Finite Group Databasebeta

← C2×#10431 #238659

Subgroups — C2×#10431 #238659

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2C2×#1834C211yesmaximal, characteristic, complemented
3C2×#1802C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic, complemented
5C22×D4⋊C8C211yesmaximal, characteristic, complemented
6C22×D4⋊C8C211yesmaximal, characteristic, complemented
7#10431C218yesmaximal, direct product factor
8C2×#1296C211yesmaximal, characteristic, complemented
9C22×#121—21no—
10C22×#121—21no—
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102—21no—
16C22×#102—21no—
17C22×C4×C8C2211yescharacteristic
18#2537—22no—
19#2537—22no—
20#2537—22no—
21#2537C2214yes—
22C2×#394—22no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#380—21no—
26C2×#380—21no—
27#1834—22no—
28#1834C2214yescomplemented
29#1802—22no—
30#1802—22no—
31#1802C2214yescomplemented
32#1802C2214yescomplemented
33C2×#340—22no—
34C2×#340—21no—
35C2×#340—21no—
36C2×D4⋊C8—24no—
37C2×D4⋊C8—24no—
38C2×D4⋊C8C2214yescomplemented
39C2×D4⋊C8C2214yescomplemented
40#1296C2214yescomplemented
41C23×D4—22no—
42C23×D4D411yescharacteristic
43C23×D4D411yescharacteristic
44C24×C4—21no—
45C24×C4—21no—
46C23×C8—21no—
47C23×C8—21no—
48C23×C8—21no—
49C2×#121—24no—
50C2×#121—24no—
51C2×#121—24no—
52C2×#121—24no—
53C2×#121—44no—
54C2×#121—44no—
55C2×#121—44no—
56C2×#121—44no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4D412yes—
70C2×C4×D4D412yes—
71C2×C4×D4C2312yes—
72C2×C4×D4C2312yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D412yes—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C42C2311yescharacteristic
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—44no—
97C2×#102—44no—
98C2×C4×C8—21no—
99C2×C4×C8—22no—
100C2×C4×C8D414yescomplemented
101C2×C4×C8C2312yes—
102#394—28no—
103#394—28no—
104#384—21no—
105#384—21no—
106#384—21no—
107#384—21no—
108#384—21no—
109#384D412yes—
110#384C2312yes—
111#384C2312yes—
112#380—24no—
113#380—24no—
114#380—24no—
115#380—24no—
116C2×#95—22no—
117#340—24no—
118#340—24no—
119#340—28no—
120D4⋊C8—28no—
121D4⋊C8—28no—
122C25—21no—
123C25—21no—
124C22×D4—28no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—48no—
138C22×D4—48no—
139C22×D4—84no—
140C22×D4D814yes—
141C22×D4SD1614yes—
142C22×D4C2×D412yes—
143C22×D4C2×D412yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—41no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C22×C8—24no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—42no—
179#121—48no—
180#121—48no—
181#121—416no—
182#121—416no—
183#121—48no—
184#121—48no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4112yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C2×C42C2411yescharacteristic
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261#102—48no—
262#102—48no—
263C4×C8—24no—
264C4×C8—22no—
265C4×C8C2×D414yescomplemented
266#95—28no—
267C24—22no—
268C24—22no—
269C24—24no—
270C24—24no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24#12011yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—28no—
297C2×D4—28no—
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—416no—
301C2×D4—416no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—88no—
311C2×D4—832no—
312C2×D4C2×D814yes—
313C2×D4C2×SD1614yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4#12011yescharacteristic
359C22×C4#12011yescharacteristic
360C22×C4#12011yescharacteristic
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4#12112yes—
364C22×C4#12312yes—
365C22×C4#12312yes—
366C22×C4#12412yes—
367C22×C4C2×D812yes—
368C22×C4C2×SD1612yes—
369C22×C4#13512yes—
370C22×C4#13512yes—
371C22×C4AGL(1,ℤ/8)12yes—
372C22×C4AGL(1,ℤ/8)12yes—
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4#13712yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C2×C8—24no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—48no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—84no—
389C2×C8—84no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C22⋊C4—416no—
413C22⋊C4—416no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C42#12112yes—
421C42#12112yes—
422C42#12112yes—
423C42#12312yes—
424C42#12312yes—
425C42#12412yes—
426C42C22×D411yescharacteristic
427C42C2×#4111yescharacteristic
428C42C2×#4111yescharacteristic
429C42C2×#4111yescharacteristic
430C23—24no—
431C23—24no—
432C23—42no—
433C23—42no—
434C23—48no—
435C23—48no—
436C23—48no—
437C23—48no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—48no—
449C23—48no—
450C23—84no—
451C23—84no—
452C23—84no—
453C23—84no—
454C23—84no—
455C23—84no—
456C23—84no—
457C23—84no—
458C23#39412yes—
459C23#44712yes—
460C23#44912yes—
461C23#45012yes—
462C23#45212yes—
463C23#45412yes—
464C23#45512yes—
465C23C2×#12011yescharacteristic
466D4—24no—
467D4—24no—
468D4—24no—
469D4—24no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—48no—
475D4—48no—
476D4—832no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—88no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—88no—
504C2×C4—88no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—164no—
509C2×C4#39412yes—
510C2×C4#39412yes—
511C2×C4#39412yes—
512C2×C4#44712yes—
513C2×C4#44912yes—
514C2×C4#45012yes—
515C2×C4#45212yes—
516C2×C4#46514yes—
517C2×C4#46614yes—
518C2×C4#46814yes—
519C2×C4#46914yes—
520C2×C4#48014yes—
521C2×C4#48114yes—
522C2×C4#48214yes—
523C2×C4#48214yes—
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×#12111yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C2×C4C2×#12311yescharacteristic
528C2×C4C2×#12311yescharacteristic
529C2×C4C2×#12411yescharacteristic
530C2×C4C22×D811yescharacteristic
531C2×C4C22×SD1611yescharacteristic
532C2×C4C2×#13511yescharacteristic
533C2×C4C2×#13511yescharacteristic
534C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
535C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
536C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
537C2×C4C2×#13711yescharacteristic
538C8—44no—
539C8—82no—
540C8—82no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—816no—
554C22—816no—
555C22—82no—
556C22—82no—
557C22#162514yes—
558C22#163514yes—
559C22#163914yes—
560C22#164614yes—
561C22#203314yes—
562C22#203414yes—
563C22#204014yes—
564C22C2×#39411yescharacteristic
565C22C2×#44711yescharacteristic
566C22C2×#45011yescharacteristic
567C22C2×#44911yescharacteristic
568C22C2×#45211yescharacteristic
569C22C2×#45511yescharacteristic
570C22C2×#45411yescharacteristic
571C4—21no—
572C4—21no—
573C4—22no—
574C4—84no—
575C4—82no—
576C4—82no—
577C4—162no—
578C4#203314yes—
579C4#203414yes—
580C4C2×#46511yescharacteristic
581C4C2×#46611yescharacteristic
582C4C2×#46811yescharacteristic
583C4C2×#46911yescharacteristic
584C4C2×#48011yescharacteristic
585C4C2×#48211yescharacteristic
586C4C2×#48211yescharacteristic
587C4C2×#48111yescharacteristic
588C2—84no—
589C2—84no—
590C2#1043118yesdirect product factor
591C2C2×#163911yescharacteristic
592C2C2×#162511yescharacteristic
593C2C2×#163511yescharacteristic
594C2C2×#164611yescharacteristic
595C2C2×#203311yescharacteristic
596C2C2×#203411yescharacteristic
597C2C2×#204011yescharacteristic
subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#2537C2
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×#1834C2semidirect
3. C2×#1802C2semidirect
4. C2×#1802C2semidirect
5. C22×D4⋊C8C2semidirect
6. C22×D4⋊C8C2semidirect
7. #10431C2direct (N × H)
8. C2×#1296C2semidirect
28. #1834C22semidirect
31. #1802C22semidirect
32. #1802C22semidirect
38. C2×D4⋊C8C22semidirect
39. C2×D4⋊C8C22semidirect
40. #1296C22semidirect
100. C2×C4×C8D4semidirect
265. C4×C8C2×D4semidirect
590. C2#10431direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#18342
C2×#18022
C22×D4⋊C82
#104312
C2×#12962