Finite Group Databasebeta

← C2×#10409 #239587

Subgroups — C2×#10409 #239587

Order 512.

2034 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
1C2×#2541C211yesmaximal, characteristic
2C2×#1957C211yesmaximal, characteristic, complemented
3C2×#1948C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic, complemented
5C22×#331C211yesmaximal, characteristic, complemented
6C22×D4⋊C8C211yesmaximal, characteristic, complemented
7#10409C218yesmaximal, direct product factor
8C2×#1279C211yesmaximal, characteristic, complemented
9C22×#127—21no—
10C22×#127C2211yescharacteristic
11C22×#121—21no—
12C22×#121—21no—
13C22×C4×D4C2211yescharacteristic
14C22×#106—21no—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#102—21no—
18C22×#102—22no—
19C22×C4×C8C2211yescharacteristic
20#2541—22no—
21#2541—22no—
22#2541—22no—
23#2541C2214yes—
24C2×#392—22no—
25C2×#390—21no—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1957—22no—
29#1957C2214yescomplemented
30#1948—22no—
31#1948C2214yescomplemented
32#1802—22no—
33#1802C2214yescomplemented
34C2×#340—21no—
35C2×#340—21no—
36C2×#331—24no—
37C2×#331C2214yescomplemented
38C2×D4⋊C8—24no—
39C2×D4⋊C8C2214yescomplemented
40#1279C2214yescomplemented
41C23×D4—22no—
42C23×D4—22no—
43C23×D4—21no—
44C23×D4—21no—
45C23×D4D411yescharacteristic
46C23×D4D412yes—
47C24×C4—21no—
48C23×C8—21no—
49C23×C8—21no—
50C23×C8—21no—
51C2×#127—24no—
52C2×#127—24no—
53C2×#127—22no—
54C2×#127—21no—
55C2×#127—21no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127—22no—
59C2×#127—44no—
60C2×#127—44no—
61C2×#127D412yes—
62C2×#127C2312yes—
63C2×#121—24no—
64C2×#121—24no—
65C2×#121—24no—
66C2×#121—24no—
67C2×#121—44no—
68C2×#121—44no—
69C2×#121—44no—
70C2×#121—44no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4D412yes—
78C2×C4×D4C2312yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—21no—
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C2×#106—24no—
91C2×#106—44no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8D414yescomplemented
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100C2×#102—24no—
101C2×#102—28no—
102C2×#102—48no—
103C2×#102—44no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—21no—
107C2×C4×C8C2312yes—
108#392—28no—
109#392—28no—
110#390—24no—
111#390—24no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384C2312yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4C2312yes—
120C2×#95—21no—
121#340—24no—
122#340—24no—
123#331—28no—
124D4⋊C8—28no—
125C25—21no—
126C25—22no—
127C22×D4—28no—
128C22×D4—28no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—44no—
143C22×D4—44no—
144C22×D4—48no—
145C22×D4—48no—
146C22×D4—48no—
147C22×D4—48no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—84no—
161C22×D4—84no—
162C22×D4SD1614yes—
163C22×D4C2×D412yes—
164C22×D4C2×D414yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—41no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C22×C8—24no—
181C22×C8—24no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—42no—
192#127—24no—
193#127—24no—
194#127—22no—
195#127—22no—
196#127—22no—
197#127—22no—
198#127—48no—
199#127—416no—
200#127—48no—
201#127—42no—
202#127—42no—
203#127—42no—
204#127—42no—
205#121—48no—
206#121—48no—
207#121—416no—
208#121—416no—
209#121—48no—
210#121—48no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4D814yes—
240C2×C4⋊C4SD1614yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C2×D412yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4112yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264#106—48no—
265C4⋊C8—24no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—42no—
270C4⋊C8C2×D414yescomplemented
271#102—416no—
272#102—48no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—42no—
276#95—22no—
277#95—22no—
278C24—22no—
279C24—24no—
280C24—24no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—44no—
294C24#12011yescharacteristic
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—24no—
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—416no—
308C2×D4—416no—
309C2×D4—416no—
310C2×D4—416no—
311C2×D4—416no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—82no—
336C2×D4—82no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—88no—
340C2×D4—832no—
341C2×D4—832no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4C2×SD1614yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4#12011yescharacteristic
383C22×C4#12011yescharacteristic
384C22×C4#12011yescharacteristic
385C22×C4#12112yes—
386C22×C4#12112yes—
387C22×C4#12112yes—
388C22×C4#12312yes—
389C22×C4#12312yes—
390C22×C4#12412yes—
391C22×C4C2×D812yes—
392C22×C4C2×SD1612yes—
393C22×C4C2×SD1612yes—
394C22×C4#13512yes—
395C22×C4AGL(1,ℤ/8)12yes—
396C22×C4AGL(1,ℤ/8)12yes—
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4AGL(1,ℤ/8)12yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—48no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—48no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—84no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4C2×D814yes—
430C4⋊C4C2×SD1614yes—
431C22⋊C4—416no—
432C22⋊C4—416no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—44no—
436C42—42no—
437C42—42no—
438C42—44no—
439C42#12112yes—
440C42#12112yes—
441C42#12112yes—
442C42#12312yes—
443C42#12312yes—
444C42#12412yes—
445C42C22×D411yescharacteristic
446C42C2×#4111yescharacteristic
447C42C2×#4111yescharacteristic
448C42C2×#4111yescharacteristic
449C23—24no—
450C23—42no—
451C23—48no—
452C23—48no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—48no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—84no—
467C23—84no—
468C23—84no—
469C23—84no—
470C23—88no—
471C23—88no—
472C23—88no—
473C23—88no—
474C23—88no—
475C23—88no—
476C23—88no—
477C23#39412yes—
478C23#44712yes—
479C23#44912yes—
480C23#45012yes—
481C23#45012yes—
482C23#45312yes—
483C23#45512yes—
484C23C2×#12011yescharacteristic
485D4—24no—
486D4—24no—
487D4—48no—
488D4—42no—
489D4—42no—
490D4—48no—
491D4—84no—
492D4—88no—
493D4—832no—
494D4—832no—
495D4—82no—
496D4—82no—
497D4—82no—
498D4—82no—
499D4—82no—
500D4—82no—
501D4—82no—
502D4—82no—
503D4—88no—
504D4—88no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—88no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—164no—
532C2×C4#39412yes—
533C2×C4#39412yes—
534C2×C4#39412yes—
535C2×C4#44712yes—
536C2×C4#44912yes—
537C2×C4#45012yes—
538C2×C4#45012yes—
539C2×C4#45914yes—
540C2×C4#46014yes—
541C2×C4#46514yes—
542C2×C4#46914yes—
543C2×C4#48114yes—
544C2×C4#48214yes—
545C2×C4#48614yes—
546C2×C4#49014yes—
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×#12311yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4C2×#12411yescharacteristic
553C2×C4C22×D811yescharacteristic
554C2×C4C22×SD1611yescharacteristic
555C2×C4C22×SD1611yescharacteristic
556C2×C4C2×#13511yescharacteristic
557C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
558C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
559C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
560C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
561C8—44no—
562C8—44no—
563C8—82no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—82no—
568C22—82no—
569C22—816no—
570C22—82no—
571C22—82no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—1616no—
580C22#162114yes—
581C22#162914yes—
582C22#163914yes—
583C22#164214yes—
584C22#203414yes—
585C22#203714yes—
586C22#204114yes—
587C22C2×#39411yescharacteristic
588C22C2×#44711yescharacteristic
589C22C2×#45011yescharacteristic
590C22C2×#45011yescharacteristic
591C22C2×#44911yescharacteristic
592C22C2×#45311yescharacteristic
593C22C2×#45511yescharacteristic
594C4—21no—
595C4—21no—
596C4—22no—
597C4—44no—
598C4—44no—
599C4—82no—
600C4—162no—
601C4#203414yes—
602C4#204114yes—
603C4C2×#45911yescharacteristic
604C4C2×#46011yescharacteristic
605C4C2×#46511yescharacteristic
606C4C2×#46911yescharacteristic
607C4C2×#48211yescharacteristic
608C4C2×#48111yescharacteristic
609C4C2×#48611yescharacteristic
610C4C2×#49011yescharacteristic
611C2—84no—
612C2—164no—
613C2#1040918yesdirect product factor
614C2C2×#162911yescharacteristic
615C2C2×#163911yescharacteristic
616C2C2×#162111yescharacteristic
617C2C2×#164211yescharacteristic
618C2C2×#203411yescharacteristic
619C2C2×#203711yescharacteristic
620C2C2×#204111yescharacteristic
subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#2541C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1957C2semidirect
3. C2×#1948C2semidirect
4. C2×#1802C2semidirect
5. C22×#331C2semidirect
6. C22×D4⋊C8C2semidirect
7. #10409C2direct (N × H)
8. C2×#1279C2semidirect
29. #1957C22semidirect
31. #1948C22semidirect
33. #1802C22semidirect
37. C2×#331C22semidirect
39. C2×D4⋊C8C22semidirect
40. #1279C22semidirect
97. C2×C4⋊C8D4semidirect
270. C4⋊C8C2×D4semidirect
613. C2#10409direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25412
C2×#19572
C2×#19482
C2×#18022
C22×#3312
C22×D4⋊C82
#104092
C2×#12792