
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#493—24no—
149#493C2318yes—
150#486—24no—
151#486C2318yes—
152#481—28no—
153#475—24no—
154#475C2318yescomplemented
155#464—216no—
156#462—24no—
157#462C2318yescomplemented
158#460—24no—
159#460—24no—
160#460C2318yescomplemented
161#460C2318yescomplemented
162#459—216no—
163#448—28no—
164#447—216no—
165C4×SD16—216no—
166#431—22no—
167#431—22no—
168#431C2314yes—
169#431C2314yes—
170C2×#106—22no—
171#423—22no—
172#423C2314yes—
173C2×C4⋊C8—21no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2312yes—
177#417—28no—
178C2×#103—22no—
179C2×#103—22no—
180C2×#102—24no—
181C2×#102—22no—
182C2×#102—22no—
183C2×C22⋊C8—21no—
184C2×C22⋊C8—21no—
185C2×C4×C8—21no—
186C2×C4×C8C2312yes—
187#331C23116yes—
188Q8⋊C8C23116yescomplemented
189Q8⋊C8C23116yescomplemented
190C25—22no—
191C2×#41—24no—
192C2×#41—42no—
193C2×#41—44no—
194C2×#41—44no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—21no—
202C22×Q8—44no—
203C22×Q8C2×D412yes—
204C22×Q8C2×D412yes—
205C22×Q8C2×D414yes—
206C22×Q8C2×D414yes—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—48no—
221C22×D4—41no—
222C22×D4—44no—
223C22×D4—44no—
224C22×D4—84no—
225C22×D4C2×D414yes—
226C22×D4C2×D418yes—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—216no—
232C23×C4—28no—
233C23×C4—41no—
234C23×C4—88no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238#137—432no—
239C2×Q16—28no—
240C2×Q16—48no—
241C2×SD16—28no—
242C2×SD16—28no—
243C2×SD16—48no—
244C2×SD16—48no—
245C2×D8—216no—
246C2×D8—88no—
247C2×M16—24no—
248C2×M16—44no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8C2412yes—
256#127—24no—
257#127—24no—
258#127—24no—
259#127—24no—
260#127—24no—
261#127—22no—
262#127—22no—
263#127—21no—
264#127—21no—
265#127—21no—
266#127C2×D418yes—
267#127C2×D414yes—
268#127C2412yes—
269#127C2414yes—
270#124—216no—
271#124—216no—
272#124—232no—
273#124—232no—
274#124—216no—
275#124—216no—
276#124—24no—
277#124—42no—
278#123—232no—
279#123—264no—
280#123—232no—
281#121—24no—
282#121—24no—
283#121—42no—
284#121—42no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8C2×D418yes—
300C4×Q8C2×D414yes—
301C4×Q8C2×D414yes—
302C4×Q8C2×D418yes—
303C4×Q8C2414yes—
304C4×Q8C2414yes—
305C4×D4—216no—
306C4×D4—216no—
307C4×D4—232no—
308C4×D4—232no—
309C4×D4—216no—
310C4×D4—216no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—42no—
314C4×D4—42no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D414yes—
333C2×C22⋊C4—216no—
334C2×C22⋊C4—216no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—216no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—41no—
340C2×C22⋊C4—41no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357#106—28no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—22no—
361C4⋊C8—22no—
362C4⋊C8C2414yes—
363C4⋊C8C2414yes—
364#103—28no—
365#103—28no—
366#102—216no—
367#102—28no—
368#102—28no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C4×C8—24no—
372C4×C8—22no—
373C4×C8C2414yes—
374C24—28no—
375C24—24no—
376C24—48no—
377C24—48no—
378C24—48no—
379C24—41no—
380C24#12011yescharacteristic
381#41—48no—
382#41—48no—
383#41—88no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—28no—
390C2×Q8—24no—
391C2×Q8—28no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—48no—
396C2×Q8—44no—
397C2×Q8—48no—
398C2×Q8—41no—
399C2×Q8—44no—
400C2×Q8—48no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8C22×D414yes—
404C2×Q8C22×D414yes—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—28no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—48no—
417C2×D4—416no—
418C2×D4—416no—
419C2×D4—48no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—44no—
426C2×D4—48no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—48no—
430C2×D4—42no—
431C2×D4—48no—
432C2×D4—44no—
433C2×D4—48no—
434C2×D4—416no—
435C2×D4—84no—
436C2×D4—88no—
437C2×D4—88no—
438C2×D4—84no—
439C2×D4C22×D418yes—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—232no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—41no—
456C22×C4—44no—
457C22×C4—832no—
458C22×C4—816no—
459C22×C4—84no—
460C22×C4#12012yes—
461C22×C4#12011yescharacteristic
462C22×C4#12011yescharacteristic
463C22×C4#12012yes—
464C22×C4#12012yes—
465C22×C4#12011yescharacteristic
466C22×C4#13514yes—
467C22×C4AGL(1,ℤ/8)14yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474Q16—416no—
475SD16—416no—
476SD16—416no—
477D8—816no—
478M16—48no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—44no—
483C2×C8—44no—
484C2×C8—44no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4C22×D414yes—
500C4⋊C4C22×D414yes—
501C22⋊C4—232no—
502C22⋊C4—232no—
503C22⋊C4—232no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C42—28no—
507C42—28no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—24no—
512C42#12012yes—
513C42#12012yes—
514C42#12012yes—
515C42C22×D412yes—
516C42C22×D411yescharacteristic
517C42C22×D412yes—
518C42C22×D412yes—
519C42C22×D411yescharacteristic
520C42C22×D411yescharacteristic
521C42C2511yescharacteristic
522C23—22no—
523C23—22no—
524C23—28no—
525C23—416no—
526C23—416no—
527C23—416no—
528C23—48no—
529C23—48no—
530C23—48no—
531C23—42no—
532C23—832no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23#44914yes—
537C23C2×#12012yes—
538C23C2×#12011yescharacteristic
539Q8—24no—
540Q8—24no—
541Q8—24no—
542Q8—24no—
543Q8—48no—
544Q8—48no—
545Q8—44no—
546Q8—44no—
547Q8—42no—
548Q8—42no—
549Q8—82no—
550D4—48no—
551D4—416no—
552D4—44no—
553D4—416no—
554D4—48no—
555D4—48no—
556D4—44no—
557D4—42no—
558D4—44no—
559D4—42no—
560D4—48no—
561D4—44no—
562D4—84no—
563D4—88no—
564D4—82no—
565D4—88no—
566D4—82no—
567D4—82no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—832no—
589C2×C4—82no—
590C2×C4—88no—
591C2×C4#44914yes—
592C2×C4#44918yes—
593C2×C4C2×#12012yes—
594C2×C4C2×#12012yes—
595C2×C4C2×#12011yescharacteristic
596C2×C4C2×#12012yes—
597C2×C4C2×#12011yescharacteristic
598C2×C4C2×#12011yescharacteristic
599C2×C4C2×#13512yes—
600C2×C4C2×#13514yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4C2×AGL(1,ℤ/8)14yes—
603C2×C4#57514yes—
604C2×C4C23×D411yescharacteristic
605C2×C4C23×D411yescharacteristic
606C2×C4C23×D411yescharacteristic
607C8—44no—
608C8—44no—
609C8—44no—
610C22—28no—
611C22—22no—
612C22—22no—
613C22—48no—
614C22—48no—
615C22—48no—
616C22—832no—
617C22—816no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—164no—
622C22C2×#44914yes—
623C22C2×#44912yes—
624C22#300512yes—
625C22#301912yes—
626C22C22×#12011yescharacteristic
627C4—22no—
628C4—21no—
629C4—22no—
630C4—21no—
631C4—44no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—82no—
636C4C2×#44914yes—
637C4C2×#44912yes—
638C4#300512yes—
639C4C22×#13511yescharacteristic
640C4C22×AGL(1,ℤ/8)11yescharacteristic
641C4C2×#57512yes—
642C2—24no—
643C2—88no—
644C2—162no—
645C2#1875014yesdirect product factor
646C2C22×#44911yescharacteristic
647C2C2×#300511yescharacteristic
648C2C2×#301911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC2×#3205C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#559C2semidirect
3. C2×#3103C2semidirect
4. C2×#3037C2semidirect
5. #18750C2direct (N × H)
6. C2×#1640C2semidirect
7. C2×#1620C2semidirect
8. C2×#1545C2semidirect
9. C2×#1487C2semidirect
10. C2×#1479C2semidirect
19. C2×#559C22semidirect
34. #3103C22semidirect
36. C2×#475C22semidirect
38. #3037C22semidirect
40. C2×#462C22semidirect
41. C2×#460C22semidirect
42. C2×#460C22semidirect
51. #1640C22semidirect
52. #1620C22semidirect
53. #1545C22semidirect
55. #1487C22semidirect
56. #1479C22semidirect
57. C2×Q8⋊C8C22semidirect
58. C2×Q8⋊C8C22semidirect
154. #475C23semidirect
157. #462C23semidirect
160. #460C23semidirect
161. #460C23semidirect
188. Q8⋊C8C23semidirect
189. Q8⋊C8C23semidirect
645. C2#18750direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5592
C2×#32052
C2×#31032
C2×#30372
#187502
C2×#16402
C2×#16202
C2×#15452
C2×#14872
C2×#14792