Finite Group Databasebeta

← C2×#11716 #481488

Subgroups — C2×#11716 #481488

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C22×#342C211yesmaximal, characteristic
3C22×#323C211yesmaximal, characteristic
4#11716C218yesmaximal, direct product factor
5C2×#1352C212yesmaximal, complemented
6C2×#1351C212yesmaximal, complemented
7C22×C22⋊Q8C411yescharacteristic, complemented
8C23×C22⋊C4—21no—
9C23×C22⋊C4C2211yescharacteristic
10C22×C22⋊C8—21no—
11#2626—22no—
12#2626—22no—
13#2626C414yescomplemented
14#2626C2214yes—
15C2×#399C411yescharacteristic, complemented
16C2×#399C2211yescharacteristic
17C2×#396—21no—
18C2×#396—21no—
19C2×#393C411yescharacteristic, complemented
20C2×#393C2211yescharacteristic
21C2×#385—21no—
22C2×#380C411yescharacteristic, complemented
23C2×#379—21no—
24C2×#342—22no—
25C2×#342—21no—
26C2×#342C2212yes—
27C2×#342C2214yes—
28C2×#323—22no—
29C2×#323—21no—
30C2×#323C2212yes—
31C2×#323C2214yes—
32#1352C2218yescomplemented
33#1351C2218yescomplemented
34C26D411yescharacteristic
35C23×Q8C2×C411yescharacteristic, complemented
36C24×C4—41no—
37C24×C4C2×C411yescharacteristic
38C24×C4D411yescharacteristic
39C23×C8—41no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—42no—
42C2×C22⋊Q8—41no—
43C2×C22⋊Q8—41no—
44C2×C22⋊Q8C2×C414yescomplemented
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C411yescharacteristic, complemented
50C22×C22⋊C4—28no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—41no—
60C22×C22⋊C4—41no—
61C22×C22⋊C4—42no—
62C22×C22⋊C4—42no—
63C22×C22⋊C4—41no—
64C22×C22⋊C4—44no—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4D412yes—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4D412yes—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2311yescharacteristic
73C2×C22⋊C8—22no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—41no—
77#399—22no—
78#399—21no—
79#399—21no—
80#399C2×C412yescomplemented
81#399C2×C414yescomplemented
82#399C2312yes—
83#396—24no—
84#396—24no—
85#396—24no—
86#396—24no—
87#393—22no—
88#393—21no—
89#393—21no—
90#393C2×C412yescomplemented
91#393C2×C414yescomplemented
92#393C2312yes—
93#385—24no—
94#385—24no—
95#380—22no—
96#380C2×C414yescomplemented
97#379—24no—
98#379—24no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95C2×C411yescharacteristic, complemented
102C2×#95C2×C411yescharacteristic, complemented
103#342—22no—
104#342—24no—
105#342C2314yes—
106#323—22no—
107#323—24no—
108#323C2314yes—
109C25—21no—
110C25—22no—
111C25—21no—
112C25—24no—
113C25—82no—
114C25—82no—
115C25C22⋊C411yescharacteristic
116C25C22⋊C412yes—
117C25C22⋊C411yescharacteristic
118C25SD1614yes—
119C25Q1614yes—
120C25C2×D411yescharacteristic
121C25C2×D412yes—
122C22×Q8—21no—
123C22×Q8—41no—
124C22×Q8—41no—
125C22×Q8—82no—
126C22×Q8C22×C412yescomplemented
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—48no—
138C23×C4—44no—
139C23×C4—42no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4—84no—
143C23×C4—84no—
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C22×C8—44no—
153C22×C8—42no—
154C22×C8—84no—
155C22⋊Q8—44no—
156C22⋊Q8—44no—
157C22⋊Q8—44no—
158C22⋊Q8—44no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C22×C412yescomplemented
173C2×C22⋊C4—28no—
174C2×C22⋊C4—216no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—416no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—48no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22×C411yescharacteristic
218C2×C22⋊C4C22×C412yes—
219C2×C22⋊C4C2×D411yescharacteristic
220C2×C22⋊C4C2×D412yes—
221C2×C22⋊C4C2×D411yescharacteristic
222C2×C22⋊C4C2×D412yes—
223C2×C22⋊C4C2411yescharacteristic
224C22⋊C8—24no—
225C22⋊C8—42no—
226C22⋊C8—44no—
227#95—24no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95C22×C412yescomplemented
233#95C22×C412yescomplemented
234C24—24no—
235C24—24no—
236C24—22no—
237C24—22no—
238C24—21no—
239C24—21no—
240C24—24no—
241C24—28no—
242C24—24no—
243C24—28no—
244C24—21no—
245C24—22no—
246C24—21no—
247C24—22no—
248C24—22no—
249C24—24no—
250C24—22no—
251C24—24no—
252C24—42no—
253C24—42no—
254C24—44no—
255C24—44no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—82no—
261C24—82no—
262C24—82no—
263C24—82no—
264C24—88no—
265C24—88no—
266C24—82no—
267C24—82no—
268C24—84no—
269C24—84no—
270C24—816no—
271C24#9911yescharacteristic
272C24#9911yescharacteristic
273C24#9912yes—
274C24#10314yes—
275C24#10318yes—
276C24#10314yes—
277C24#10414yes—
278C24C2×C22⋊C411yescharacteristic
279C24C2×C22⋊C412yes—
280C24C2×C22⋊C411yescharacteristic
281C24C2×C22⋊C412yes—
282C24C2×SD1612yes—
283C24C2×SD1614yes—
284C24C2×Q1612yes—
285C24C2×Q1614yes—
286C24C22×D411yescharacteristic
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—82no—
290C2×Q8—82no—
291C2×Q8—84no—
292C2×Q8—84no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—416no—
320C22×C4—48no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—84no—
341C22×C4—84no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—88no—
345C22×C4—88no—
346C22×C4—832no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C412yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C2×C8—44no—
356C2×C8—88no—
357C2×C8—84no—
358C4⋊C4—42no—
359C4⋊C4—48no—
360C4⋊C4—42no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C22⋊C4—28no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—41no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—48no—
377C22⋊C4—416no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—48no—
382C22⋊C4—84no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—84no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—21no—
395C23—22no—
396C23—21no—
397C23—24no—
398C23—22no—
399C23—28no—
400C23—24no—
401C23—24no—
402C23—22no—
403C23—21no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—28no—
410C23—22no—
411C23—24no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—48no—
417C23—48no—
418C23—44no—
419C23—44no—
420C23—416no—
421C23—44no—
422C23—44no—
423C23—416no—
424C23—42no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—48no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—84no—
442C23—84no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—88no—
448C23—88no—
449C23—88no—
450C23—84no—
451C23—84no—
452C23—816no—
453C23—816no—
454C23—88no—
455C23—88no—
456C23#32814yes—
457C23#32814yes—
458C23#32818yes—
459C23C2×#9912yes—
460C23C2×#9912yes—
461C23C2×#9911yescharacteristic
462C23C2×#9911yescharacteristic
463C23C2×#10312yes—
464C23C2×#10314yes—
465C23C2×#10312yes—
466C23C2×#10314yes—
467C23C2×#10412yes—
468C23C2×#10414yes—
469C23C22×C22⋊C411yescharacteristic
470C23C22×SD1611yescharacteristic
471C23C22×Q1611yescharacteristic
472Q8—84no—
473Q8—84no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—48no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—832no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—816no—
523C2×C4—82no—
524C2×C4—82no—
525C8—84no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—22no—
532C22—21no—
533C22—22no—
534C22—24no—
535C22—28no—
536C22—24no—
537C22—28no—
538C22—21no—
539C22—21no—
540C22—24no—
541C22—22no—
542C22—48no—
543C22—44no—
544C22—44no—
545C22—41no—
546C22—41no—
547C22—44no—
548C22—42no—
549C22—48no—
550C22—416no—
551C22—48no—
552C22—41no—
553C22—42no—
554C22—41no—
555C22—41no—
556C22—42no—
557C22—41no—
558C22—44no—
559C22—48no—
560C22—44no—
561C22—81no—
562C22—82no—
563C22—81no—
564C22—82no—
565C22—84no—
566C22—82no—
567C22—84no—
568C22—88no—
569C22—84no—
570C22C2×#32814yes—
571C22C2×#32814yes—
572C22C2×#32812yes—
573C22C2×#32812yes—
574C22#212112yes—
575C22#212712yes—
576C22C22×#9911yescharacteristic
577C22C22×#10311yescharacteristic
578C22C22×#10411yescharacteristic
579C4—42no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—82no—
586C4—82no—
587C4—88no—
588C4—82no—
589C4—82no—
590C2—21no—
591C2—21no—
592C2—24no—
593C2—22no—
594C2—24no—
595C2—41no—
596C2—41no—
597C2—44no—
598C2—42no—
599C2#1171614yesdirect product factor
600C2C22×#32811yescharacteristic
601C2C2×#212111yescharacteristic
602C2C2×#212711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11716C2direct (N × H)
5. C2×#1352C2semidirect
6. C2×#1351C2semidirect
7. C22×C22⋊Q8C4semidirect
13. #2626C4semidirect
15. C2×#399C4semidirect
19. C2×#393C4semidirect
22. C2×#380C4semidirect
32. #1352C22semidirect
33. #1351C22semidirect
35. C23×Q8C2×C4semidirect
44. C2×C22⋊Q8C2×C4semidirect
49. C22×C4⋊C4C2×C4semidirect
80. #399C2×C4semidirect
81. #399C2×C4semidirect
90. #393C2×C4semidirect
91. #393C2×C4semidirect
96. #380C2×C4semidirect
101. C2×#95C2×C4semidirect
102. C2×#95C2×C4semidirect
126. C22×Q8C22×C4semidirect
172. C2×C4⋊C4C22×C4semidirect
232. #95C22×C4semidirect
233. #95C22×C4semidirect
599. C2#11716direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26262
C22×#3422
C22×#3232
#117162
C2×#13522
C2×#13512