Finite Group Databasebeta

← C2×#18768 #6348683

Subgroups — C2×#18768 #6348683

Order 512.

3186 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q82C211yesmaximal, characteristic, complemented
2C2×#3206C211yesmaximal, characteristic
3C2×#3103C211yesmaximal, characteristic
4C2×#3037C211yesmaximal, characteristic
5#18768C2116yesmaximal, direct product factor
6C2×#1645C214yesmaximal, complemented
7C2×#1623C214yesmaximal, complemented
8C2×#1548C211yesmaximal, characteristic, complemented
9C2×#1487C211yesmaximal, characteristic, complemented
10C2×#1479C211yesmaximal, characteristic, complemented
11C22×#137—21no—
12C2×Q82—22no—
13C2×Q82—24no—
14C2×Q82—22no—
15C2×Q82—22no—
16C2×Q82—24no—
17C2×Q82—24no—
18C2×Q82C2214yes—
19C2×Q82C2218yescomplemented
20C2×#532C2211yescharacteristic
21C22×C4⋊Q8—22no—
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8—21no—
24C22×C4⋊Q8C2211yescharacteristic
25C22×C4×Q8—21no—
26C22×C4×Q8—21no—
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29#3206—24no—
30#3206C2218yes—
31C2×#492C2212yes—
32C2×#487C2212yes—
33#3103—24no—
34#3103C2218yes—
35C2×#481—21no—
36C2×#475C2212yes—
37#3037—24no—
38#3037C2218yes—
39C2×#464—22no—
40C2×#462C2212yes—
41C2×#460C2212yes—
42C2×#460C2212yes—
43C2×#461—22no—
44C2×#448—22no—
45C2×#448—21no—
46C2×C4×SD16—22no—
47C2×#431C2211yescharacteristic
48C2×#431C2211yescharacteristic
49C2×#423C2211yescharacteristic
50C2×#417—21no—
51#1645C22132yescomplemented
52#1623C22132yescomplemented
53#1548C2218yescomplemented
54C2×#333C2214yes—
55#1487C2218yescomplemented
56#1479C2218yescomplemented
57C2×Q8⋊C8C2214yes—
58C2×Q8⋊C8C2214yes—
59C22×#41—21no—
60C23×Q8—22no—
61C23×Q8D411yescharacteristic
62C23×Q8D411yescharacteristic
63C23×Q8D412yes—
64C2×#137—28no—
65C2×#137—48no—
66C22×Q16—22no—
67C22×SD16—22no—
68C22×SD16—22no—
69C22×SD16—24no—
70C22×M16—21no—
71Q82—216no—
72Q82—216no—
73Q82—216no—
74Q82—28no—
75Q82—216no—
76Q82—28no—
77Q82—28no—
78Q82—216no—
79Q82—216no—
80Q82C23116yes—
81#532—22no—
82#532—22no—
83#532—22no—
84#532C2314yes—
85C2×C4⋊Q8—216no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—216no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8D414yes—
102C2×C4⋊Q8D412yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2314yes—
106C2×#127C2311yescharacteristic
107C2×#124—21no—
108C2×#121—21no—
109C2×#121—21no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—24no—
113C2×C4×Q8—24no—
114C2×C4×Q8—24no—
115C2×C4×Q8—24no—
116C2×C4×Q8—28no—
117C2×C4×Q8—28no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8—22no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—22no—
125C2×C4×Q8—22no—
126C2×C4×Q8D414yes—
127C2×C4×Q8D412yes—
128C2×C4×Q8D412yes—
129C2×C4×Q8D414yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2314yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2314yes—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4D411yescharacteristic
143C22×C4⋊C4D411yescharacteristic
144C22×C42—22no—
145C22×C42—21no—
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#492—24no—
149#492C2318yes—
150#487—24no—
151#487C2318yes—
152#481—28no—
153#475—24no—
154#475C2318yes—
155#464—216no—
156#462—24no—
157#462C2318yes—
158#461—216no—
159#460—24no—
160#460—24no—
161#460C2318yes—
162#460C2318yes—
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#333C23116yes—
188Q8⋊C8C23116yes—
189Q8⋊C8C23116yes—
190C2×#41—24no—
191C2×#41—42no—
192C2×#41—44no—
193C2×#41—44no—
194C22×Q8—28no—
195C22×Q8—216no—
196C22×Q8—28no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—21no—
200C22×Q8—21no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—21no—
205C22×Q8—84no—
206C22×Q8—88no—
207C22×Q8SD1618yescomplemented
208C22×Q8C2×D414yes—
209C22×Q8C2×D412yes—
210C22×Q8C2×D414yes—
211C22×Q8C2×D412yes—
212C22×Q8C2×D414yes—
213C22×Q8C2×D418yes—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—41no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—22no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—41no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227#137—432no—
228C2×Q16—28no—
229C2×Q16—48no—
230C2×SD16—28no—
231C2×SD16—28no—
232C2×SD16—216no—
233C2×SD16—48no—
234C2×SD16—48no—
235C2×SD16—88no—
236C2×M16—24no—
237C2×M16—44no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C4⋊Q8—232no—
242C4⋊Q8—264no—
243C4⋊Q8—216no—
244C4⋊Q8—216no—
245C4⋊Q8—232no—
246C4⋊Q8—232no—
247C4⋊Q8—232no—
248C4⋊Q8—216no—
249C4⋊Q8—216no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8C2×D418yes—
261C4⋊Q8C2×D414yes—
262C4⋊Q8C2412yes—
263C4⋊Q8C2414yes—
264#127—21no—
265#127—21no—
266#127—21no—
267#127C2412yes—
268#124—24no—
269#124—42no—
270#121—24no—
271#121—24no—
272#121—42no—
273#121—42no—
274C4×Q8—216no—
275C4×Q8—216no—
276C4×Q8—232no—
277C4×Q8—232no—
278C4×Q8—216no—
279C4×Q8—216no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8C2×D418yes—
295C4×Q8C2×D414yes—
296C4×Q8C2×D414yes—
297C4×Q8C2×D418yes—
298C4×Q8C2414yes—
299C4×Q8C2414yes—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—42no—
303C4×D4—42no—
304C2×C4⋊C4—216no—
305C2×C4⋊C4—216no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2×D414yes—
334C2×C22⋊C4—41no—
335C2×C22⋊C4—41no—
336C2×C42—216no—
337C2×C42—28no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354#106—28no—
355C4⋊C8—24no—
356C4⋊C8—24no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8C2414yes—
360C4⋊C8C2414yes—
361#103—28no—
362#103—216no—
363#103—28no—
364#102—28no—
365#102—28no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368C4×C8—24no—
369C4×C8—22no—
370C4×C8C2414yes—
371C24—41no—
372C24#12011yescharacteristic
373#41—48no—
374#41—48no—
375#41—88no—
376C2×Q8—216no—
377C2×Q8—28no—
378C2×Q8—216no—
379C2×Q8—232no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—28no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—28no—
388C2×Q8—24no—
389C2×Q8—28no—
390C2×Q8—216no—
391C2×Q8—22no—
392C2×Q8—48no—
393C2×Q8—41no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—84no—
397C2×Q8—88no—
398C2×Q8—88no—
399C2×Q8—832no—
400C2×Q8—816no—
401C2×Q8C2×SD1618yes—
402C2×Q8C2×SD16116yescomplemented
403C2×Q8#13718yescomplemented
404C2×Q8C22×D414yes—
405C2×Q8C22×D414yes—
406C2×Q8C22×D418yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—84no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—41no—
437C22×C4—44no—
438C22×C4—84no—
439C22×C4#12012yes—
440C22×C4#12011yescharacteristic
441C22×C4#12011yescharacteristic
442C22×C4#12012yes—
443C22×C4#12012yes—
444C22×C4#12011yescharacteristic
445C22×C4C2×SD1614yes—
446C22×C4#13714yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453Q16—416no—
454SD16—416no—
455SD16—416no—
456SD16—816no—
457M16—48no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—44no—
462C2×C8—44no—
463C2×C8—44no—
464C4⋊C4—232no—
465C4⋊C4—232no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4C22×D414yes—
487C4⋊C4C22×D414yes—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C42—232no—
491C42—28no—
492C42—28no—
493C42—24no—
494C42—24no—
495C42—24no—
496C42—24no—
497C42#12012yes—
498C42#12012yes—
499C42#12012yes—
500C42C22×D412yes—
501C42C22×D411yescharacteristic
502C42C22×D412yes—
503C42C22×D412yes—
504C42C22×D411yescharacteristic
505C42C22×D411yescharacteristic
506C42C2511yescharacteristic
507C23—22no—
508C23—22no—
509C23—42no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23#44814yes—
514C23C2×#12012yes—
515C23C2×#12011yescharacteristic
516Q8—216no—
517Q8—28no—
518Q8—216no—
519Q8—28no—
520Q8—24no—
521Q8—24no—
522Q8—48no—
523Q8—42no—
524Q8—42no—
525Q8—84no—
526Q8—88no—
527Q8—832no—
528Q8—82no—
529Q8C22×SD1618yes—
530Q8C2×#13718yescomplemented
531D4—44no—
532D4—42no—
533D4—44no—
534D4—42no—
535D4—82no—
536D4—88no—
537D4—82no—
538D4—82no—
539C2×C4—216no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—216no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—416no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—82no—
563C2×C4—88no—
564C2×C4#44814yes—
565C2×C4#44818yes—
566C2×C4C2×#12012yes—
567C2×C4C2×#12012yes—
568C2×C4C2×#12011yescharacteristic
569C2×C4C2×#12012yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4C2×#12011yescharacteristic
572C2×C4C22×SD1612yes—
573C2×C4C22×SD1614yes—
574C2×C4C2×#13712yes—
575C2×C4C2×#13714yes—
576C2×C4C2×#13714yes—
577C2×C4C23×D411yescharacteristic
578C2×C4C23×D411yescharacteristic
579C2×C4C23×D411yescharacteristic
580C8—44no—
581C8—44no—
582C8—44no—
583C22—28no—
584C22—22no—
585C22—22no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—164no—
590C22C2×#44814yes—
591C22C2×#44812yes—
592C22#300412yes—
593C22#301912yes—
594C22C22×#12011yescharacteristic
595C4—28no—
596C4—28no—
597C4—22no—
598C4—21no—
599C4—21no—
600C4—22no—
601C4—48no—
602C4—44no—
603C4—44no—
604C4—82no—
605C4C2×#44814yes—
606C4C2×#44812yes—
607C4#300412yes—
608C4C23×SD1611yescharacteristic
609C4C22×#13711yescharacteristic
610C4C22×#13712yes—
611C2—24no—
612C2—162no—
613C2#1876814yesdirect product factor
614C2C22×#44811yescharacteristic
615C2C2×#300411yescharacteristic
616C2C2×#301911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC2×#532C22
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×Q82C2semidirect
5. #18768C2direct (N × H)
6. C2×#1645C2semidirect
7. C2×#1623C2semidirect
8. C2×#1548C2semidirect
9. C2×#1487C2semidirect
10. C2×#1479C2semidirect
19. C2×Q82C22semidirect
51. #1645C22semidirect
52. #1623C22semidirect
53. #1548C22semidirect
55. #1487C22semidirect
56. #1479C22semidirect
207. C22×Q8SD16semidirect
402. C2×Q8C2×SD16semidirect
403. C2×Q8#137semidirect
530. Q8C2×#137semidirect
613. C2#18768direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q822
C2×#32062
C2×#31032
C2×#30372
#187682
C2×#16452
C2×#16232
C2×#15482
C2×#14872
C2×#14792