Finite Group Databasebeta

← C2×#18748 #6348661

Subgroups — C2×#18748 #6348661

Order 512.

3338 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×#554C211yesmaximal, characteristic, complemented
2C2×#3206C211yesmaximal, characteristic, complemented
3C2×#3103C211yesmaximal, characteristic, complemented
4C2×#3036C211yesmaximal, characteristic
5#18748C2116yesmaximal, direct product factor
6C2×#1643C214yesmaximal, complemented
7C2×#1622C214yesmaximal, complemented
8C2×#1548C211yesmaximal, characteristic, complemented
9C2×#1487C211yesmaximal, characteristic, complemented
10C2×#1478C211yesmaximal, characteristic, complemented
11C22×AGL(1,ℤ/8)—21no—
12C2×#554—22no—
13C2×#554—24no—
14C2×#554—22no—
15C2×#554—24no—
16C2×#554—22no—
17C2×#554—24no—
18C2×#554C2214yes—
19C2×#554C2218yescomplemented
20C2×#532C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125—21no—
23C22×C22⋊Q8—21no—
24C22×C22⋊Q8—22no—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4—21no—
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4—21no—
29#3206—24no—
30#3206C2218yescomplemented
31C2×#492C2212yescomplemented
32C2×#487C2212yescomplemented
33#3103—24no—
34#3103C2218yescomplemented
35C2×#481—21no—
36C2×#475C2212yescomplemented
37#3036—24no—
38#3036C2218yes—
39C2×#460C2212yescomplemented
40C2×#463—22no—
41C2×#461—22no—
42C2×#461C2212yes—
43C2×#459C2212yes—
44C2×#448—22no—
45C2×#448—21no—
46C2×C4×SD16—22no—
47C2×#431C2211yescharacteristic
48C2×#431C2211yescharacteristic
49C2×#423C2211yescharacteristic
50C2×#417—21no—
51#1643C22132yescomplemented
52#1622C22132yescomplemented
53#1548C2218yescomplemented
54C2×#333C2214yescomplemented
55#1487C2218yescomplemented
56#1478C2218yescomplemented
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yes—
59C22×#41—21no—
60C23×Q8D411yescharacteristic
61C23×Q8D412yes—
62C23×D4D411yescharacteristic
63C24×C4—22no—
64C24×C4—21no—
65C2×AGL(1,ℤ/8)—28no—
66C2×AGL(1,ℤ/8)—48no—
67C22×SD16—22no—
68C22×SD16—22no—
69C22×SD16—24no—
70C22×D8—22no—
71C22×M16—21no—
72#554—216no—
73#554—216no—
74#554—216no—
75#554—28no—
76#554—216no—
77#554—28no—
78#554—216no—
79#554—28no—
80#554—216no—
81#554C23116yes—
82#532—22no—
83#532—22no—
84#532—22no—
85#532C2314yes—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8D414yes—
91C2×C4⋊Q8D412yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2314yes—
95C2×#127C2311yescharacteristic
96C2×#125—28no—
97C2×#125—24no—
98C2×#125—24no—
99C2×#125—28no—
100C2×#124—21no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—216no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—28no—
108C2×C22⋊Q8—216no—
109C2×#121—21no—
110C2×#121—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8D414yes—
116C2×C4×Q8D412yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2314yes—
119C2×C4×D4—28no—
120C2×C4×D4—24no—
121C2×C4×D4—24no—
122C2×C4×D4—28no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—22no—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4D412yes—
130C2×C4×D4D414yes—
131C2×C4×D4C2312yes—
132C2×C4×D4C2314yes—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—28no—
135C22×C4⋊C4—24no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—24no—
141C22×C4⋊C4—28no—
142C22×C4⋊C4D411yescharacteristic
143C22×C4⋊C4D411yescharacteristic
144C22×C22⋊C4—22no—
145C22×C22⋊C4—21no—
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#492—24no—
149#492C2318yescomplemented
150#487—24no—
151#487C2318yescomplemented
152#481—28no—
153#475—24no—
154#475C2318yescomplemented
155#463—216no—
156#461—216no—
157#461—24no—
158#461C2318yes—
159#460—24no—
160#460C2318yescomplemented
161#459—24no—
162#459C2318yes—
163#448—28no—
164#448—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—24no—
180C2×#103—22no—
181C2×#102—22no—
182C2×#102—22no—
183C2×C22⋊C8—21no—
184C2×C22⋊C8—21no—
185C2×C4×C8—21no—
186C2×C4×C8C2312yes—
187#333C23116yescomplemented
188Q8⋊C8C23116yescomplemented
189D4⋊C8C23116yes—
190C25—21no—
191C2×#41—24no—
192C2×#41—42no—
193C2×#41—44no—
194C2×#41—44no—
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—21no—
200C22×Q8—44no—
201C22×Q8—84no—
202C22×Q8C2×D412yes—
203C22×Q8C2×D414yes—
204C22×Q8C2×D414yes—
205C22×Q8C2×D418yes—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—24no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—41no—
213C22×D4SD1618yescomplemented
214C22×D4C2×D412yes—
215C22×D4C2×D414yes—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—216no—
221C23×C4—28no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—48no—
227C23×C4—41no—
228C23×C4—88no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232AGL(1,ℤ/8)—432no—
233C2×SD16—28no—
234C2×SD16—28no—
235C2×SD16—216no—
236C2×SD16—48no—
237C2×SD16—48no—
238C2×SD16—88no—
239C2×D8—28no—
240C2×D8—48no—
241C2×M16—24no—
242C2×M16—44no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8C2×D418yes—
257C4⋊Q8C2×D414yes—
258C4⋊Q8C2412yes—
259C4⋊Q8C2414yes—
260#127—21no—
261#127—21no—
262#127—21no—
263#127C2412yes—
264#125—216no—
265#125—232no—
266#125—216no—
267#124—24no—
268#124—42no—
269C22⋊Q8—216no—
270C22⋊Q8—232no—
271C22⋊Q8—232no—
272C22⋊Q8—264no—
273C22⋊Q8—216no—
274C22⋊Q8—232no—
275#121—24no—
276#121—24no—
277#121—42no—
278#121—42no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—22no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—22no—
286C4×Q8C2×D418yes—
287C4×Q8C2×D414yes—
288C4×Q8C2414yes—
289C4×D4—216no—
290C4×D4—232no—
291C4×D4—216no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—42no—
302C4×D4—42no—
303C4×D4C2×D414yes—
304C4×D4C2×D418yes—
305C4×D4C2414yes—
306C2×C4⋊C4—216no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—216no—
309C2×C4⋊C4—232no—
310C2×C4⋊C4—216no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—216no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D414yes—
335C2×C22⋊C4—216no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—41no—
342C2×C22⋊C4—41no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355#106—28no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360C4⋊C8C2414yes—
361C4⋊C8C2414yes—
362#103—28no—
363#103—216no—
364#103—28no—
365#102—28no—
366#102—28no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369C4×C8—24no—
370C4×C8—22no—
371C4×C8C2414yes—
372C24—24no—
373C24—22no—
374C24—28no—
375C24—44no—
376C24—44no—
377C24—41no—
378C24#12011yescharacteristic
379#41—48no—
380#41—48no—
381#41—88no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—28no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—48no—
390C2×Q8—44no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×Q8—41no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—84no—
397C2×Q8—88no—
398C2×Q8—88no—
399C2×Q8C22×D414yes—
400C2×Q8C22×D418yes—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—28no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—28no—
409C2×D4—216no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—84no—
419C2×D4C2×SD1618yes—
420C2×D4C2×SD16116yescomplemented
421C2×D4C22×D414yes—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—232no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—41no—
445C22×C4—44no—
446C22×C4—832no—
447C22×C4—816no—
448C22×C4—84no—
449C22×C4#12012yes—
450C22×C4#12011yescharacteristic
451C22×C4#12011yescharacteristic
452C22×C4#12012yes—
453C22×C4#12012yes—
454C22×C4#12011yescharacteristic
455C22×C4C2×SD1614yes—
456C22×C4#13714yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463SD16—416no—
464SD16—416no—
465SD16—816no—
466D8—416no—
467M16—48no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—44no—
474C4⋊C4—232no—
475C4⋊C4—232no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4C22×D414yes—
494C4⋊C4C22×D414yes—
495C22⋊C4—232no—
496C22⋊C4—28no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C42—28no—
502C42—24no—
503C42—24no—
504C42#12012yes—
505C42#12012yes—
506C42#12012yes—
507C42C22×D412yes—
508C42C22×D411yescharacteristic
509C42C22×D412yes—
510C42C22×D411yescharacteristic
511C42C22×D412yes—
512C42C22×D411yescharacteristic
513C42C2511yescharacteristic
514C23—22no—
515C23—22no—
516C23—24no—
517C23—28no—
518C23—216no—
519C23—48no—
520C23—48no—
521C23—432no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23#44814yes—
529C23C2×#12012yes—
530C23C2×#12011yescharacteristic
531Q8—24no—
532Q8—24no—
533Q8—48no—
534Q8—48no—
535Q8—44no—
536Q8—42no—
537Q8—42no—
538Q8—84no—
539Q8—88no—
540Q8—82no—
541D4—28no—
542D4—216no—
543D4—24no—
544D4—44no—
545D4—42no—
546D4—42no—
547D4—44no—
548D4—82no—
549D4—88no—
550D4—82no—
551D4—82no—
552D4C22×SD1618yes—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—416no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—416no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—832no—
577C2×C4—82no—
578C2×C4—88no—
579C2×C4#44814yes—
580C2×C4#44818yes—
581C2×C4C2×#12012yes—
582C2×C4C2×#12011yescharacteristic
583C2×C4C2×#12012yes—
584C2×C4C2×#12012yes—
585C2×C4C2×#12011yescharacteristic
586C2×C4C2×#12011yescharacteristic
587C2×C4C22×SD1612yes—
588C2×C4C22×SD1614yes—
589C2×C4C2×#13712yes—
590C2×C4C2×#13714yes—
591C2×C4#57514yes—
592C2×C4C23×D411yescharacteristic
593C2×C4C23×D411yescharacteristic
594C2×C4C23×D411yescharacteristic
595C8—44no—
596C8—44no—
597C8—44no—
598C22—28no—
599C22—22no—
600C22—22no—
601C22—28no—
602C22—432no—
603C22—416no—
604C22—44no—
605C22—44no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—164no—
610C22C2×#44814yes—
611C22C2×#44812yes—
612C22#300512yes—
613C22#301912yes—
614C22C22×#12011yescharacteristic
615C4—22no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—48no—
623C4—82no—
624C4C2×#44814yes—
625C4C2×#44812yes—
626C4#300512yes—
627C4C23×SD1611yescharacteristic
628C4C22×#13711yescharacteristic
629C4C2×#57512yes—
630C2—24no—
631C2—48no—
632C2—162no—
633C2#1874814yesdirect product factor
634C2C22×#44811yescharacteristic
635C2C2×#300511yescharacteristic
636C2C2×#301911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC2×#3036C2
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×#554C2semidirect
2. C2×#3206C2semidirect
3. C2×#3103C2semidirect
5. #18748C2direct (N × H)
6. C2×#1643C2semidirect
7. C2×#1622C2semidirect
8. C2×#1548C2semidirect
9. C2×#1487C2semidirect
10. C2×#1478C2semidirect
19. C2×#554C22semidirect
30. #3206C22semidirect
31. C2×#492C22semidirect
32. C2×#487C22semidirect
34. #3103C22semidirect
36. C2×#475C22semidirect
39. C2×#460C22semidirect
51. #1643C22semidirect
52. #1622C22semidirect
53. #1548C22semidirect
54. C2×#333C22semidirect
55. #1487C22semidirect
56. #1478C22semidirect
57. C2×Q8⋊C8C22semidirect
149. #492C23semidirect
151. #487C23semidirect
154. #475C23semidirect
160. #460C23semidirect
187. #333C23semidirect
188. Q8⋊C8C23semidirect
213. C22×D4SD16semidirect
420. C2×D4C2×SD16semidirect
633. C2#18748direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#32062
C2×#31032
C2×#30362
#187482
C2×#16432
C2×#16222
C2×#15482
C2×#14872
C2×#14782