Finite Group Databasebeta

← C2×#18759 #6348673

Subgroups — C2×#18759 #6348673

Order 512.

3964 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic
2C2×#3230C211yesmaximal, characteristic, complemented
3C2×#3104C211yesmaximal, characteristic, complemented
4C2×#3077C211yesmaximal, characteristic
5#18759C2116yesmaximal, direct product factor
6C2×#1649C214yesmaximal, complemented
7C2×#1637C214yesmaximal, complemented
8C2×#1549C211yesmaximal, characteristic, complemented
9C2×#1487C211yesmaximal, characteristic, complemented
10C2×#1478C211yesmaximal, characteristic
11C24×Q8—21no—
12C2×D4×Q8—22no—
13C2×D4×Q8—24no—
14C2×D4×Q8—24no—
15C2×D4×Q8—22no—
16C2×D4×Q8—22no—
17C2×D4×Q8—24no—
18C2×D4×Q8C2214yes—
19C2×D4×Q8C2218yes—
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8—21no—
22C22×C4⋊Q8C2211yescharacteristic
23C22×C22⋊Q8—21no—
24C22×C22⋊Q8—22no—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4—21no—
27C22×C4×D4C2211yescharacteristic
28#3230—24no—
29#3230C2218yescomplemented
30C2×#500C2212yescomplemented
31C2×#487C2212yescomplemented
32#3104—24no—
33#3104C2218yescomplemented
34C2×#484—21no—
35C2×#484—22no—
36#3077—24no—
37#3077C2218yes—
38C2×#477—22no—
39C2×#477C2212yescomplemented
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#1649C22132yescomplemented
52#1637C22132yescomplemented
53#1549C2218yescomplemented
54C2×#332C2214yescomplemented
55#1487C2218yescomplemented
56#1478C2218yes—
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yes—
59C23×Q8—22no—
60C23×Q8—28no—
61C23×Q8—24no—
62C23×Q8—28no—
63C23×Q8—24no—
64C23×Q8—21no—
65C23×Q8—416no—
66C23×Q8D411yescharacteristic
67C23×D4D411yescharacteristic
68C24×C4—22no—
69C24×C4—21no—
70C22×Q16—22no—
71C22×M16—21no—
72D4×Q8—216no—
73D4×Q8—216no—
74D4×Q8—216no—
75D4×Q8—216no—
76D4×Q8—28no—
77D4×Q8—216no—
78D4×Q8—28no—
79D4×Q8—28no—
80D4×Q8—216no—
81D4×Q8C23116yes—
82#533—22no—
83#533—22no—
84#533—22no—
85#533C2314yes—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8D412yes—
95C2×C4⋊Q8D414yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2314yes—
100C2×#125—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×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8D414yes—
118C2×C4×Q8D412yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2314yes—
121C2×C4×D4—28no—
122C2×C4×D4—24no—
123C2×C4×D4—24no—
124C2×C4×D4—28no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—22no—
128C2×C4×D4—22no—
129C2×C4×D4D412yes—
130C2×C4×D4D414yes—
131C2×C4×D4C2314yes—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4D411yescharacteristic
138C22×C4⋊C4D412yes—
139C22×C4⋊C4D411yescharacteristic
140C22×C22⋊C4—22no—
141C22×C22⋊C4—21no—
142C22×C42—21no—
143C22×C42C2311yescharacteristic
144#500—24no—
145#500C2318yescomplemented
146#487—24no—
147#487C2318yescomplemented
148#484—28no—
149#484—216no—
150#478—216no—
151#477—216no—
152#477—24no—
153#477C2318yescomplemented
154#476—24no—
155#476C2318yes—
156#474—24no—
157#474C2318yes—
158#462—24no—
159#462C2318yescomplemented
160#451—28no—
161C4×Q16—216no—
162#431—22no—
163#431—22no—
164#431C2314yes—
165#431C2314yes—
166#428—28no—
167C2×#107—22no—
168C2×#107—24no—
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—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#332C23116yescomplemented
188Q8⋊C8C23116yescomplemented
189D4⋊C8C23116yes—
190C25—21no—
191C22×Q8—28no—
192C22×Q8—216no—
193C22×Q8—28no—
194C22×Q8—232no—
195C22×Q8—216no—
196C22×Q8—216no—
197C22×Q8—22no—
198C22×Q8—21no—
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8—416no—
206C22×Q8—48no—
207C22×Q8—464no—
208C22×Q8—432no—
209C22×Q8—464no—
210C22×Q8—416no—
211C22×Q8—416no—
212C22×Q8C2×D414yes—
213C22×Q8C2×D412yes—
214C22×D4—22no—
215C22×D4—21no—
216C22×D4—24no—
217C22×D4Q1618yescomplemented
218C22×D4C2×D412yes—
219C22×D4C2×D414yes—
220C23×C4—21no—
221C23×C4—216no—
222C23×C4—28no—
223C23×C4—21no—
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—416no—
231C23×C4—48no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C2×Q16—28no—
236C2×Q16—48no—
237C2×M16—24no—
238C2×M16—44no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C4⋊Q8—232no—
243C4⋊Q8—216no—
244C4⋊Q8—216no—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—22no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C2×D418yes—
259C4⋊Q8C2×D414yes—
260C4⋊Q8C2412yes—
261C4⋊Q8C2412yes—
262C4⋊Q8C2414yes—
263#125—24no—
264#125—42no—
265C22⋊Q8—232no—
266C22⋊Q8—216no—
267C22⋊Q8—264no—
268C22⋊Q8—232no—
269C22⋊Q8—216no—
270C22⋊Q8—232no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—42no—
274C22⋊Q8—42no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—22no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—22no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×Q8—42no—
286C4×Q8C2×D418yes—
287C4×Q8C2×D414yes—
288C4×Q8C2414yes—
289C4×D4—232no—
290C4×D4—216no—
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×D4C2×D414yes—
300C4×D4C2×D418yes—
301C4×D4C2414yes—
302#117—24no—
303#117—42no—
304C2×C4⋊C4—216no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
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—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—41no—
325C2×C4⋊C4—41no—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D414yes—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4C2×D418yes—
332C2×C22⋊C4—216no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—41no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—41no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352#107—28no—
353#107—28no—
354#107—216no—
355#106—28no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360C4⋊C8C2414yes—
361C4⋊C8C2414yes—
362#103—28no—
363#103—28no—
364#103—216no—
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#12011yescharacteristic
378C2×Q8—216no—
379C2×Q8—28no—
380C2×Q8—216no—
381C2×Q8—232no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—28no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—464no—
391C2×Q8—432no—
392C2×Q8—48no—
393C2×Q8—416no—
394C2×Q8—416no—
395C2×Q8—44no—
396C2×Q8—4128no—
397C2×Q8—464no—
398C2×Q8—432no—
399C2×Q8—432no—
400C2×Q8—41no—
401C2×Q8—41no—
402C2×Q8—48no—
403C2×Q8—44no—
404C2×Q8AGL(1,ℤ/8)18yes—
405C2×Q8C22×D414yes—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—28no—
410C2×D4—216no—
411C2×D4—28no—
412C2×D4C2×Q1618yes—
413C2×D4C2×Q16116yescomplemented
414C2×D4C22×D414yes—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—232no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—42no—
430C22×C4—464no—
431C22×C4—432no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4#12012yes—
442C22×C4#12011yescharacteristic
443C22×C4#12011yescharacteristic
444C22×C4#12012yes—
445C22×C4#12012yes—
446C22×C4#12011yescharacteristic
447C22×C4C2×Q1614yes—
448C22×C4#13714yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455Q16—416no—
456M16—48no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—44no—
461C2×C8—44no—
462C2×C8—44no—
463C4⋊C4—232no—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4C22×D414yes—
487C4⋊C4C22×D418yes—
488C4⋊C4C22×D414yes—
489C22⋊C4—232no—
490C22⋊C4—28no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—44no—
494C42—28no—
495C42—24no—
496C42—24no—
497C42—44no—
498C42#12012yes—
499C42#12012yes—
500C42#12012yes—
501C42C22×D412yes—
502C42C22×D411yescharacteristic
503C42C22×D412yes—
504C42C22×D411yescharacteristic
505C42C22×D412yes—
506C42C22×D411yescharacteristic
507C42C2511yescharacteristic
508C23—22no—
509C23—22no—
510C23—24no—
511C23—28no—
512C23—216no—
513C23—48no—
514C23—432no—
515C23—48no—
516C23—44no—
517C23—44no—
518C23#45114yes—
519C23C2×#12012yes—
520C23C2×#12011yescharacteristic
521Q8—216no—
522Q8—28no—
523Q8—24no—
524Q8—464no—
525Q8—416no—
526Q8—44no—
527Q8—48no—
528Q8—48no—
529Q8—44no—
530Q8—82no—
531Q8—82no—
532Q8C2×AGL(1,ℤ/8)18yes—
533D4—28no—
534D4—216no—
535D4—24no—
536D4C22×Q1618yes—
537C2×C4—216no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—464no—
551C2×C4—416no—
552C2×C4—48no—
553C2×C4—416no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4#45118yes—
565C2×C4#45114yes—
566C2×C4C2×#12012yes—
567C2×C4C2×#12011yescharacteristic
568C2×C4C2×#12012yes—
569C2×C4C2×#12012yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4C2×#12011yescharacteristic
572C2×C4C22×Q1612yes—
573C2×C4C22×Q1614yes—
574C2×C4C2×AGL(1,ℤ/8)14yes—
575C2×C4C2×#13712yes—
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—28no—
587C22—432no—
588C22—416no—
589C22—44no—
590C22—44no—
591C22C2×#45114yes—
592C22C2×#45112yes—
593C22#300312yes—
594C22#302012yes—
595C22C22×#12011yescharacteristic
596C4—28no—
597C4—22no—
598C4—21no—
599C4—22no—
600C4—21no—
601C4—48no—
602C4—44no—
603C4—44no—
604C4—82no—
605C4—82no—
606C4C2×#45114yes—
607C4C2×#45112yes—
608C4#300312yes—
609C4C23×Q1611yescharacteristic
610C4C22×AGL(1,ℤ/8)12yes—
611C4C22×#13711yescharacteristic
612C2—24no—
613C2—48no—
614C2#1875914yesdirect product factor
615C2C22×#45111yescharacteristic
616C2C2×#300311yescharacteristic
617C2C2×#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×#3230C2semidirect
3. C2×#3104C2semidirect
5. #18759C2direct (N × H)
6. C2×#1649C2semidirect
7. C2×#1637C2semidirect
8. C2×#1549C2semidirect
9. C2×#1487C2semidirect
29. #3230C22semidirect
30. C2×#500C22semidirect
31. C2×#487C22semidirect
33. #3104C22semidirect
39. C2×#477C22semidirect
43. C2×#462C22semidirect
51. #1649C22semidirect
52. #1637C22semidirect
53. #1549C22semidirect
54. C2×#332C22semidirect
55. #1487C22semidirect
57. C2×Q8⋊C8C22semidirect
145. #500C23semidirect
147. #487C23semidirect
153. #477C23semidirect
159. #462C23semidirect
187. #332C23semidirect
188. Q8⋊C8C23semidirect
217. C22×D4Q16semidirect
413. C2×D4C2×Q16semidirect
614. C2#18759direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C2×#32302
C2×#31042
C2×#30772
#187592
C2×#16492
C2×#16372
C2×#15492
C2×#14872
C2×#14782