Finite Group Databasebeta

← C2×#18540 #6348087

Subgroups — C2×#18540 #6348087

Order 512.

3986 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic
2C2×#3165C211yesmaximal, characteristic, complemented
3C2×#3113C211yesmaximal, characteristic, complemented
4C2×#3110C211yesmaximal, characteristic
5#18540C2116yesmaximal, direct product factor
6C2×#1649C214yesmaximal, complemented
7C2×#1651C214yesmaximal, complemented
8C2×#1556C211yesmaximal, characteristic, complemented
9C2×#1491C211yesmaximal, characteristic, complemented
10C2×#1490C211yesmaximal, characteristic
11C24×Q8—21no—
12C2×D4×Q8—22no—
13C2×D4×Q8—24no—
14C2×D4×Q8—24no—
15C2×D4×Q8—22no—
16C2×D4×Q8—24no—
17C2×D4×Q8—22no—
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#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#1651C22132yescomplemented
52#1649C22132yescomplemented
53#1556C2218yescomplemented
54C2×#332C2214yescomplemented
55#1491C2218yescomplemented
56#1490C2218yes—
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×SD16—22no—
72C22×M16—21no—
73D4×Q8—216no—
74D4×Q8—216no—
75D4×Q8—216no—
76D4×Q8—28no—
77D4×Q8—216no—
78D4×Q8—216no—
79D4×Q8—28no—
80D4×Q8—216no—
81D4×Q8—28no—
82D4×Q8C23116yes—
83#533—22no—
84#533—22no—
85#533—22no—
86#533C2314yes—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8D412yes—
96C2×C4⋊Q8D414yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2314yes—
101C2×#125—21no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—216no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—28no—
108C2×C22⋊Q8—28no—
109C2×C22⋊Q8—216no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8D414yes—
119C2×C4×Q8D412yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2314yes—
122C2×C4×D4—28no—
123C2×C4×D4—24no—
124C2×C4×D4—28no—
125C2×C4×D4—24no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—22no—
129C2×C4×D4—22no—
130C2×C4×D4D414yes—
131C2×C4×D4D412yes—
132C2×C4×D4C2314yes—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4D411yescharacteristic
139C22×C4⋊C4D411yescharacteristic
140C22×C4⋊C4D412yes—
141C22×C22⋊C4—22no—
142C22×C22⋊C4—21no—
143C22×C42—21no—
144C22×C42C2311yescharacteristic
145#501—216no—
146#500—24no—
147#500C2318yescomplemented
148#498—24no—
149#498C2318yescomplemented
150#484—28no—
151#483—28no—
152#475—24no—
153#475C2318yescomplemented
154#474—24no—
155#474C2318yes—
156#462—24no—
157#462C2318yescomplemented
158#461—24no—
159#461C2318yes—
160#451—28no—
161#450—28no—
162#441—216no—
163#440—216no—
164#432—22no—
165#432—22no—
166#432C2314yes—
167#432C2314yes—
168#428—216no—
169C2×#107—22no—
170C2×#107—24no—
171C2×#106—22no—
172C2×#106—24no—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×#103—22no—
176C2×#103—22no—
177C2×#103—22no—
178C2×#102—22no—
179C2×C22⋊C8—21no—
180C2×C22⋊C8—21no—
181C4×M16—22no—
182C4×M16C2314yes—
183C2×#97—21no—
184C2×#97—21no—
185C2×#97—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—28no—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4C2×D414yes—
218C22×D4C2×D412yes—
219C23×C4—21no—
220C23×C4—216no—
221C23×C4—28no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—416no—
230C23×C4—48no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C2×Q16—28no—
235C2×Q16—48no—
236C2×SD16—28no—
237C2×SD16—48no—
238C2×M16—28no—
239C2×M16—24no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C4⋊Q8—232no—
244C4⋊Q8—216no—
245C4⋊Q8—216no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8C2×D414yes—
260C4⋊Q8C2×D418yes—
261C4⋊Q8C2412yes—
262C4⋊Q8C2412yes—
263C4⋊Q8C2414yes—
264#125—24no—
265#125—42no—
266C22⋊Q8—232no—
267C22⋊Q8—216no—
268C22⋊Q8—264no—
269C22⋊Q8—232no—
270C22⋊Q8—216no—
271C22⋊Q8—232no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—42no—
275C22⋊Q8—42no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—22no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—22no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—42no—
286C4×Q8—42no—
287C4×Q8C2×D418yes—
288C4×Q8C2×D414yes—
289C4×Q8C2414yes—
290C4×D4—232no—
291C4×D4—216no—
292C4×D4—216no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4C2×D418yes—
301C4×D4C2×D414yes—
302C4×D4C2414yes—
303#117—24no—
304#117—42no—
305C2×C4⋊C4—216no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—41no—
326C2×C4⋊C4—41no—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D418yes—
333C2×C22⋊C4—216no—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—41no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—41no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353#107—216no—
354#107—28no—
355#106—216no—
356#106—28no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8C2414yes—
360C4⋊C8C2414yes—
361#103—28no—
362#103—28no—
363#103—28no—
364#102—28no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367#97—24no—
368#97—24no—
369#97—24no—
370C4×C8—22no—
371C4×C8C2414yes—
372C24—28no—
373C24—24no—
374C24—22no—
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—4128no—
396C2×Q8—464no—
397C2×Q8—432no—
398C2×Q8—432no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—48no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8AGL(1,ℤ/8)18yes—
405C2×Q8C22×D414yes—
406C2×D4—28no—
407C2×D4—216no—
408C2×D4—216no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4#13718yes—
413C2×D4C22×D414yes—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—232no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—464no—
431C22×C4—432no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4#12012yes—
442C22×C4#12012yes—
443C22×C4#12012yes—
444C22×C4#12011yescharacteristic
445C22×C4#12011yescharacteristic
446C22×C4#12011yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453Q16—416no—
454SD16—416no—
455M16—216no—
456C2×C8—28no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—44no—
461C2×C8—44no—
462C4⋊C4—232no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4C22×D414yes—
486C4⋊C4C22×D414yes—
487C4⋊C4C22×D418yes—
488C22⋊C4—232no—
489C22⋊C4—28no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—44no—
493C42—28no—
494C42—24no—
495C42—24no—
496C42—44no—
497C42#12012yes—
498C42#12012yes—
499C42#12012yes—
500C42C22×D411yescharacteristic
501C42C22×D412yes—
502C42C22×D412yes—
503C42C22×D412yes—
504C42C22×D411yescharacteristic
505C42C22×D411yescharacteristic
506C42C2511yescharacteristic
507C23—22no—
508C23—22no—
509C23—216no—
510C23—24no—
511C23—28no—
512C23—48no—
513C23—432no—
514C23—44no—
515C23—48no—
516C23—44no—
517C23#45614yes—
518C23C2×#12012yes—
519C23C2×#12011yescharacteristic
520Q8—216no—
521Q8—28no—
522Q8—24no—
523Q8—464no—
524Q8—416no—
525Q8—48no—
526Q8—48no—
527Q8—44no—
528Q8—44no—
529Q8—82no—
530Q8—82no—
531Q8C2×AGL(1,ℤ/8)18yes—
532D4—216no—
533D4—28no—
534D4—24no—
535D4C2×#13718yes—
536C2×C4—216no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—464no—
553C2×C4—416no—
554C2×C4—416no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4C2×#12012yes—
567C2×C4C2×#12012yes—
568C2×C4C2×#12011yescharacteristic
569C2×C4C2×#12012yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4C2×#12011yescharacteristic
572C2×C4C2×AGL(1,ℤ/8)14yes—
573C2×C4C2×#13714yes—
574C2×C4C23×D411yescharacteristic
575C2×C4C23×D411yescharacteristic
576C2×C4C23×D411yescharacteristic
577C8—28no—
578C8—44no—
579C8—44no—
580C22—28no—
581C22—22no—
582C22—22no—
583C22—28no—
584C22—432no—
585C22—416no—
586C22—44no—
587C22—44no—
588C22#300312yes—
589C22#300412yes—
590C22C2×#45614yes—
591C22C2×#45612yes—
592C22C22×#12011yescharacteristic
593C4—28no—
594C4—22no—
595C4—22no—
596C4—21no—
597C4—21no—
598C4—22no—
599C4—48no—
600C4—44no—
601C4—44no—
602C4—82no—
603C4—82no—
604C4#300312yes—
605C4#300412yes—
606C4C22×AGL(1,ℤ/8)12yes—
607C4C22×#13712yes—
608C2—24no—
609C2—48no—
610C2#1854014yesdirect product factor
611C2C2×#300311yescharacteristic
612C2C2×#300411yescharacteristic
613C2C22×#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. #18540C2direct (N × H)
6. C2×#1649C2semidirect
7. C2×#1651C2semidirect
8. C2×#1556C2semidirect
9. C2×#1491C2semidirect
29. #3165C22semidirect
31. C2×#500C22semidirect
32. C2×#498C22semidirect
34. #3113C22semidirect
39. C2×#475C22semidirect
41. C2×#462C22semidirect
51. #1651C22semidirect
52. #1649C22semidirect
53. #1556C22semidirect
54. C2×#332C22semidirect
55. #1491C22semidirect
57. C2×Q8⋊C8C22semidirect
147. #500C23semidirect
149. #498C23semidirect
153. #475C23semidirect
157. #462C23semidirect
187. #332C23semidirect
188. Q8⋊C8C23semidirect
610. C2#18540direct (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×#31652
C2×#31132
C2×#31102
#185402
C2×#16492
C2×#16512
C2×#15562
C2×#14912
C2×#14902