Finite Group Databasebeta

← C2×#18542 #6348089

Subgroups — C2×#18542 #6348089

Order 512.

2998 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#3165C211yesmaximal, characteristic, complemented
3C2×#3113C211yesmaximal, characteristic, complemented
4C2×#3110C211yesmaximal, characteristic
5#18542C2116yesmaximal, direct product factor
6C2×#1650C214yesmaximal, complemented
7C2×#1652C214yesmaximal, complemented
8C2×#1556C211yesmaximal, characteristic, complemented
9C2×#1491C211yesmaximal, characteristic, complemented
10C2×#1490C211yesmaximal, characteristic
11C2×#556—22no—
12C2×#556—24no—
13C2×#556—24no—
14C2×#556—22no—
15C2×#556—24no—
16C2×#556—22no—
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#3165—24no—
29#3165C2218yescomplemented
30C2×#501—22no—
31C2×#500C2212yescomplemented
32C2×#498C2212yescomplemented
33#3113—24no—
34#3113C2218yescomplemented
35#3110—24no—
36#3110C2218yes—
37C2×#484—21no—
38C2×#483—21no—
39C2×#475C2212yescomplemented
40C2×#474C2212yes—
41C2×#462C2212yescomplemented
42C2×#461C2212yes—
43C2×#451—21no—
44C2×#450—21no—
45C2×#441—22no—
46C2×#440—22no—
47C2×#432C2211yescharacteristic
48C2×#432C2211yescharacteristic
49C2×#428—22no—
50C2×C4×M16C2211yescharacteristic
51#1652C22132yescomplemented
52#1650C22132yescomplemented
53#1556C2218yescomplemented
54C2×#332C2214yescomplemented
55#1491C2218yescomplemented
56#1490C2218yes—
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yes—
59C23×Q8—21no—
60C23×Q8D411yescharacteristic
61C23×D4D411yescharacteristic
62C24×C4—21no—
63C22×Q16—22no—
64C22×SD16—22no—
65C22×M16—21no—
66#556—216no—
67#556—216no—
68#556—216no—
69#556—28no—
70#556—216no—
71#556—216no—
72#556—28no—
73#556—216no—
74#556—28no—
75#556C23116yes—
76#533—22no—
77#533—22no—
78#533—22no—
79#533C2314yes—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8D412yes—
85C2×C4⋊Q8D414yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2314yes—
90C2×#126—216no—
91C2×#126—28no—
92C2×#126—28no—
93C2×#126—216no—
94C2×#125—21no—
95C2×#124—28no—
96C2×#124—24no—
97C2×#124—24no—
98C2×#124—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C4×Q8—28no—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8—28no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8D414yes—
116C2×C4×Q8D412yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2314yes—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4D414yes—
124C2×C4×D4D412yes—
125C2×C4×D4C2314yes—
126C2×C4×D4C2312yes—
127C2×#117—28no—
128C2×#117—24no—
129C2×#117—24no—
130C2×#117—28no—
131C2×#117—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4D411yescharacteristic
136C22×C4⋊C4D411yescharacteristic
137C22×C4⋊C4D412yes—
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C22⋊C4—21no—
141C22×C42—22no—
142C22×C42—21no—
143C22×C42C2311yescharacteristic
144#501—216no—
145#500—24no—
146#500C2318yescomplemented
147#498—24no—
148#498C2318yescomplemented
149#484—28no—
150#483—28no—
151#475—24no—
152#475C2318yescomplemented
153#474—24no—
154#474C2318yes—
155#462—24no—
156#462C2318yescomplemented
157#461—24no—
158#461C2318yes—
159#451—28no—
160#450—28no—
161#441—216no—
162#440—216no—
163#432—22no—
164#432—22no—
165#432C2314yes—
166#432C2314yes—
167#428—216no—
168C2×#107—22no—
169C2×#107—24no—
170C2×#106—22no—
171C2×#106—24no—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2312yes—
174C2×#103—22no—
175C2×#103—22no—
176C2×#103—22no—
177C2×#102—22no—
178C2×C22⋊C8—21no—
179C2×C22⋊C8—21no—
180C4×M16—22no—
181C4×M16C2314yes—
182C2×#97—21no—
183C2×#97—21no—
184C2×#97—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—48no—
200C22×Q8C2×D412yes—
201C22×Q8C2×D414yes—
202C22×D4—22no—
203C22×D4—21no—
204C22×D4—44no—
205C22×D4C2×D412yes—
206C22×D4C2×D414yes—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—48no—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C2×Q16—28no—
220C2×Q16—48no—
221C2×SD16—28no—
222C2×SD16—48no—
223C2×M16—28no—
224C2×M16—24no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230C4⋊Q8—24no—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8C2×D414yes—
242C4⋊Q8C2×D418yes—
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—22no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4C2×D418yes—
286C4×D4C2×D414yes—
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—216no—
346#107—28no—
347#106—216no—
348#106—28no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351C4⋊C8C2414yes—
352C4⋊C8C2414yes—
353#103—28no—
354#103—28no—
355#103—28no—
356#102—28no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359#97—24no—
360#97—24no—
361#97—24no—
362C4×C8—22no—
363C4×C8C2414yes—
364C24—24no—
365C24—22no—
366C24—44no—
367C24—44no—
368C24—44no—
369C24#12011yescharacteristic
370C2×Q8—24no—
371C2×Q8—22no—
372C2×Q8—28no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—48no—
380C2×Q8—44no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—416no—
384C2×Q8—416no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—48no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8C22×D414yes—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—48no—
396C2×D4—44no—
397C2×D4—48no—
398C2×D4C22×D414yes—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—48no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4#12012yes—
424C22×C4#12012yes—
425C22×C4#12012yes—
426C22×C4#12011yescharacteristic
427C22×C4#12011yescharacteristic
428C22×C4#12011yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435Q16—416no—
436SD16—416no—
437M16—216no—
438C2×C8—28no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—44no—
443C2×C8—44no—
444C4⋊C4—232no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—28no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4C22×D414yes—
468C4⋊C4C22×D414yes—
469C4⋊C4C22×D418yes—
470C22⋊C4—232no—
471C22⋊C4—232no—
472C22⋊C4—28no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—44no—
476C42—232no—
477C42—28no—
478C42—24no—
479C42—24no—
480C42—44no—
481C42#12012yes—
482C42#12012yes—
483C42#12012yes—
484C42C22×D411yescharacteristic
485C42C22×D412yes—
486C42C22×D412yes—
487C42C22×D412yes—
488C42C22×D411yescharacteristic
489C42C22×D411yescharacteristic
490C42C2511yescharacteristic
491C23—22no—
492C23—22no—
493C23—24no—
494C23—48no—
495C23—48no—
496C23—44no—
497C23—44no—
498C23—48no—
499C23—44no—
500C23—816no—
501C23#45614yes—
502C23C2×#12012yes—
503C23C2×#12011yescharacteristic
504Q8—24no—
505Q8—24no—
506Q8—48no—
507Q8—416no—
508Q8—44no—
509Q8—48no—
510Q8—48no—
511Q8—44no—
512Q8—44no—
513Q8—82no—
514Q8—82no—
515D4—24no—
516D4—24no—
517D4—48no—
518D4—44no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—416no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—416no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4C2×#12012yes—
549C2×C4C2×#12012yes—
550C2×C4C2×#12011yescharacteristic
551C2×C4C2×#12012yes—
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12011yescharacteristic
554C2×C4#57514yes—
555C2×C4#57514yes—
556C2×C4C23×D411yescharacteristic
557C2×C4C23×D411yescharacteristic
558C2×C4C23×D411yescharacteristic
559C8—28no—
560C8—44no—
561C8—44no—
562C22—28no—
563C22—22no—
564C22—22no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—816no—
569C22—88no—
570C22#300512yes—
571C22#300512yes—
572C22C2×#45614yes—
573C22C2×#45612yes—
574C22C22×#12011yescharacteristic
575C4—22no—
576C4—22no—
577C4—21no—
578C4—21no—
579C4—22no—
580C4—44no—
581C4—44no—
582C4—48no—
583C4—44no—
584C4—82no—
585C4—82no—
586C4#300512yes—
587C4#300512yes—
588C4C2×#57512yes—
589C4C2×#57512yes—
590C2—24no—
591C2—84no—
592C2#1854214yesdirect product factor
593C2C2×#300511yescharacteristic
594C2C2×#300511yescharacteristic
595C2C22×#45611yescharacteristic
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×#3165C2semidirect
3. C2×#3113C2semidirect
5. #18542C2direct (N × H)
6. C2×#1650C2semidirect
7. C2×#1652C2semidirect
8. C2×#1556C2semidirect
9. C2×#1491C2semidirect
29. #3165C22semidirect
31. C2×#500C22semidirect
32. C2×#498C22semidirect
34. #3113C22semidirect
39. C2×#475C22semidirect
41. C2×#462C22semidirect
51. #1652C22semidirect
52. #1650C22semidirect
53. #1556C22semidirect
54. C2×#332C22semidirect
55. #1491C22semidirect
57. C2×Q8⋊C8C22semidirect
146. #500C23semidirect
148. #498C23semidirect
152. #475C23semidirect
156. #462C23semidirect
186. #332C23semidirect
187. Q8⋊C8C23semidirect
592. C2#18542direct (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×#31652
C2×#31132
C2×#31102
#185422
C2×#16502
C2×#16522
C2×#15562
C2×#14912
C2×#14902