Finite Group Databasebeta

← C2×#9231 #198012

Subgroups — C2×#9231 #198012

Order 512.

2000 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2C2×C42×D4C211yesmaximal, characteristic
3C2×#1994C211yesmaximal, characteristic, complemented
4C2×#1503C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6#9231C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic, complemented
8C2×#1276C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×C4×D4—21no—
11C22×C4×D4C2211yescharacteristic
12C23×C42—21no—
13#3101—22no—
14#3101—22no—
15#3101C414yescomplemented
16#3101C2214yes—
17C2×#483—21no—
18C2×#476C412yescomplemented
19C2×#461C412yescomplemented
20C2×#450—21no—
21C2×C4×SD16—22no—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C42×D4—22no—
26C42×D4—22no—
27C42×D4—22no—
28C42×D4C2214yes—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4—21no—
32C2×C4×C22⋊C4—22no—
33C2×C43C2211yescharacteristic
34#1994—22no—
35#1994C2214yescomplemented
36C2×#339—21no—
37C2×#339—21no—
38#1503—22no—
39#1503C2214yescomplemented
40C2×#328—22no—
41#1486—22no—
42#1486C2214yes—
43C2×D4⋊C8—22no—
44#1303C2214yescomplemented
45#1276C2214yescomplemented
46C23×D4D411yescharacteristic
47C24×C4—22no—
48C24×C4—21no—
49C22×SD16—22no—
50C22×M16—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533C2×C412yescomplemented
58#533C2312yes—
59C2×C4⋊Q8C2×C411yescharacteristic, complemented
60C2×C4⋊Q8C2×C411yescharacteristic, complemented
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—216no—
67C2×C4×D4—24no—
68C2×C4×D4—24no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C412yes—
76C2×C4×D4C2×C412yes—
77C2×C4×D4D412yes—
78C2×C4×D4D412yes—
79C2×C4×D4D412yes—
80C2×C4×D4C2312yes—
81C2×#117—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—24no—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42—24no—
92C22×C42—44no—
93C22×C42—44no—
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42C2311yescharacteristic
97#483—24no—
98#483—24no—
99#476—24no—
100#476C2×C418yescomplemented
101#461—24no—
102#461C2×C418yescomplemented
103#450—24no—
104#450—24no—
105C4×SD16—28no—
106C4×SD16—28no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431C2×C412yescomplemented
113#431C2312yes—
114#431C2312yes—
115C2×#106—22no—
116#423—21no—
117#423—21no—
118#423—21no—
119#423C2312yes—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2×C412yescomplemented
124C2×#103—22no—
125C2×#102—22no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4D412yes—
135C4×C4⋊C4C2312yes—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—24no—
139C4×C22⋊C4—28no—
140C2×#95—22no—
141C2×#95D412yes—
142C43—21no—
143C43—21no—
144C43D412yes—
145C43C2312yes—
146#339—24no—
147#339—24no—
148#328—28no—
149D4⋊C8—28no—
150C25—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×D4—28no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C22⋊C412yes—
158C22×D4C22⋊C412yes—
159C22×D4C2×D412yes—
160C23×C4—21no—
161C23×C4—216no—
162C23×C4—28no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—48no—
173C23×C4—48no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C2×SD16—28no—
182C2×SD16—48no—
183C2×M16—22no—
184C2×M16—41no—
185C2×M16—42no—
186C2×M16—42no—
187C22×C8—22no—
188C22×C8—41no—
189C22×C8—41no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8C22×C412yescomplemented
197C4⋊Q8C22×C412yescomplemented
198#125—22no—
199#125—22no—
200#125—41no—
201#125—41no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×D4—232no—
219C4×D4—232no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C22⋊C414yes—
233C4×D4C22⋊C414yes—
234C4×D4C22×C414yes—
235C4×D4C2×D414yes—
236#117—22no—
237#117—22no—
238#117—41no—
239#117—41no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4C22⋊C412yes—
253C2×C4⋊C4C22⋊C412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—41no—
264C2×C42—28no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—28no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—48no—
278C2×C42—416no—
279C2×C42—44no—
280C2×C42—48no—
281C2×C42—41no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—48no—
285C2×C42SD1614yescomplemented
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42#4111yescharacteristic
296C2×C42#4111yescharacteristic
297C2×C42#4111yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42C2411yescharacteristic
300#106—28no—
301C4⋊C8—22no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—41no—
306C4⋊C8—41no—
307C4⋊C8—41no—
308C4⋊C8C22×C414yescomplemented
309#103—28no—
310#102—28no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—41no—
315C4×C8—22no—
316C4×C8—22no—
317C4×C8—22no—
318C4×C8—41no—
319#95—28no—
320#95—22no—
321#95C2×D414yes—
322C24—28no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—44no—
327C24—42no—
328C24—42no—
329C24#12011yescharacteristic
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—44no—
335C2×Q8—82no—
336C2×D4—28no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4#10418yes—
346C2×D4C2×C22⋊C414yes—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—232no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—42no—
365C22×C4—416no—
366C22×C4—416no—
367C22×C4—464no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4#12011yescharacteristic
397C22×C4#12011yescharacteristic
398C22×C4#12011yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12311yescharacteristic
404C22×C4#12311yescharacteristic
405C22×C4C2×SD1612yes—
406C22×C4#13512yes—
407C22×C4AGL(1,ℤ/8)12yes—
408C22×C4#13712yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412SD16—416no—
413M16—42no—
414M16—42no—
415M16—82no—
416C2×C8—24no—
417C2×C8—44no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—82no—
421C2×C8—82no—
422C4⋊C4—28no—
423C4⋊C4—28no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4C2×C22⋊C414yes—
440C22⋊C4—232no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C42—28no—
448C42—28no—
449C42—22no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—44no—
454C42—44no—
455C42—416no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—44no—
461C42—42no—
462C42C2×C22⋊C411yescharacteristic
463C42C2×C22⋊C411yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42C4×D411yescharacteristic
467C42C4×D411yescharacteristic
468C42#12111yescharacteristic
469C42#12111yescharacteristic
470C42#12111yescharacteristic
471C42#12111yescharacteristic
472C42#12311yescharacteristic
473C42#12311yescharacteristic
474C42C2×SD1614yescomplemented
475C42C23×C411yescharacteristic
476C42C22×D411yescharacteristic
477C42C2×#4111yescharacteristic
478C42C2×#4111yescharacteristic
479C23—22no—
480C23—22no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—24no—
485C23—44no—
486C23—416no—
487C23—44no—
488C23—416no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23#38611yescharacteristic
495C23#38611yescharacteristic
496C23#45012yes—
497C23#45212yes—
498C23C2×#12011yescharacteristic
499Q8—44no—
500Q8—82no—
501Q8—82no—
502Q8—82no—
503D4—28no—
504D4—28no—
505D4—24no—
506D4—28no—
507D4C2×#10418yes—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—216no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—464no—
527C2×C4—42no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—416no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#38611yescharacteristic
548C2×C4#38611yescharacteristic
549C2×C4#38611yescharacteristic
550C2×C4#38611yescharacteristic
551C2×C4#38611yescharacteristic
552C2×C4#38611yescharacteristic
553C2×C4C2×#10414yes—
554C2×C4C4×SD1612yes—
555C2×C4C4×SD1612yes—
556C2×C4#44012yes—
557C2×C4#44012yes—
558C2×C4#45012yes—
559C2×C4#45212yes—
560C2×C4#46112yes—
561C2×C4#46112yes—
562C2×C4#46312yes—
563C2×C4#46312yes—
564C2×C4C22×C22⋊C411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×#12311yescharacteristic
570C2×C4C22×SD1611yescharacteristic
571C2×C4C2×#13511yescharacteristic
572C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
573C2×C4C2×#13711yescharacteristic
574C8—44no—
575C8—82no—
576C8—82no—
577C22—21no—
578C22—21no—
579C22—24no—
580C22—24no—
581C22—21no—
582C22—21no—
583C22—28no—
584C22—48no—
585C22—48no—
586C22—416no—
587C22—44no—
588C22—416no—
589C22—44no—
590C22#187712yes—
591C22#187712yes—
592C22#189612yes—
593C22#190612yes—
594C22C2×#38611yescharacteristic
595C22C2×#45011yescharacteristic
596C22C2×#45211yescharacteristic
597C4—28no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—44no—
605C4—48no—
606C4—44no—
607C4—82no—
608C4—82no—
609C4#187712yes—
610C4#187712yes—
611C4#189612yes—
612C4C22×#10412yes—
613C4C2×C4×SD1611yescharacteristic
614C4C2×#44011yescharacteristic
615C4C2×#46111yescharacteristic
616C4C2×#46311yescharacteristic
617C2—21no—
618C2—21no—
619C2—22no—
620C2—48no—
621C2#923114yesdirect product factor
622C2C2×#187711yescharacteristic
623C2C2×#189611yescharacteristic
624C2C2×#190611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23#386
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1994C2semidirect
4. C2×#1503C2semidirect
6. #9231C2direct (N × H)
7. C2×#1303C2semidirect
8. C2×#1276C2semidirect
15. #3101C4semidirect
18. C2×#476C4semidirect
19. C2×#461C4semidirect
35. #1994C22semidirect
39. #1503C22semidirect
44. #1303C22semidirect
45. #1276C22semidirect
57. #533C2×C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
60. C2×C4⋊Q8C2×C4semidirect
100. #476C2×C4semidirect
102. #461C2×C4semidirect
112. #431C2×C4semidirect
123. C2×C4⋊C8C2×C4semidirect
196. C4⋊Q8C22×C4semidirect
197. C4⋊Q8C22×C4semidirect
285. C2×C42SD16semidirect
308. C4⋊C8C22×C4semidirect
474. C42C2×SD16semidirect
621. C2#9231direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31012
C2×C42×D42
C2×#19942
C2×#15032
C2×#14862
#92312
C2×#13032
C2×#12762