Finite Group Databasebeta

← C2×#18767 #6348682

Subgroups — C2×#18767 #6348682

Order 512.

4276 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#550C211yesmaximal, characteristic
2C2×#3208C211yesmaximal, characteristic, complemented
3C2×#3101C211yesmaximal, characteristic, complemented
4C2×#3078C211yesmaximal, characteristic, complemented
5#18767C2116yesmaximal, direct product factor
6C2×#1642C214yesmaximal, complemented
7C2×#1631C214yesmaximal, complemented
8C2×#1545C211yesmaximal, characteristic, complemented
9C2×#1486C211yesmaximal, characteristic, complemented
10C2×#1479C211yesmaximal, characteristic, complemented
11C23×#41—21no—
12C2×#550—24no—
13C2×#550—22no—
14C2×#550—22no—
15C2×#550—24no—
16C2×#550—24no—
17C2×#550—22no—
18C2×#550C2214yes—
19C2×#550C2218yes—
20C2×#533C2211yescharacteristic
21C22×#127—21no—
22C22×#127C2211yescharacteristic
23C22×#121—22no—
24C22×#121—21no—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4—21no—
27C22×C4×D4C2211yescharacteristic
28#3208—24no—
29#3208C2218yescomplemented
30C2×#492C2212yescomplemented
31C2×#486C2212yescomplemented
32#3101—24no—
33#3101C2218yescomplemented
34C2×#483—21no—
35#3078—24no—
36#3078C2218yescomplemented
37C2×#479—22no—
38C2×#477C2212yescomplemented
39C2×#475C2212yescomplemented
40C2×#476C2212yescomplemented
41C2×#460—22no—
42C2×#461C2212yescomplemented
43C2×#450—22no—
44C2×#450—21no—
45C2×C4×SD16—22no—
46C2×#431C2211yescharacteristic
47C2×#431C2211yescharacteristic
48C2×#428—21no—
49C2×#423C2211yescharacteristic
50C2×#419—21no—
51#1642C22132yescomplemented
52#1631C22132yescomplemented
53#1545C2218yescomplemented
54C2×#331C2214yescomplemented
55#1486C2218yescomplemented
56#1479C2218yescomplemented
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yescomplemented
59C22×#41—28no—
60C22×#41—24no—
61C22×#41—24no—
62C22×#41—28no—
63C22×#41—48no—
64C22×#41—48no—
65C23×Q8D411yescharacteristic
66C23×D4—22no—
67C23×D4—22no—
68C23×D4—21no—
69C23×D4—21no—
70C23×D4D411yescharacteristic
71C23×D4D412yes—
72C24×C4—22no—
73C24×C4—21no—
74C22×SD16—22no—
75C22×SD16—24no—
76C22×M16—21no—
77#550—216no—
78#550—216no—
79#550—216no—
80#550—216no—
81#550—28no—
82#550—28no—
83#550—216no—
84#550—216no—
85#550—28no—
86#550C23116yes—
87#533—22no—
88#533—22no—
89#533—22no—
90#533C2314yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127—28no—
94C2×#127—24no—
95C2×#127—24no—
96C2×#127—28no—
97C2×#127—21no—
98C2×#127—22no—
99C2×#127—21no—
100C2×#127—22no—
101C2×#127D414yes—
102C2×#127D412yes—
103C2×#127C2312yes—
104C2×#127C2314yes—
105C2×#125—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×#121—28no—
109C2×#121—24no—
110C2×#121—216no—
111C2×#121—28no—
112C2×#121—28no—
113C2×#121—216no—
114C2×#121—24no—
115C2×#121—28no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8D412yes—
123C2×C4×Q8D414yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2314yes—
126C2×C4×D4—28no—
127C2×C4×D4—24no—
128C2×C4×D4—24no—
129C2×C4×D4—28no—
130C2×C4×D4—21no—
131C2×C4×D4—22no—
132C2×C4×D4—21no—
133C2×C4×D4—22no—
134C2×C4×D4D414yes—
135C2×C4×D4D412yes—
136C2×C4×D4C2314yes—
137C2×C4×D4C2312yes—
138C2×#117—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4D411yescharacteristic
141C22×C4⋊C4D411yescharacteristic
142C22×C22⋊C4—22no—
143C22×C22⋊C4—21no—
144C22×C42—21no—
145C22×C42C2311yescharacteristic
146#492—24no—
147#492C2318yescomplemented
148#486—24no—
149#486C2318yescomplemented
150#483—28no—
151#479—216no—
152#477—24no—
153#477C2318yescomplemented
154#476—24no—
155#476C2318yescomplemented
156#475—24no—
157#475C2318yescomplemented
158#461—24no—
159#461C2318yescomplemented
160#460—216no—
161#450—28no—
162#450—216no—
163C4×SD16—216no—
164#431—22no—
165#431—22no—
166#431C2314yes—
167#431C2314yes—
168#428—28no—
169C2×#107—22no—
170C2×#106—22no—
171C2×#106—22no—
172#423—22no—
173#423C2314yes—
174C2×C4⋊C8—21no—
175C2×C4⋊C8—21no—
176C2×C4⋊C8C2312yes—
177C2×C4⋊C8C2312yes—
178#419—28no—
179C2×#103—22no—
180C2×#103—22no—
181C2×#103—22no—
182C2×#102—24no—
183C2×#102—22no—
184C2×C22⋊C8—21no—
185C2×C22⋊C8—21no—
186C2×C4×C8—21no—
187C2×C4×C8C2312yes—
188#331C23116yescomplemented
189Q8⋊C8C23116yescomplemented
190D4⋊C8C23116yescomplemented
191C25—22no—
192C25—21no—
193C2×#41—232no—
194C2×#41—216no—
195C2×#41—216no—
196C2×#41—416no—
197C2×#41—432no—
198C2×#41—432no—
199C2×#41—464no—
200C2×#41—416no—
201C2×#41—432no—
202C22×Q8—22no—
203C22×Q8—21no—
204C22×Q8—24no—
205C22×Q8—21no—
206C22×Q8—21no—
207C22×Q8SD1618yescomplemented
208C22×Q8C2×D412yes—
209C22×Q8C2×D414yes—
210C22×D4—216no—
211C22×D4—28no—
212C22×D4—216no—
213C22×D4—28no—
214C22×D4—22no—
215C22×D4—21no—
216C22×D4—24no—
217C22×D4—22no—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—22no—
224C22×D4—24no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4—48no—
228C22×D4—416no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4—84no—
232C22×D4—88no—
233C22×D4C2×D412yes—
234C22×D4C2×D414yes—
235C22×D4C2×D414yes—
236C22×D4C2×D418yes—
237C23×C4—21no—
238C23×C4—216no—
239C23×C4—28no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4—22no—
246C23×C4—416no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C2×SD16—28no—
253C2×SD16—216no—
254C2×SD16—48no—
255C2×SD16—88no—
256C2×M16—24no—
257C2×M16—44no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8C2412yes—
268C4⋊Q8C2412yes—
269#127—232no—
270#127—216no—
271#127—216no—
272#127—24no—
273#127—24no—
274#127—24no—
275#127—24no—
276#127—24no—
277#127—22no—
278#127—22no—
279#127C2×D418yes—
280#127C2×D414yes—
281#127C2414yes—
282#125—24no—
283#125—42no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—42no—
287C22⋊Q8—42no—
288#121—232no—
289#121—216no—
290#121—264no—
291#121—232no—
292#121—232no—
293#121—216no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—22no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—22no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—42no—
304C4×Q8—42no—
305C4×Q8C2×D414yes—
306C4×Q8C2×D418yes—
307C4×Q8C2414yes—
308C4×D4—232no—
309C4×D4—216no—
310C4×D4—216no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—22no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—22no—
318C4×D4C2×D418yes—
319C4×D4C2×D414yes—
320C4×D4C2414yes—
321#117—24no—
322#117—42no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—41no—
336C2×C4⋊C4—41no—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D414yes—
341C2×C22⋊C4—216no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—41no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—41no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361#107—28no—
362#106—28no—
363#106—28no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—22no—
367C4⋊C8—22no—
368C4⋊C8C2414yes—
369C4⋊C8C2414yes—
370#103—28no—
371#103—28no—
372#103—28no—
373#102—28no—
374#102—216no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377C4×C8—24no—
378C4×C8—22no—
379C4×C8C2414yes—
380C24—28no—
381C24—24no—
382C24—22no—
383C24—24no—
384C24—44no—
385C24—48no—
386C24—48no—
387C24—44no—
388C24—48no—
389C24—44no—
390C24#12011yescharacteristic
391#41—464no—
392#41—4128no—
393#41—432no—
394#41—432no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—216no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—41no—
405C2×Q8—41no—
406C2×Q8—44no—
407C2×Q8C2×SD1618yes—
408C2×Q8C2×SD16116yescomplemented
409C2×Q8C22×D414yes—
410C2×D4—232no—
411C2×D4—232no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—28no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—24no—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—48no—
425C2×D4—44no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—416no—
429C2×D4—416no—
430C2×D4—432no—
431C2×D4—464no—
432C2×D4—48no—
433C2×D4—44no—
434C2×D4—48no—
435C2×D4—44no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—416no—
439C2×D4—48no—
440C2×D4—84no—
441C2×D4—88no—
442C2×D4—88no—
443C2×D4—832no—
444C2×D4—816no—
445C2×D4C22×D414yes—
446C2×D4C22×D418yes—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—232no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—42no—
460C22×C4—432no—
461C22×C4—464no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—416no—
469C22×C4—48no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4#12012yes—
473C22×C4#12011yescharacteristic
474C22×C4#12011yescharacteristic
475C22×C4#12012yes—
476C22×C4#12012yes—
477C22×C4#12011yescharacteristic
478C22×C4C2×SD1614yes—
479C22×C4AGL(1,ℤ/8)14yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486SD16—416no—
487SD16—816no—
488M16—48no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—44no—
493C2×C8—44no—
494C2×C8—44no—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4C22×D414yes—
513C4⋊C4C22×D414yes—
514C22⋊C4—232no—
515C22⋊C4—28no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—44no—
519C42—28no—
520C42—24no—
521C42—24no—
522C42—44no—
523C42#12012yes—
524C42#12012yes—
525C42#12012yes—
526C42C22×D411yescharacteristic
527C42C22×D412yes—
528C42C22×D412yes—
529C42C22×D412yes—
530C42C22×D411yescharacteristic
531C42C22×D411yescharacteristic
532C42C2511yescharacteristic
533C23—22no—
534C23—22no—
535C23—24no—
536C23—28no—
537C23—48no—
538C23—416no—
539C23—416no—
540C23—48no—
541C23—416no—
542C23—48no—
543C23—48no—
544C23—44no—
545C23—44no—
546C23—48no—
547C23—48no—
548C23—44no—
549C23—816no—
550C23—832no—
551C23#45014yes—
552C23C2×#12012yes—
553C23C2×#12011yescharacteristic
554Q8—28no—
555Q8—216no—
556Q8—24no—
557Q8—44no—
558Q8—44no—
559Q8—82no—
560Q8—82no—
561Q8C22×SD1618yes—
562D4—24no—
563D4—24no—
564D4—48no—
565D4—48no—
566D4—416no—
567D4—464no—
568D4—416no—
569D4—48no—
570D4—44no—
571D4—48no—
572D4—44no—
573D4—44no—
574D4—48no—
575D4—84no—
576D4—88no—
577D4—832no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—216no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—464no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—416no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4#45018yes—
604C2×C4#45014yes—
605C2×C4C2×#12012yes—
606C2×C4C2×#12012yes—
607C2×C4C2×#12011yescharacteristic
608C2×C4C2×#12012yes—
609C2×C4C2×#12011yescharacteristic
610C2×C4C2×#12011yescharacteristic
611C2×C4C22×SD1612yes—
612C2×C4C22×SD1614yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4C2×AGL(1,ℤ/8)14yes—
615C2×C4#57514yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C23×D411yescharacteristic
618C2×C4C23×D411yescharacteristic
619C8—44no—
620C8—44no—
621C8—44no—
622C22—28no—
623C22—22no—
624C22—22no—
625C22—48no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—48no—
630C22—44no—
631C22—816no—
632C22—832no—
633C22—88no—
634C22—816no—
635C22C2×#45014yes—
636C22C2×#45012yes—
637C22#300512yes—
638C22#301912yes—
639C22C22×#12011yescharacteristic
640C4—28no—
641C4—22no—
642C4—21no—
643C4—22no—
644C4—21no—
645C4—44no—
646C4—44no—
647C4—82no—
648C4—82no—
649C4C2×#45014yes—
650C4C2×#45012yes—
651C4#300512yes—
652C4C23×SD1611yescharacteristic
653C4C22×AGL(1,ℤ/8)11yescharacteristic
654C4C2×#57512yes—
655C2—24no—
656C2—84no—
657C2—88no—
658C2#1876714yesdirect product factor
659C2C22×#45011yescharacteristic
660C2C2×#300511yescharacteristic
661C2C2×#301911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC22×#550C2
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
2. C2×#3208C2semidirect
3. C2×#3101C2semidirect
4. C2×#3078C2semidirect
5. #18767C2direct (N × H)
6. C2×#1642C2semidirect
7. C2×#1631C2semidirect
8. C2×#1545C2semidirect
9. C2×#1486C2semidirect
10. C2×#1479C2semidirect
29. #3208C22semidirect
30. C2×#492C22semidirect
31. C2×#486C22semidirect
33. #3101C22semidirect
36. #3078C22semidirect
38. C2×#477C22semidirect
39. C2×#475C22semidirect
40. C2×#476C22semidirect
42. C2×#461C22semidirect
51. #1642C22semidirect
52. #1631C22semidirect
53. #1545C22semidirect
54. C2×#331C22semidirect
55. #1486C22semidirect
56. #1479C22semidirect
57. C2×Q8⋊C8C22semidirect
58. C2×D4⋊C8C22semidirect
147. #492C23semidirect
149. #486C23semidirect
153. #477C23semidirect
155. #476C23semidirect
157. #475C23semidirect
159. #461C23semidirect
188. #331C23semidirect
189. Q8⋊C8C23semidirect
190. D4⋊C8C23semidirect
207. C22×Q8SD16semidirect
408. C2×Q8C2×SD16semidirect
658. C2#18767direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5502
C2×#32082
C2×#31012
C2×#30782
#187672
C2×#16422
C2×#16312
C2×#15452
C2×#14862
C2×#14792