Finite Group Databasebeta

← C2×#11759 #481675

Subgroups — C2×#11759 #481675

Order 512.

1730 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C22×#342C211yesmaximal, characteristic
3C2×#1462C211yesmaximal, characteristic
4#11759C218yesmaximal, direct product factor
5C2×#1351C212yesmaximal, complemented
6C2×#1344C212yesmaximal, complemented
7C22×C22⋊Q8C411yescharacteristic, complemented
8C23×C22⋊C4—21no—
9C23×C22⋊C4C2211yescharacteristic
10C2×#407—21no—
11#2561—22no—
12#2561—22no—
13#2561C414yescomplemented
14#2561C2214yes—
15C2×#397C411yescharacteristic, complemented
16C2×#397C2211yescharacteristic
17C2×#393C411yescharacteristic, complemented
18C2×#393C2211yescharacteristic
19C2×#388—21no—
20C2×#388—21no—
21C2×#381—21no—
22C2×#379—21no—
23C2×C4×C22⋊C4C411yescharacteristic, complemented
24C2×#342—22no—
25C2×#342—21no—
26C2×#342C2212yes—
27C2×#342C2214yes—
28#1462—22no—
29#1462C2214yes—
30C2×#323—21no—
31C2×#323C2212yes—
32#1351C2218yescomplemented
33#1344C2218yescomplemented
34C26D411yescharacteristic
35C23×Q8C2×C411yescharacteristic, complemented
36C24×C4—41no—
37C24×C4C2×C411yescharacteristic
38C24×C4D411yescharacteristic
39C22×M16—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⋊C4C2×C411yescharacteristic, complemented
47C22×C22⋊C4—28no—
48C22×C22⋊C4—24no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—41no—
57C22×C22⋊C4—41no—
58C22×C22⋊C4—42no—
59C22×C22⋊C4—42no—
60C22×C22⋊C4—41no—
61C22×C22⋊C4—44no—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4D412yes—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4D412yes—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4C2312yes—
69C22×C22⋊C4C2311yescharacteristic
70C22×C42—21no—
71#407—24no—
72#407—42no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—41no—
75#397—22no—
76#397—21no—
77#397—21no—
78#397C2×C412yescomplemented
79#397C2×C414yescomplemented
80#397C2312yes—
81#393—22no—
82#393—21no—
83#393—21no—
84#393C2×C412yescomplemented
85#393C2×C414yescomplemented
86#393C2312yes—
87#388—24no—
88#388—24no—
89#388—24no—
90#388—24no—
91#381—24no—
92#381—24no—
93#379—24no—
94#379—24no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4C2×C414yescomplemented
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95C2×C411yescharacteristic, complemented
102C2×#95C2×C411yescharacteristic, complemented
103#342—24no—
104#342—22no—
105#342C2314yes—
106#323—24no—
107#323—22no—
108#323C2314yes—
109C25—21no—
110C25—22no—
111C25—21no—
112C25—24no—
113C25—22no—
114C25—22no—
115C25—44no—
116C25—82no—
117C25C22⋊C412yes—
118C25C22⋊C411yescharacteristic
119C25C22⋊C411yescharacteristic
120C25C2×D412yes—
121C25C2×D411yescharacteristic
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—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—44no—
138C23×C4—48no—
139C23×C4—44no—
140C23×C4—42no—
141C23×C4—82no—
142C23×C4—84no—
143C23×C4—84no—
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C2×M16—44no—
153C2×M16—84no—
154C22×C8—42no—
155C22⋊Q8—44no—
156C22⋊Q8—44no—
157C22⋊Q8—44no—
158C22⋊Q8—44no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4C22×C412yescomplemented
166C2×C22⋊C4—216no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—48no—
195C2×C22⋊C4—416no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—48no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22×C411yescharacteristic
212C2×C22⋊C4C22×C412yes—
213C2×C22⋊C4C2×D412yes—
214C2×C22⋊C4C2×D411yescharacteristic
215C2×C22⋊C4C2×D412yes—
216C2×C22⋊C4C2×D411yescharacteristic
217C2×C22⋊C4C2411yescharacteristic
218C2×C42—24no—
219C2×C42—82no—
220C22⋊C8—24no—
221C22⋊C8—42no—
222C22⋊C8—44no—
223#95—24no—
224#95—22no—
225#95—24no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95C22×C412yescomplemented
232#95C22×C412yescomplemented
233C24—24no—
234C24—24no—
235C24—28no—
236C24—24no—
237C24—28no—
238C24—24no—
239C24—24no—
240C24—22no—
241C24—22no—
242C24—21no—
243C24—21no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24—22no—
251C24—21no—
252C24—24no—
253C24—22no—
254C24—24no—
255C24—22no—
256C24—24no—
257C24—42no—
258C24—44no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—44no—
264C24—44no—
265C24—44no—
266C24—84no—
267C24—82no—
268C24—82no—
269C24—82no—
270C24—82no—
271C24—88no—
272C24—816no—
273C24—88no—
274C24—84no—
275C24#9911yescharacteristic
276C24#9912yes—
277C24#9911yescharacteristic
278C24#10414yes—
279C24C2×C22⋊C412yes—
280C24C2×C22⋊C411yescharacteristic
281C24C2×C22⋊C411yescharacteristic
282C24C2×C22⋊C412yes—
283C24AGL(1,ℤ/8)12yes—
284C24#13712yes—
285C24C22×D411yescharacteristic
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—82no—
289C2×Q8—82no—
290C2×Q8—84no—
291C2×Q8—84no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
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—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—416no—
322C22×C4—44no—
323C22×C4—48no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—82no—
337C22×C4—84no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—84no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—88no—
344C22×C4—832no—
345C22×C4—84no—
346C22×C4—88no—
347C22×C4—84no—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C412yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354M16—88no—
355C2×C8—44no—
356C2×C8—84no—
357C4⋊C4—42no—
358C4⋊C4—48no—
359C4⋊C4—42no—
360C22⋊C4—28no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—41no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—48no—
372C22⋊C4—416no—
373C22⋊C4—48no—
374C22⋊C4—42no—
375C22⋊C4—44no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—48no—
381C22⋊C4—82no—
382C22⋊C4—84no—
383C22⋊C4—82no—
384C42—84no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—24no—
390C23—24no—
391C23—28no—
392C23—24no—
393C23—24no—
394C23—28no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—24no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—22no—
407C23—24no—
408C23—22no—
409C23—22no—
410C23—21no—
411C23—21no—
412C23—22no—
413C23—24no—
414C23—22no—
415C23—42no—
416C23—42no—
417C23—48no—
418C23—48no—
419C23—44no—
420C23—416no—
421C23—44no—
422C23—416no—
423C23—41no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—41no—
429C23—48no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—42no—
436C23—44no—
437C23—42no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—84no—
443C23—84no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—84no—
449C23—84no—
450C23—88no—
451C23—88no—
452C23—88no—
453C23—816no—
454C23—816no—
455C23—88no—
456C23—88no—
457C23#35414yes—
458C23C2×#9912yes—
459C23C2×#9912yes—
460C23C2×#9911yescharacteristic
461C23C2×#9911yescharacteristic
462C23#41712yes—
463C23#41712yes—
464C23C2×#10412yes—
465C23C2×#10414yes—
466C23C22×C22⋊C411yescharacteristic
467C23C2×AGL(1,ℤ/8)11yescharacteristic
468C23C2×#13711yescharacteristic
469Q8—84no—
470Q8—84no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—48no—
494C2×C4—42no—
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—81no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—82no—
515C2×C4—832no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—816no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C8—84no—
523C22—21no—
524C22—21no—
525C22—22no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—24no—
530C22—24no—
531C22—28no—
532C22—22no—
533C22—22no—
534C22—21no—
535C22—21no—
536C22—22no—
537C22—22no—
538C22—24no—
539C22—44no—
540C22—44no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—42no—
545C22—44no—
546C22—48no—
547C22—416no—
548C22—48no—
549C22—41no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—48no—
558C22—44no—
559C22—82no—
560C22—81no—
561C22—81no—
562C22—82no—
563C22—84no—
564C22—82no—
565C22—84no—
566C22—84no—
567C22—88no—
568C22#150412yes—
569C22#150412yes—
570C22#212112yes—
571C22C2×#35414yes—
572C22C2×#35412yes—
573C22C22×#9911yescharacteristic
574C22C2×#41711yescharacteristic
575C22C22×#10411yescharacteristic
576C4—42no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—42no—
582C4—82no—
583C4—82no—
584C4—88no—
585C4—82no—
586C4—82no—
587C2—21no—
588C2—21no—
589C2—24no—
590C2—22no—
591C2—41no—
592C2—42no—
593C2—41no—
594C2—42no—
595C2—44no—
596C2#1175914yesdirect product factor
597C2C2×#150411yescharacteristic
598C2C2×#212111yescharacteristic
599C2C22×#35411yescharacteristic
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. #11759C2direct (N × H)
5. C2×#1351C2semidirect
6. C2×#1344C2semidirect
7. C22×C22⋊Q8C4semidirect
13. #2561C4semidirect
15. C2×#397C4semidirect
17. C2×#393C4semidirect
23. C2×C4×C22⋊C4C4semidirect
32. #1351C22semidirect
33. #1344C22semidirect
35. C23×Q8C2×C4semidirect
44. C2×C22⋊Q8C2×C4semidirect
46. C22×C4⋊C4C2×C4semidirect
78. #397C2×C4semidirect
79. #397C2×C4semidirect
84. #393C2×C4semidirect
85. #393C2×C4semidirect
96. C4×C22⋊C4C2×C4semidirect
101. C2×#95C2×C4semidirect
102. C2×#95C2×C4semidirect
126. C22×Q8C22×C4semidirect
165. C2×C4⋊C4C22×C4semidirect
231. #95C22×C4semidirect
232. #95C22×C4semidirect
596. C2#11759direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25612
C22×#3422
C2×#14622
#117592
C2×#13512
C2×#13442