Finite Group Databasebeta

← C2×#18771 #6348686

Subgroups — C2×#18771 #6348686

Order 512.

4276 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#550C211yesmaximal, characteristic
2C2×#3208C211yesmaximal, characteristic, complemented
3C2×#3104C211yesmaximal, characteristic, complemented
4C2×#3077C211yesmaximal, characteristic, complemented
5#18771C2116yesmaximal, direct product factor
6C2×#1641C214yesmaximal, complemented
7C2×#1631C214yesmaximal, complemented
8C2×#1545C211yesmaximal, characteristic, complemented
9C2×#1487C211yesmaximal, characteristic, complemented
10C2×#1478C211yesmaximal, characteristic, complemented
11C23×#41—21no—
12C2×#550—24no—
13C2×#550—22no—
14C2×#550—24no—
15C2×#550—22no—
16C2×#550—22no—
17C2×#550—24no—
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#3104—24no—
33#3104C2218yescomplemented
34C2×#484—21no—
35#3077—24no—
36#3077C2218yescomplemented
37C2×#477C2212yescomplemented
38C2×#478—22no—
39C2×#476C2212yescomplemented
40C2×#474C2212yescomplemented
41C2×#462C2212yescomplemented
42C2×#460—22no—
43C2×#451—21no—
44C2×#450—22no—
45C2×C4×Q16—22no—
46C2×#431C2211yescharacteristic
47C2×#431C2211yescharacteristic
48C2×#428—21no—
49C2×#423C2211yescharacteristic
50C2×#418—21no—
51#1641C22132yescomplemented
52#1631C22132yescomplemented
53#1545C2218yescomplemented
54C2×#331C2214yescomplemented
55#1487C2218yescomplemented
56#1478C2218yescomplemented
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yescomplemented
59C22×#41—28no—
60C22×#41—24no—
61C22×#41—28no—
62C22×#41—24no—
63C22×#41—416no—
64C23×Q8D411yescharacteristic
65C23×D4—22no—
66C23×D4—22no—
67C23×D4—21no—
68C23×D4—21no—
69C23×D4D411yescharacteristic
70C23×D4D412yes—
71C24×C4—22no—
72C24×C4—21no—
73C22×Q16—22no—
74C22×SD16—24no—
75C22×M16—21no—
76#550—216no—
77#550—216no—
78#550—216no—
79#550—216no—
80#550—28no—
81#550—216no—
82#550—28no—
83#550—28no—
84#550—216no—
85#550C23116yes—
86#533—22no—
87#533—22no—
88#533—22no—
89#533C2314yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127—28no—
93C2×#127—24no—
94C2×#127—24no—
95C2×#127—28no—
96C2×#127—21no—
97C2×#127—22no—
98C2×#127—21no—
99C2×#127—22no—
100C2×#127D414yes—
101C2×#127D412yes—
102C2×#127C2312yes—
103C2×#127C2314yes—
104C2×#125—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×#121—28no—
108C2×#121—24no—
109C2×#121—216no—
110C2×#121—28no—
111C2×#121—28no—
112C2×#121—216no—
113C2×#121—24no—
114C2×#121—28no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8D414yes—
122C2×C4×Q8D412yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2314yes—
125C2×C4×D4—28no—
126C2×C4×D4—24no—
127C2×C4×D4—24no—
128C2×C4×D4—28no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—22no—
132C2×C4×D4—22no—
133C2×C4×D4D412yes—
134C2×C4×D4D414yes—
135C2×C4×D4C2314yes—
136C2×C4×D4C2312yes—
137C2×#117—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4D411yescharacteristic
140C22×C4⋊C4D411yescharacteristic
141C22×C22⋊C4—22no—
142C22×C22⋊C4—21no—
143C22×C42—21no—
144C22×C42C2311yescharacteristic
145#492—24no—
146#492C2318yescomplemented
147#486—24no—
148#486C2318yescomplemented
149#484—28no—
150#478—216no—
151#477—24no—
152#477C2318yescomplemented
153#476—24no—
154#476C2318yescomplemented
155#474—24no—
156#474C2318yescomplemented
157#462—24no—
158#462C2318yescomplemented
159#460—216no—
160#451—28no—
161#450—216no—
162C4×Q16—216no—
163#431—22no—
164#431—22no—
165#431C2314yes—
166#431C2314yes—
167#428—28no—
168C2×#107—22no—
169C2×#107—22no—
170C2×#106—22no—
171#423—22no—
172#423C2314yes—
173C2×C4⋊C8—21no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2312yes—
177#418—28no—
178C2×#103—22no—
179C2×#103—22no—
180C2×#102—22no—
181C2×#102—22no—
182C2×#102—24no—
183C2×C22⋊C8—21no—
184C2×C22⋊C8—21no—
185C2×C4×C8—21no—
186C2×C4×C8C2312yes—
187#331C23116yescomplemented
188Q8⋊C8C23116yescomplemented
189D4⋊C8C23116yescomplemented
190C25—21no—
191C25—22no—
192C2×#41—232no—
193C2×#41—216no—
194C2×#41—216no—
195C2×#41—464no—
196C2×#41—432no—
197C2×#41—464no—
198C2×#41—416no—
199C2×#41—416no—
200C22×Q8—28no—
201C22×Q8—22no—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8C2×D414yes—
206C22×Q8C2×D412yes—
207C22×D4—216no—
208C22×D4—28no—
209C22×D4—216no—
210C22×D4—28no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—24no—
214C22×D4—22no—
215C22×D4—24no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—24no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—22no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—416no—
226C22×D4—48no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—84no—
230C22×D4—88no—
231C22×D4C2×D412yes—
232C22×D4C2×D414yes—
233C22×D4C2×D414yes—
234C22×D4C2×D418yes—
235C23×C4—21no—
236C23×C4—216no—
237C23×C4—28no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—24no—
241C23×C4—22no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—416no—
245C23×C4—48no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C2×Q16—28no—
250C2×Q16—48no—
251C2×SD16—216no—
252C2×SD16—88no—
253C2×M16—24no—
254C2×M16—44no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8C2412yes—
265C4⋊Q8C2412yes—
266#127—232no—
267#127—216no—
268#127—216no—
269#127—24no—
270#127—24no—
271#127—24no—
272#127—24no—
273#127—22no—
274#127—24no—
275#127—22no—
276#127C2×D418yes—
277#127C2×D414yes—
278#127C2414yes—
279#125—24no—
280#125—42no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—42no—
284C22⋊Q8—42no—
285#121—232no—
286#121—216no—
287#121—264no—
288#121—232no—
289#121—232no—
290#121—216no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—22no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—22no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—42no—
301C4×Q8—42no—
302C4×Q8C2×D418yes—
303C4×Q8C2×D414yes—
304C4×Q8C2414yes—
305C4×D4—232no—
306C4×D4—216no—
307C4×D4—216no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—22no—
313C4×D4—24no—
314C4×D4—22no—
315C4×D4C2×D414yes—
316C4×D4C2×D418yes—
317C4×D4C2414yes—
318#117—24no—
319#117—42no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D414yes—
338C2×C22⋊C4—216no—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—41no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—41no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358#107—28no—
359#107—28no—
360#106—28no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363C4⋊C8—22no—
364C4⋊C8—22no—
365C4⋊C8C2414yes—
366C4⋊C8C2414yes—
367#103—28no—
368#103—28no—
369#102—28no—
370#102—28no—
371#102—216no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374C4×C8—24no—
375C4×C8—22no—
376C4×C8C2414yes—
377C24—24no—
378C24—28no—
379C24—22no—
380C24—24no—
381C24—48no—
382C24—44no—
383C24—48no—
384C24—44no—
385C24—44no—
386C24—48no—
387C24#12011yescharacteristic
388#41—4128no—
389#41—464no—
390#41—432no—
391#41—432no—
392C2×Q8—216no—
393C2×Q8—28no—
394C2×Q8—216no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—41no—
402C2×Q8—41no—
403C2×Q8—44no—
404C2×Q8AGL(1,ℤ/8)18yes—
405C2×Q8C22×D414yes—
406C2×D4—232no—
407C2×D4—232no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—28no—
418C2×D4—24no—
419C2×D4—28no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—464no—
423C2×D4—432no—
424C2×D4—48no—
425C2×D4—416no—
426C2×D4—416no—
427C2×D4—48no—
428C2×D4—44no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—48no—
432C2×D4—44no—
433C2×D4—48no—
434C2×D4—44no—
435C2×D4—416no—
436C2×D4—48no—
437C2×D4—84no—
438C2×D4—88no—
439C2×D4—88no—
440C2×D4—832no—
441C2×D4—816no—
442C2×D4C22×D414yes—
443C2×D4C22×D418yes—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—232no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—464no—
458C22×C4—432no—
459C22×C4—48no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4#12012yes—
468C22×C4#12011yescharacteristic
469C22×C4#12011yescharacteristic
470C22×C4#12012yes—
471C22×C4#12012yes—
472C22×C4#12011yescharacteristic
473C22×C4#13514yes—
474C22×C4#13714yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481Q16—416no—
482SD16—816no—
483M16—48no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—44no—
488C2×C8—44no—
489C2×C8—44no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4C22×D414yes—
508C4⋊C4C22×D414yes—
509C22⋊C4—232no—
510C22⋊C4—28no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—44no—
514C42—28no—
515C42—24no—
516C42—24no—
517C42—44no—
518C42#12012yes—
519C42#12012yes—
520C42#12012yes—
521C42C22×D412yes—
522C42C22×D411yescharacteristic
523C42C22×D412yes—
524C42C22×D411yescharacteristic
525C42C22×D412yes—
526C42C22×D411yescharacteristic
527C42C2511yescharacteristic
528C23—22no—
529C23—22no—
530C23—24no—
531C23—28no—
532C23—416no—
533C23—48no—
534C23—416no—
535C23—48no—
536C23—48no—
537C23—44no—
538C23—416no—
539C23—48no—
540C23—48no—
541C23—44no—
542C23—48no—
543C23—44no—
544C23—816no—
545C23—832no—
546C23#45214yes—
547C23C2×#12012yes—
548C23C2×#12011yescharacteristic
549Q8—216no—
550Q8—28no—
551Q8—24no—
552Q8—44no—
553Q8—44no—
554Q8—82no—
555Q8—82no—
556Q8C2×AGL(1,ℤ/8)18yes—
557D4—24no—
558D4—24no—
559D4—48no—
560D4—464no—
561D4—416no—
562D4—48no—
563D4—44no—
564D4—416no—
565D4—48no—
566D4—48no—
567D4—44no—
568D4—48no—
569D4—44no—
570D4—84no—
571D4—88no—
572D4—832no—
573C2×C4—216no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—464no—
586C2×C4—416no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4#45218yes—
598C2×C4#45214yes—
599C2×C4C2×#12012yes—
600C2×C4C2×#12011yescharacteristic
601C2×C4C2×#12012yes—
602C2×C4C2×#12012yes—
603C2×C4C2×#12011yescharacteristic
604C2×C4C2×#12011yescharacteristic
605C2×C4C2×#13512yes—
606C2×C4C2×#13514yes—
607C2×C4C2×AGL(1,ℤ/8)14yes—
608C2×C4C2×#13714yes—
609C2×C4C2×#13712yes—
610C2×C4C23×D411yescharacteristic
611C2×C4C23×D411yescharacteristic
612C2×C4C23×D411yescharacteristic
613C8—44no—
614C8—44no—
615C8—44no—
616C22—28no—
617C22—22no—
618C22—22no—
619C22—44no—
620C22—48no—
621C22—48no—
622C22—44no—
623C22—48no—
624C22—44no—
625C22—816no—
626C22—832no—
627C22—88no—
628C22—816no—
629C22C2×#45214yes—
630C22C2×#45212yes—
631C22#300312yes—
632C22#301912yes—
633C22C22×#12011yescharacteristic
634C4—28no—
635C4—22no—
636C4—21no—
637C4—22no—
638C4—21no—
639C4—44no—
640C4—44no—
641C4—82no—
642C4—82no—
643C4C2×#45214yes—
644C4C2×#45212yes—
645C4#300312yes—
646C4C22×#13511yescharacteristic
647C4C22×AGL(1,ℤ/8)12yes—
648C4C22×#13711yescharacteristic
649C2—24no—
650C2—84no—
651C2—88no—
652C2#1877114yesdirect product factor
653C2C22×#45211yescharacteristic
654C2C2×#300311yescharacteristic
655C2C2×#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×#3104C2semidirect
4. C2×#3077C2semidirect
5. #18771C2direct (N × H)
6. C2×#1641C2semidirect
7. C2×#1631C2semidirect
8. C2×#1545C2semidirect
9. C2×#1487C2semidirect
10. C2×#1478C2semidirect
29. #3208C22semidirect
30. C2×#492C22semidirect
31. C2×#486C22semidirect
33. #3104C22semidirect
36. #3077C22semidirect
37. C2×#477C22semidirect
39. C2×#476C22semidirect
40. C2×#474C22semidirect
41. C2×#462C22semidirect
51. #1641C22semidirect
52. #1631C22semidirect
53. #1545C22semidirect
54. C2×#331C22semidirect
55. #1487C22semidirect
56. #1478C22semidirect
57. C2×Q8⋊C8C22semidirect
58. C2×D4⋊C8C22semidirect
146. #492C23semidirect
148. #486C23semidirect
152. #477C23semidirect
154. #476C23semidirect
156. #474C23semidirect
158. #462C23semidirect
187. #331C23semidirect
188. Q8⋊C8C23semidirect
189. D4⋊C8C23semidirect
652. C2#18771direct (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×#31042
C2×#30772
#187712
C2×#16412
C2×#16312
C2×#15452
C2×#14872
C2×#14782