Finite Group Databasebeta

← C2×#24091 #6452968

Subgroups — C2×#24091 #6452968

Order 512.

3780 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#553C211yesmaximal, characteristic
2C22×#480C211yesmaximal, characteristic, complemented
3C23×#106C211yesmaximal, characteristic, complemented
4C2×#2486C211yesmaximal, characteristic, complemented
5#24091C2116yesmaximal, direct product factor
6C2×#2059C212yesmaximal, complemented
7C2×#2034C212yesmaximal, complemented
8C2×#1893C211yesmaximal, characteristic, complemented
9C2×#1869C211yesmaximal, characteristic, complemented
10C2×#1864C212yesmaximal, complemented
11C2×#1854C212yesmaximal, complemented
12C2×#1824C211yesmaximal, characteristic, complemented
13C24×C8C2211yescharacteristic
14C2×#553—28no—
15C2×#553—24no—
16C2×#553—28no—
17C2×#553C2218yes—
18C22×#126—21no—
19C22×#125C2211yescharacteristic
20C22×#123—21no—
21C22×#123—21no—
22C22×#121—21no—
23C22×#121C2211yescharacteristic
24C22×#121C2211yescharacteristic
25C22×C4×D4—21no—
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C22×#117C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31C2×#480—28no—
32C2×#480C2218yescomplemented
33C22×#107C2212yes—
34C22×#106—28no—
35C22×#106C2212yes—
36C22×#106C2212yes—
37C22×#106C2218yescomplemented
38C22×#102C2212yes—
39C22×#102C2212yes—
40C22×C22⋊C8C2211yescharacteristic
41C22×C22⋊C8C2212yes—
42#2486—24no—
43#2486C2218yescomplemented
44C2×#398—21no—
45C2×#397—21no—
46C2×#389C2212yes—
47C2×#381—21no—
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50#2059C22116yescomplemented
51#2034C22116yescomplemented
52#1893C2218yescomplemented
53#1869C2218yescomplemented
54#1864C22116yescomplemented
55#1854C22116yescomplemented
56#1824C2218yescomplemented
57C2×#340C2212yes—
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C23×D4—21no—
62C23×D4—21no—
63C23×D4D411yescharacteristic
64C23×D4C2311yescharacteristic
65C24×C4—21no—
66C24×C4—21no—
67C24×C4—21no—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C23×C8—24no—
71C23×C8—28no—
72C23×C8C2312yes—
73C23×C8C2314yes—
74C23×C8C2312yes—
75#553—264no—
76#553—232no—
77#553—232no—
78C2×#126—216no—
79C2×#126—28no—
80C2×#125—24no—
81C2×#125—24no—
82C2×#125—22no—
83C2×#125C2314yes—
84C2×#123—216no—
85C2×#123—28no—
86C2×#123—28no—
87C2×#123—48no—
88C2×#121—216no—
89C2×#121—28no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121—22no—
93C2×#121—22no—
94C2×#121—24no—
95C2×#121D418yes—
96C2×#121C2314yes—
97C2×#121C2314yes—
98C2×C4×D4—216no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—28no—
104C2×C4×D4—22no—
105C2×C4×D4D414yes—
106C2×C4×D4C2314yes—
107C2×C4×D4C2314yes—
108C2×#117—22no—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117C2314yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—24no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D418yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—22no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4D411yescharacteristic
138C22×C22⋊C4D411yescharacteristic
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141#480—232no—
142C2×#107—28no—
143C2×#107C2318yes—
144C2×#106—28no—
145C2×#106—28no—
146C2×#106—232no—
147C2×#106C2318yes—
148C2×#106C2318yes—
149C2×#102—28no—
150C2×#102—28no—
151C2×#102C2318yes—
152C2×#102C2318yes—
153C2×C22⋊C8—24no—
154C2×C22⋊C8—28no—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2318yes—
157#398—28no—
158#397—28no—
159#389—24no—
160#389C2318yes—
161#381—28no—
162#380—24no—
163#380—22no—
164#380C2314yes—
165#380C2318yes—
166C2×#95—21no—
167C2×#95—22no—
168C2×#95—22no—
169#340C2318yes—
170#340C2318yes—
171#340C2318yes—
172#336C2318yes—
173C25—21no—
174C25C2×D411yescharacteristic
175C25C2×D411yescharacteristic
176C22×D4—24no—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—84no—
190C22×D4C2×D414yes—
191C22×D4C2×D414yes—
192C22×D4C2412yes—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—84no—
213C23×C4—84no—
214C23×C4C2×D412yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C22×C8—232no—
226C22×C8—28no—
227C22×C8—24no—
228C22×C8—24no—
229C22×C8—216no—
230C22×C8—24no—
231C22×C8—24no—
232C22×C8—24no—
233C22×C8#4118yes—
234C22×C8C2414yes—
235C22×C8C2414yes—
236#126—264no—
237#125—216no—
238#125—28no—
239#125—28no—
240#123—264no—
241#123—432no—
242#121—264no—
243#121—216no—
244#121—28no—
245#121—216no—
246#121—28no—
247#121—28no—
248#121C2×D4116yes—
249C4×D4—264no—
250C4×D4—216no—
251C4×D4—28no—
252C4×D4—216no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—28no—
256#117—216no—
257#117—28no—
258#117—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D4116yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D414yes—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42C2×D412yes—
318C2×C42C2412yes—
319#107—216no—
320#106—216no—
321#106—216no—
322#102—216no—
323#102—216no—
324C22⋊C8—28no—
325C22⋊C8—216no—
326#95—28no—
327#95—24no—
328#95—28no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—44no—
339C24—42no—
340C24—42no—
341C24—44no—
342C24#12014yes—
343C24#13514yes—
344C24AGL(1,ℤ/8)14yes—
345C24C22×D412yes—
346C24C22×D412yes—
347C24C22×D411yescharacteristic
348C2×D4—28no—
349C2×D4—216no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—216no—
353C2×D4—24no—
354C2×D4—216no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—416no—
361C2×D4—432no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—88no—
366C2×D4—816no—
367C2×D4C22×D414yes—
368C22×C4—24no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—88no—
415C22×C4—816no—
416C22×C4—88no—
417C22×C4—816no—
418C22×C4—816no—
419C22×C4#12014yes—
420C22×C4#12018yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C2×#4114yes—
431C22×C42−1+414yes—
432C22×C4C2511yescharacteristic
433C2×C8—232no—
434C2×C8—216no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—48no—
441C2×C8C2×#4118yes—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—216no—
447C4⋊C4—24no—
448C4⋊C4—28no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—416no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4C22×D414yes—
461C4⋊C4C22×D414yes—
462C4⋊C4C22×D414yes—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—28no—
467C22⋊C4—216no—
468C22⋊C4—28no—
469C22⋊C4—416no—
470C22⋊C4—48no—
471C42—216no—
472C42—24no—
473C42—22no—
474C42—22no—
475C23—216no—
476C23—216no—
477C23—28no—
478C23—24no—
479C23—28no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—48no—
488C23—48no—
489C23—44no—
490C23—48no—
491C23—44no—
492C23—44no—
493C23—48no—
494C23—48no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—88no—
500C23—88no—
501C23#449116yes—
502C23C2×#12014yes—
503C23C2×#12012yes—
504C23#54714yes—
505C23C2×#13514yes—
506C23C2×#13512yes—
507C23C2×AGL(1,ℤ/8)14yes—
508C23C2×AGL(1,ℤ/8)12yes—
509C23#57714yes—
510C23C23×D411yescharacteristic
511D4—28no—
512D4—416no—
513D4—432no—
514D4—48no—
515D4—42no—
516D4—42no—
517D4—816no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—816no—
555C2×C4—816no—
556C2×C4—88no—
557C2×C4—816no—
558C2×C4—88no—
559C2×C4C2×#12014yes—
560C2×C4C2×#12012yes—
561C2×C4#52514yes—
562C2×C4#53614yes—
563C2×C4#54714yes—
564C2×C4#54718yes—
565C2×C4C23×D411yescharacteristic
566C2×C4C23×D411yescharacteristic
567C2×C4C22×#4112yes—
568C2×C4C2×2−1+412yes—
569C8—28no—
570C8—44no—
571C22—232no—
572C22—28no—
573C22—28no—
574C22—24no—
575C22—24no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—44no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—88no—
585C22—816no—
586C22—84no—
587C22—84no—
588C22#240614yes—
589C22C2×#44918yes—
590C22C2×#44914yes—
591C22#300914yes—
592C22#333718yes—
593C22#334518yes—
594C22C22×#12011yescharacteristic
595C22C2×#54712yes—
596C22C22×#13511yescharacteristic
597C22C22×AGL(1,ℤ/8)11yescharacteristic
598C22C2×#57712yes—
599C4—24no—
600C4—42no—
601C4—42no—
602C4—44no—
603C4—42no—
604C4—44no—
605C4—44no—
606C4—84no—
607C4—84no—
608C4#240614yes—
609C4C2×#52511yescharacteristic
610C4C2×#53611yescharacteristic
611C4C2×#54712yes—
612C2—28no—
613C2—82no—
614C2—84no—
615C2#2409118yesdirect product factor
616C2C2×#240611yescharacteristic
617C2C22×#44911yescharacteristic
618C2C2×#300911yescharacteristic
619C2C2×#333712yes—
620C2C2×#334512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#553C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#480C2semidirect
3. C23×#106C2semidirect
4. C2×#2486C2semidirect
5. #24091C2direct (N × H)
6. C2×#2059C2semidirect
7. C2×#2034C2semidirect
8. C2×#1893C2semidirect
9. C2×#1869C2semidirect
10. C2×#1864C2semidirect
11. C2×#1854C2semidirect
12. C2×#1824C2semidirect
32. C2×#480C22semidirect
37. C22×#106C22semidirect
43. #2486C22semidirect
50. #2059C22semidirect
51. #2034C22semidirect
52. #1893C22semidirect
53. #1869C22semidirect
54. #1864C22semidirect
55. #1854C22semidirect
56. #1824C22semidirect
615. C2#24091direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5532
C22×#4802
C23×#1062
C2×#24862
#240912
C2×#20592
C2×#20342
C2×#18932
C2×#18692
C2×#18642
C2×#18542
C2×#18242