Finite Group Databasebeta

← C2×#18781 #6348697

Subgroups — C2×#18781 #6348697

Order 512.

2976 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#3229C211yesmaximal, characteristic, complemented
3C2×#3104C211yesmaximal, characteristic, complemented
4C2×#3077C211yesmaximal, characteristic
5#18781C2116yesmaximal, direct product factor
6C2×#1652C214yesmaximal, complemented
7C2×#1638C214yesmaximal, complemented
8C2×#1546C211yesmaximal, characteristic, complemented
9C2×#1487C211yesmaximal, characteristic, complemented
10C2×#1478C211yesmaximal, characteristic
11C2×#556—22no—
12C2×#556—24no—
13C2×#556—24no—
14C2×#556—22no—
15C2×#556—22no—
16C2×#556—24no—
17C2×#556C2214yes—
18C2×#556C2218yes—
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126—22no—
22C22×#124—21no—
23C22×C22⋊Q8—21no—
24C22×C4×Q8—21no—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×#117—21no—
28#3229—24no—
29#3229C2218yescomplemented
30C2×#498C2212yescomplemented
31C2×#487C2212yescomplemented
32#3104—24no—
33#3104C2218yescomplemented
34C2×#484—21no—
35C2×#483—22no—
36#3077—24no—
37#3077C2218yes—
38C2×#477C2212yescomplemented
39C2×#475—22no—
40C2×#478—22no—
41C2×#476C2212yes—
42C2×#474C2212yes—
43C2×#462C2212yescomplemented
44C2×#451—21no—
45C2×C4×Q16—22no—
46C2×#431C2211yescharacteristic
47C2×#431C2211yescharacteristic
48C2×#428—21no—
49C2×#423C2211yescharacteristic
50C2×#418—21no—
51#1652C22132yescomplemented
52#1638C22132yescomplemented
53#1546C2218yescomplemented
54C2×#332C2214yescomplemented
55#1487C2218yescomplemented
56#1478C2218yes—
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yes—
59C23×Q8—21no—
60C23×Q8D411yescharacteristic
61C23×D4D411yescharacteristic
62C24×C4—21no—
63C22×Q16—22no—
64C22×M16—21no—
65#556—216no—
66#556—216no—
67#556—216no—
68#556—216no—
69#556—28no—
70#556—216no—
71#556—28no—
72#556—28no—
73#556—216no—
74#556C23116yes—
75#533—22no—
76#533—22no—
77#533—22no—
78#533C2314yes—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8D412yes—
84C2×C4⋊Q8D414yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2314yes—
89C2×#126—216no—
90C2×#126—28no—
91C2×#126—28no—
92C2×#126—216no—
93C2×#125—21no—
94C2×#124—28no—
95C2×#124—24no—
96C2×#124—24no—
97C2×#124—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C4×Q8—28no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—28no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8D414yes—
115C2×C4×Q8D412yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2314yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4D412yes—
123C2×C4×D4D414yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2312yes—
126C2×#117—28no—
127C2×#117—24no—
128C2×#117—24no—
129C2×#117—28no—
130C2×#117—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4D412yes—
135C22×C4⋊C4D411yescharacteristic
136C22×C4⋊C4D411yescharacteristic
137C22×C22⋊C4—22no—
138C22×C22⋊C4—22no—
139C22×C22⋊C4—21no—
140C22×C42—22no—
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143#498—24no—
144#498C2318yescomplemented
145#487—24no—
146#487C2318yescomplemented
147#484—28no—
148#483—216no—
149#478—216no—
150#477—24no—
151#477C2318yescomplemented
152#476—24no—
153#476C2318yes—
154#475—216no—
155#474—24no—
156#474C2318yes—
157#462—24no—
158#462C2318yescomplemented
159#451—28no—
160C4×Q16—216no—
161#431—22no—
162#431—22no—
163#431C2314yes—
164#431C2314yes—
165#428—28no—
166C2×#107—22no—
167C2×#107—22no—
168C2×#106—22no—
169C2×#106—24no—
170#423—22no—
171#423C2314yes—
172C2×C4⋊C8—21no—
173C2×C4⋊C8—21no—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176#418—28no—
177C2×#103—22no—
178C2×#103—24no—
179C2×#103—22no—
180C2×#102—22no—
181C2×#102—22no—
182C2×C22⋊C8—21no—
183C2×C22⋊C8—21no—
184C2×C4×C8—21no—
185C2×C4×C8C2312yes—
186#332C23116yescomplemented
187Q8⋊C8C23116yescomplemented
188D4⋊C8C23116yes—
189C25—21no—
190C22×Q8—22no—
191C22×Q8—21no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—44no—
201C22×Q8C2×D412yes—
202C22×Q8C2×D414yes—
203C22×D4—22no—
204C22×D4—21no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4C2×D412yes—
208C22×D4C2×D414yes—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—48no—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C2×Q16—28no—
222C2×Q16—48no—
223C2×M16—24no—
224C2×M16—44no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230C4⋊Q8—24no—
231C4⋊Q8—24no—
232C4⋊Q8—22no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8C2×D418yes—
242C4⋊Q8C2×D414yes—
243C4⋊Q8C2412yes—
244C4⋊Q8C2412yes—
245C4⋊Q8C2414yes—
246#126—264no—
247#126—232no—
248#126—232no—
249#125—24no—
250#125—42no—
251#124—232no—
252#124—216no—
253#124—216no—
254C22⋊Q8—232no—
255C22⋊Q8—216no—
256C22⋊Q8—216no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—42no—
260C22⋊Q8—42no—
261C4×Q8—232no—
262C4×Q8—216no—
263C4×Q8—216no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—22no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—42no—
274C4×Q8—42no—
275C4×Q8C2×D418yes—
276C4×Q8C2×D414yes—
277C4×Q8C2414yes—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4C2×D414yes—
286C4×D4C2×D418yes—
287C4×D4C2414yes—
288#117—232no—
289#117—216no—
290#117—216no—
291#117—24no—
292#117—42no—
293C2×C4⋊C4—216no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—41no—
314C2×C4⋊C4—41no—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D418yes—
321C2×C22⋊C4—216no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—216no—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—41no—
330C2×C42—216no—
331C2×C42—28no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—41no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345#107—28no—
346#107—28no—
347#106—28no—
348#106—216no—
349C4⋊C8—24no—
350C4⋊C8—24no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353C4⋊C8C2414yes—
354C4⋊C8C2414yes—
355#103—28no—
356#103—28no—
357#103—216no—
358#102—28no—
359#102—28no—
360C22⋊C8—24no—
361C22⋊C8—24no—
362C4×C8—24no—
363C4×C8—22no—
364C4×C8C2414yes—
365C24—24no—
366C24—22no—
367C24—44no—
368C24—44no—
369C24—44no—
370C24#12011yescharacteristic
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—28no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—44no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—44no—
390C2×Q8—416no—
391C2×Q8—48no—
392C2×Q8—44no—
393C2×Q8C22×D414yes—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—28no—
400C2×D4—48no—
401C2×D4—44no—
402C2×D4C22×D414yes—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4#12012yes—
427C22×C4#12011yescharacteristic
428C22×C4#12011yescharacteristic
429C22×C4#12012yes—
430C22×C4#12012yes—
431C22×C4#12011yescharacteristic
432C22×C4#13514yes—
433C22×C4#13714yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440Q16—416no—
441M16—48no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C4⋊C4—232no—
449C4⋊C4—28no—
450C4⋊C4—24no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4C22×D418yes—
472C4⋊C4C22×D414yes—
473C4⋊C4C22×D414yes—
474C22⋊C4—232no—
475C22⋊C4—232no—
476C22⋊C4—28no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—44no—
480C42—232no—
481C42—28no—
482C42—24no—
483C42—24no—
484C42—44no—
485C42#12012yes—
486C42#12012yes—
487C42#12012yes—
488C42C22×D412yes—
489C42C22×D411yescharacteristic
490C42C22×D412yes—
491C42C22×D411yescharacteristic
492C42C22×D412yes—
493C42C22×D411yescharacteristic
494C42C2511yescharacteristic
495C23—22no—
496C23—22no—
497C23—24no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—816no—
505C23#45214yes—
506C23C2×#12012yes—
507C23C2×#12011yescharacteristic
508Q8—24no—
509Q8—24no—
510Q8—48no—
511Q8—44no—
512Q8—416no—
513Q8—48no—
514Q8—44no—
515Q8—44no—
516Q8—44no—
517Q8—44no—
518Q8—82no—
519Q8—82no—
520D4—24no—
521D4—24no—
522D4—48no—
523D4—44no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—416no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—416no—
540C2×C4—48no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4#45218yes—
551C2×C4#45214yes—
552C2×C4C2×#12012yes—
553C2×C4C2×#12011yescharacteristic
554C2×C4C2×#12012yes—
555C2×C4C2×#12012yes—
556C2×C4C2×#12011yescharacteristic
557C2×C4C2×#12011yescharacteristic
558C2×C4C2×#13512yes—
559C2×C4C2×#13514yes—
560C2×C4C2×#13714yes—
561C2×C4C2×#13712yes—
562C2×C4#57514yes—
563C2×C4C23×D411yescharacteristic
564C2×C4C23×D411yescharacteristic
565C2×C4C23×D411yescharacteristic
566C8—44no—
567C8—44no—
568C8—44no—
569C22—28no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—816no—
576C22—88no—
577C22C2×#45214yes—
578C22C2×#45212yes—
579C22#300512yes—
580C22#302012yes—
581C22C22×#12011yescharacteristic
582C4—22no—
583C4—21no—
584C4—22no—
585C4—21no—
586C4—48no—
587C4—44no—
588C4—44no—
589C4—44no—
590C4—82no—
591C4—82no—
592C4C2×#45214yes—
593C4C2×#45212yes—
594C4#300512yes—
595C4C22×#13511yescharacteristic
596C4C22×#13711yescharacteristic
597C4C2×#57512yes—
598C2—24no—
599C2—84no—
600C2#1878114yesdirect product factor
601C2C22×#45211yescharacteristic
602C2C2×#300511yescharacteristic
603C2C2×#302011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC23×D4D4
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×#3229C2semidirect
3. C2×#3104C2semidirect
5. #18781C2direct (N × H)
6. C2×#1652C2semidirect
7. C2×#1638C2semidirect
8. C2×#1546C2semidirect
9. C2×#1487C2semidirect
29. #3229C22semidirect
30. C2×#498C22semidirect
31. C2×#487C22semidirect
33. #3104C22semidirect
38. C2×#477C22semidirect
43. C2×#462C22semidirect
51. #1652C22semidirect
52. #1638C22semidirect
53. #1546C22semidirect
54. C2×#332C22semidirect
55. #1487C22semidirect
57. C2×Q8⋊C8C22semidirect
144. #498C23semidirect
146. #487C23semidirect
151. #477C23semidirect
158. #462C23semidirect
186. #332C23semidirect
187. Q8⋊C8C23semidirect
600. C2#18781direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#32292
C2×#31042
C2×#30772
#187812
C2×#16522
C2×#16382
C2×#15462
C2×#14872
C2×#14782