Finite Group Databasebeta

← C2×#25678 #6450044

Subgroups — C2×#25678 #6450044

Order 512.

3598 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C22×#501C211yesmaximal, characteristic, complemented
3C22×#489C211yesmaximal, characteristic, complemented
4C2×#2857C211yesmaximal, characteristic, complemented
5#25678C2116yesmaximal, direct product factor
6C2×#2036C212yesmaximal, complemented
7C2×#2034C212yesmaximal
8C2×#1992C211yesmaximal, characteristic, complemented
9C2×#1959C211yesmaximal, characteristic, complemented
10C2×#1865C212yesmaximal, complemented
11C2×#1862C212yesmaximal, complemented
12C2×#1749C211yesmaximal, characteristic, complemented
13C22×C4⋊Q8C2211yescharacteristic
14C2×#529—24no—
15C2×#529—28no—
16C2×#529—28no—
17C2×#529C2218yes—
18C22×#126—21no—
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×#123—21no—
22C22×C22⋊Q8—21no—
23C22×C22⋊Q8C2211yescharacteristic
24C22×#121C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4—21no—
27C22×C4×D4C2211yescharacteristic
28C22×#117—21no—
29C23×C42C2211yescharacteristic
30C2×#501—28no—
31C2×#501C2218yescomplemented
32C2×#489—28no—
33C2×#489C2218yescomplemented
34C22×#107C2212yes—
35C22×#106C2212yes—
36C22×#103C2212yes—
37C22×#103C2212yes—
38C22×#102C2212yes—
39C22×#102C2212yes—
40C22×C22⋊C8C2212yes—
41C22×#97C2211yescharacteristic
42C22×#97C2211yescharacteristic
43#2857—24no—
44#2857C2218yescomplemented
45C2×#391—21no—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×#380C2212yes—
51#2036C22116yescomplemented
52#2034C22116yes—
53#1992C2218yescomplemented
54#1959C2218yescomplemented
55#1865C22116yescomplemented
56#1862C22116yescomplemented
57#1749C2218yescomplemented
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C23×Q8—21no—
62C23×Q8C2311yescharacteristic
63C23×D4—21no—
64C23×D4C2311yescharacteristic
65C24×C4—21no—
66C24×C4D411yescharacteristic
67C24×C4D411yescharacteristic
68C24×C4C2311yescharacteristic
69C23×C8C2312yes—
70C23×C8C2312yes—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—24no—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8C2314yes—
75#529—264no—
76#529—232no—
77#529—232no—
78C2×#126—216no—
79C2×#126—28no—
80C2×#125—22no—
81C2×#125—24no—
82C2×#125D418yes—
83C2×#125C2314yes—
84C2×#124—22no—
85C2×#124—24no—
86C2×#124D418yes—
87C2×#124C2314yes—
88C2×#123—216no—
89C2×#123—28no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—48no—
95C2×C22⋊Q8C2314yes—
96C2×#121—24no—
97C2×#121—22no—
98C2×#121—24no—
99C2×#121C2314yes—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—22no—
103C2×C4×Q8C2314yes—
104C2×C4×D4—216no—
105C2×C4×D4—24no—
106C2×C4×D4—28no—
107C2×C4×D4—24no—
108C2×C4×D4—22no—
109C2×C4×D4C2314yes—
110C2×#117—216no—
111C2×#117—28no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C42—21no—
131C22×C42—24no—
132C22×C42—24no—
133C22×C42D414yes—
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#501—232no—
140#489—232no—
141C2×#107—28no—
142C2×#107C2318yes—
143C2×#106—28no—
144C2×#106C2318yes—
145C2×#103—28no—
146C2×#103—28no—
147C2×#103C2318yes—
148C2×#103C2318yes—
149C2×#102—28no—
150C2×#102—28no—
151C2×#102C2318yes—
152C2×#102C2318yes—
153C2×C22⋊C8—28no—
154C2×C22⋊C8C2318yes—
155C2×#97—24no—
156C2×#97—24no—
157C2×#97C2314yes—
158C2×#97C2314yes—
159#391—28no—
160#389—24no—
161#389—28no—
162#389C2318yes—
163#384—22no—
164#384C2314yes—
165#380—24no—
166#380—28no—
167#380C2318yes—
168C2×#95—22no—
169C2×#95—22no—
170#340C2318yes—
171#340C2318yes—
172#336C2318yes—
173C25—21no—
174C25C2×D411yescharacteristic
175C22×Q8—22no—
176C22×Q8—21no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8C2×D414yes—
184C22×Q8C2412yes—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—21no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4C2×D414yes—
194C22×D4C2412yes—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D412yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C22×C8—24no—
225C22×C8—24no—
226C22×C8—24no—
227C22×C8—24no—
228C22×C8—24no—
229C22×C8#4118yes—
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—28no—
233C4⋊Q8—28no—
234C4⋊Q8—48no—
235#126—264no—
236#125—216no—
237#125—28no—
238#125C2×D4116yes—
239#124—216no—
240#124—28no—
241#124C2×D4116yes—
242#123—264no—
243C22⋊Q8—216no—
244C22⋊Q8—28no—
245C22⋊Q8—28no—
246C22⋊Q8—432no—
247#121—216no—
248#121—28no—
249#121—28no—
250C4×Q8—216no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×D4—264no—
254C4×D4—216no—
255C4×D4—28no—
256C4×D4—28no—
257#117—264no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—48no—
302C2×C42—28no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—216no—
309C2×C42—28no—
310C2×C42—28no—
311C2×C42C2×D414yes—
312C2×C42C2×D412yes—
313C2×C42C2×D414yes—
314C2×C42C2×D412yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2412yes—
318C2×C42C2412yes—
319#107—216no—
320#106—216no—
321#103—216no—
322#103—216no—
323#102—216no—
324#102—216no—
325C22⋊C8—216no—
326#97—28no—
327#97—28no—
328#95—28no—
329#95—28no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24#13714yes—
345C24C22×D412yes—
346C24C22×D411yescharacteristic
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—216no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—48no—
358C2×Q8—816no—
359C2×Q8C22×D414yes—
360C2×D4—216no—
361C2×D4—24no—
362C2×D4—216no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—432no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—48no—
372C2×D4C22×D414yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—216no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—432no—
410C22×C4—48no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4#12014yes—
427C22×C4#12014yes—
428C22×C4AGL(1,ℤ/8)14yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C2×#4114yes—
438C22×C42−1+414yes—
439C22×C4C2511yescharacteristic
440C2×C8—216no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—48no—
447C2×C8C2×#4118yes—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—416no—
466C4⋊C4C22×D414yes—
467C4⋊C4C22×D414yes—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—216no—
472C22⋊C4—216no—
473C22⋊C4—416no—
474C42—216no—
475C42—28no—
476C42—24no—
477C42—28no—
478C42—22no—
479C42—22no—
480C42#12018yes—
481C42C22×D414yes—
482C42C22×D414yes—
483C23—22no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—28no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23C2×#12012yes—
511C23#53212yes—
512C23#53612yes—
513C23C2×AGL(1,ℤ/8)12yes—
514C23C2×#13714yes—
515C23C2×#13712yes—
516C23#57512yes—
517C23#57512yes—
518C23C23×D411yescharacteristic
519Q8—28no—
520Q8—48no—
521Q8—44no—
522Q8—816no—
523D4—28no—
524D4—432no—
525D4—48no—
526D4—44no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—416no—
546C2×C4—416no—
547C2×C4—432no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—88no—
569C2×C4—88no—
570C2×C4C2×#12014yes—
571C2×C4C2×#12014yes—
572C2×C4C2×#12012yes—
573C2×C4#53212yes—
574C2×C4#53612yes—
575C2×C4#54718yes—
576C2×C4#54818yes—
577C2×C4C2×AGL(1,ℤ/8)14yes—
578C2×C4C23×D411yescharacteristic
579C2×C4C23×D411yescharacteristic
580C2×C4C22×#4112yes—
581C2×C4C2×2−1+412yes—
582C8—28no—
583C8—44no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—42no—
588C22—416no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—82no—
596C22—82no—
597C22—816no—
598C22—82no—
599C22—82no—
600C22#243214yes—
601C22#300614yes—
602C22#300714yes—
603C22#317114yes—
604C22#318114yes—
605C22#323914yes—
606C22#324014yes—
607C22C22×#12011yescharacteristic
608C22C2×#53211yescharacteristic
609C22C2×#53611yescharacteristic
610C22C22×AGL(1,ℤ/8)11yescharacteristic
611C22C22×#13711yescharacteristic
612C22C2×#57511yescharacteristic
613C22C2×#57511yescharacteristic
614C4—24no—
615C4—24no—
616C4—24no—
617C4—42no—
618C4—44no—
619C4—44no—
620C4—42no—
621C4—44no—
622C4—84no—
623C4—84no—
624C4#243214yes—
625C4C2×#54712yes—
626C4C2×#54812yes—
627C2—44no—
628C2—84no—
629C2#2567818yesdirect product factor
630C2C2×#243211yescharacteristic
631C2C2×#300611yescharacteristic
632C2C2×#300711yescharacteristic
633C2C2×#317111yescharacteristic
634C2C2×#318111yescharacteristic
635C2C2×#323911yescharacteristic
636C2C2×#324011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#501C2semidirect
3. C22×#489C2semidirect
4. C2×#2857C2semidirect
5. #25678C2direct (N × H)
6. C2×#2036C2semidirect
8. C2×#1992C2semidirect
9. C2×#1959C2semidirect
10. C2×#1865C2semidirect
11. C2×#1862C2semidirect
12. C2×#1749C2semidirect
31. C2×#501C22semidirect
33. C2×#489C22semidirect
44. #2857C22semidirect
51. #2036C22semidirect
53. #1992C22semidirect
54. #1959C22semidirect
55. #1865C22semidirect
56. #1862C22semidirect
57. #1749C22semidirect
629. C2#25678direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5292
C22×#5012
C22×#4892
C2×#28572
#256782
C2×#20362
C2×#20342
C2×#19922
C2×#19592
C2×#18652
C2×#18622
C2×#17492