Finite Group Databasebeta

← C2×#18665 #6348244

Subgroups — C2×#18665 #6348244

Order 512.

3008 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#3249C211yesmaximal, characteristic, complemented
3C2×#3170C211yesmaximal, characteristic, complemented
4C2×#3077C211yesmaximal, characteristic
5#18665C2116yesmaximal, direct product factor
6C2×#1649C212yesmaximal, complemented
7C2×#1637C212yesmaximal, complemented
8C2×#1644C212yesmaximal, complemented
9C2×#1632C212yesmaximal, complemented
10C2×#1550C211yesmaximal, characteristic, complemented
11C2×#1493C211yesmaximal, characteristic, complemented
12C22×D4⋊C8C211yesmaximal, characteristic
13#3501—24no—
14#3501—24no—
15#3501—24no—
16#3501C2218yes—
17C2×#554—21no—
18C2×#554—22no—
19C2×#554—22no—
20C2×#554C2212yes—
21C2×D4×Q8—21no—
22C2×D4×Q8C2212yes—
23C2×#544—21no—
24C2×#543—22no—
25C2×#533C2211yescharacteristic
26C2×#533C2211yescharacteristic
27C2×#533C2211yescharacteristic
28C22×C22⋊Q8—22no—
29C2×#518—21no—
30C22×C4×D4C2211yescharacteristic
31#3249—24no—
32#3249C2218yescomplemented
33#3170—24no—
34#3170C2218yescomplemented
35C2×#500C2211yescharacteristic, complemented
36C2×#494C2211yescharacteristic, complemented
37C2×#487C2211yescharacteristic, complemented
38C2×#487C2211yescharacteristic, complemented
39#3077—24no—
40#3077C2218yes—
41C2×#477—21no—
42C2×#477C2212yescomplemented
43C2×#478—22no—
44C2×#476C2212yes—
45C2×#474C2212yes—
46C2×#467—22no—
47C2×#467—22no—
48C2×#462—21no—
49C2×#462C2212yescomplemented
50C2×C4×Q16—22no—
51C2×#428—21no—
52C2×#423C2211yescharacteristic
53C22×C4⋊C8C2211yescharacteristic
54C2×#418—21no—
55C22×C4×C8C2211yescharacteristic
56#1649C22116yescomplemented
57#1644C22116yescomplemented
58#1637C22116yescomplemented
59#1632C22116yescomplemented
60#1550C2218yescomplemented
61C2×#333C2212yescomplemented
62C2×#332C2212yescomplemented
63#1493C2218yescomplemented
64C2×Q8⋊C8C2212yescomplemented
65C2×Q8⋊C8C2212yescomplemented
66C2×D4⋊C8C2212yes—
67C2×D4⋊C8C2212yes—
68C2×D4⋊C8C2218yes—
69C23×Q8—22no—
70C23×D4D411yescharacteristic
71C24×C4—21no—
72C22×Q16—22no—
73C22×Q16—22no—
74C22×M16—21no—
75C23×C8—21no—
76C23×C8—21no—
77#554—216no—
78#554—216no—
79#554—28no—
80#554—28no—
81#554—24no—
82#554—216no—
83#554—24no—
84#554—216no—
85#554—24no—
86#554C2318yes—
87D4×Q8—28no—
88D4×Q8—28no—
89D4×Q8—24no—
90D4×Q8—24no—
91D4×Q8—24no—
92D4×Q8C2318yes—
93#544—28no—
94#544—28no—
95#543—216no—
96#543—216no—
97#533—22no—
98#533—22no—
99#533—22no—
100#533—22no—
101#533—22no—
102#533—22no—
103#533—22no—
104#533D414yes—
105#533D414yes—
106#533C2314yes—
107#533C2314yes—
108#533C2314yes—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8D411yescharacteristic
112C2×C4⋊Q8D411yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×#125—22no—
118C2×#125—22no—
119C2×#125—21no—
120C2×#125—21no—
121C2×#125—21no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—216no—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8—22no—
128C2×C22⋊Q8—24no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—216no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8—24no—
134C2×C22⋊Q8—21no—
135C2×C22⋊Q8—21no—
136C2×C22⋊Q8D412yes—
137C2×C22⋊Q8D412yes—
138#518—28no—
139#518—28no—
140C2×C4×Q8—21no—
141C2×C4×Q8—21no—
142C2×C4×Q8—21no—
143C2×C4×Q8—21no—
144C2×C4×Q8D412yes—
145C2×C4×Q8C2312yes—
146C2×C4×D4—22no—
147C2×C4×D4—22no—
148C2×C4×D4—22no—
149C2×C4×D4—22no—
150C2×C4×D4—21no—
151C2×C4×D4—21no—
152C2×C4×D4—22no—
153C2×C4×D4—22no—
154C2×C4×D4D412yes—
155C2×C4×D4D414yes—
156C2×C4×D4C2314yes—
157C2×C4×D4C2312yes—
158C2×#117—21no—
159C2×#117—21no—
160C2×#117—21no—
161C22×C4⋊C4—24no—
162C22×C4⋊C4—22no—
163C22×C4⋊C4D411yescharacteristic
164C22×C22⋊C4—24no—
165C22×C22⋊C4—21no—
166C22×C42C2311yescharacteristic
167#500—22no—
168#500C2314yescomplemented
169#494—22no—
170#494C2314yescomplemented
171#487—22no—
172#487—22no—
173#487C2314yescomplemented
174#487C2314yescomplemented
175#478—216no—
176#477—28no—
177#477—24no—
178#477C2318yescomplemented
179#476—24no—
180#476C2318yes—
181#474—24no—
182#474C2318yes—
183#467—216no—
184#467—216no—
185#462—28no—
186#462—24no—
187#462C2318yescomplemented
188C4×Q16—216no—
189#428—28no—
190C2×#107—22no—
191C2×#107—22no—
192C2×#107—22no—
193C2×#106—22no—
194#423—22no—
195#423C2314yes—
196C2×C4⋊C8—21no—
197C2×C4⋊C8—21no—
198C2×C4⋊C8—22no—
199C2×C4⋊C8C2312yes—
200C2×C4⋊C8C2312yes—
201C2×C4⋊C8C2314yes—
202#418—28no—
203C2×#103—22no—
204C2×#103—22no—
205C2×#103—22no—
206C2×#103—22no—
207C2×#102—22no—
208C2×#102—22no—
209C2×C4×C8—22no—
210C2×C4×C8—21no—
211C2×C4×C8C2311yescharacteristic
212C2×C4×C8C2311yescharacteristic
213C2×C4×C8C2314yes—
214#333C2318yescomplemented
215#332C2318yescomplemented
216Q8⋊C8C2318yescomplemented
217Q8⋊C8C2318yescomplemented
218D4⋊C8C2318yes—
219D4⋊C8C2318yes—
220C25—21no—
221C22×Q8—24no—
222C22×Q8—28no—
223C22×Q8—28no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8—21no—
228C22×Q8—84no—
229C22×Q8—84no—
230C22×Q8C2×D412yes—
231C22×Q8C2×D412yes—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—24no—
235C22×D4Q1618yescomplemented
236C22×D4C2×D412yes—
237C22×D4C2×D414yes—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—82no—
246C23×C4—82no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C2×Q16—28no—
251C2×Q16—28no—
252C2×Q16—48no—
253C2×Q16—84no—
254C2×M16—24no—
255C2×M16—44no—
256C22×C8—21no—
257C22×C8—22no—
258C22×C8—21no—
259C22×C8—24no—
260C22×C8—22no—
261C22×C8—24no—
262C22×C8—44no—
263C22×C8—82no—
264C4⋊Q8—216no—
265C4⋊Q8—28no—
266C4⋊Q8—28no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8C2×D414yes—
280C4⋊Q8C2×D412yes—
281C4⋊Q8C2×D414yes—
282C4⋊Q8C2×D412yes—
283C4⋊Q8C2412yes—
284C4⋊Q8C2412yes—
285C4⋊Q8C2412yes—
286C4⋊Q8C2412yes—
287#125—216no—
288#125—28no—
289#125—28no—
290#125—24no—
291#125—24no—
292#125—24no—
293#125—24no—
294#125—24no—
295#125—42no—
296C22⋊Q8—232no—
297C22⋊Q8—28no—
298C22⋊Q8—28no—
299C22⋊Q8—232no—
300C22⋊Q8—232no—
301C22⋊Q8—216no—
302C22⋊Q8—232no—
303C22⋊Q8—216no—
304C22⋊Q8—28no—
305C22⋊Q8—216no—
306C22⋊Q8—24no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—24no—
312C22⋊Q8—28no—
313C22⋊Q8—216no—
314C22⋊Q8—24no—
315C22⋊Q8—24no—
316C22⋊Q8—42no—
317C22⋊Q8—42no—
318C22⋊Q8C2×D418yes—
319C22⋊Q8C2×D418yes—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—24no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—24no—
328C4×Q8—24no—
329C4×Q8—42no—
330C4×Q8—42no—
331C4×Q8C2×D418yes—
332C4×Q8C2×D414yes—
333C4×Q8C2414yes—
334C4×D4—216no—
335C4×D4—28no—
336C4×D4—216no—
337C4×D4—28no—
338C4×D4—28no—
339C4×D4—28no—
340C4×D4—24no—
341C4×D4—24no—
342C4×D4—24no—
343C4×D4—22no—
344C4×D4—24no—
345C4×D4—22no—
346C4×D4—24no—
347C4×D4C2×D414yes—
348C4×D4C2×D418yes—
349C4×D4C2414yes—
350#117—24no—
351#117—24no—
352#117—24no—
353#117—24no—
354#117—24no—
355#117—42no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—28no—
360C2×C4⋊C4—232no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D414yes—
383C2×C4⋊C4C2×D412yes—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—232no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—41no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—41no—
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D412yes—
400C2×C42C2×D412yes—
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D412yes—
403C2×C42C2411yescharacteristic
404C2×C42C2412yes—
405#107—28no—
406#107—28no—
407#107—28no—
408#106—28no—
409C4⋊C8—24no—
410C4⋊C8—22no—
411C4⋊C8—24no—
412C4⋊C8—22no—
413C4⋊C8C2414yes—
414C4⋊C8C2414yes—
415#103—28no—
416#103—28no—
417#103—28no—
418#103—28no—
419#102—28no—
420#102—28no—
421C4×C8—24no—
422C4×C8—21no—
423C4×C8—21no—
424C4×C8C2412yes—
425C4×C8C2412yes—
426C24—22no—
427C24—24no—
428C24—28no—
429C24—82no—
430C24—82no—
431C24#12011yescharacteristic
432C2×Q8—216no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—82no—
451C2×Q8—88no—
452C2×Q8—88no—
453C2×Q8—88no—
454C2×Q8—816no—
455C2×Q8C22×D414yes—
456C2×Q8C22×D414yes—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—28no—
461C2×D4—216no—
462C2×D4—28no—
463C2×D4C2×Q1618yes—
464C2×D4C2×Q16116yescomplemented
465C2×D4C22×D414yes—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—216no—
474C22×C4—216no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—28no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—88no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4—84no—
488C22×C4—82no—
489C22×C4—88no—
490C22×C4—88no—
491C22×C4#12012yes—
492C22×C4#12011yescharacteristic
493C22×C4#12011yescharacteristic
494C22×C4#12012yes—
495C22×C4#12012yes—
496C22×C4#12011yescharacteristic
497C22×C4C2×Q1614yes—
498C22×C4#13714yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505Q16—416no—
506Q16—88no—
507M16—48no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—48no—
514C2×C8—42no—
515C2×C8—44no—
516C2×C8—44no—
517C2×C8—84no—
518C4⋊C4—216no—
519C4⋊C4—216no—
520C4⋊C4—216no—
521C4⋊C4—22no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—28no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4C22×D414yes—
550C4⋊C4C22×D414yes—
551C4⋊C4C22×D414yes—
552C22⋊C4—216no—
553C22⋊C4—216no—
554C22⋊C4—28no—
555C22⋊C4—28no—
556C22⋊C4—28no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—44no—
560C42—28no—
561C42—24no—
562C42—24no—
563C42—44no—
564C42#12012yes—
565C42#12012yes—
566C42#12012yes—
567C42C22×D412yes—
568C42C22×D411yescharacteristic
569C42C22×D412yes—
570C42C22×D411yescharacteristic
571C42C22×D412yes—
572C42C22×D411yescharacteristic
573C42C2511yescharacteristic
574C23—22no—
575C23—22no—
576C23—24no—
577C23—28no—
578C23—216no—
579C23—44no—
580C23—44no—
581C23—84no—
582C23—84no—
583C23—816no—
584C23#45114yes—
585C23C2×#12012yes—
586C23C2×#12011yescharacteristic
587Q8—24no—
588Q8—24no—
589Q8—44no—
590Q8—44no—
591Q8—44no—
592Q8—44no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—42no—
597Q8—82no—
598Q8—84no—
599Q8—82no—
600Q8—816no—
601Q8—82no—
602D4—28no—
603D4—216no—
604D4—24no—
605D4C22×Q1618yes—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—88no—
630C2×C4—84no—
631C2×C4—816no—
632C2×C4—816no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4#45118yes—
636C2×C4#45114yes—
637C2×C4C2×#12012yes—
638C2×C4C2×#12011yescharacteristic
639C2×C4C2×#12012yes—
640C2×C4C2×#12012yes—
641C2×C4C2×#12011yescharacteristic
642C2×C4C2×#12011yescharacteristic
643C2×C4C22×Q1612yes—
644C2×C4C22×Q1614yes—
645C2×C4C2×#13712yes—
646C2×C4C2×#13714yes—
647C2×C4#57612yes—
648C2×C4#57712yes—
649C2×C4C23×D411yescharacteristic
650C2×C4C23×D411yescharacteristic
651C2×C4C23×D411yescharacteristic
652C8—24no—
653C8—42no—
654C8—44no—
655C8—44no—
656C22—22no—
657C22—22no—
658C22—28no—
659C22—28no—
660C22—44no—
661C22—44no—
662C22—416no—
663C22—816no—
664C22C2×#45114yes—
665C22C2×#45112yes—
666C22#301212yes—
667C22#301812yes—
668C22C22×#12011yescharacteristic
669C4—22no—
670C4—21no—
671C4—22no—
672C4—21no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—82no—
679C4—82no—
680C4C2×#45114yes—
681C4C2×#45112yes—
682C4#301212yes—
683C4C23×Q1611yescharacteristic
684C4C22×#13711yescharacteristic
685C4C2×#57611yescharacteristic
686C4C2×#57711yescharacteristic
687C2—24no—
688C2—48no—
689C2#1866514yesdirect product factor
690C2C22×#45111yescharacteristic
691C2C2×#301211yescharacteristic
692C2C2×#301811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3249C2semidirect
3. C2×#3170C2semidirect
5. #18665C2direct (N × H)
6. C2×#1649C2semidirect
7. C2×#1637C2semidirect
8. C2×#1644C2semidirect
9. C2×#1632C2semidirect
10. C2×#1550C2semidirect
11. C2×#1493C2semidirect
32. #3249C22semidirect
34. #3170C22semidirect
35. C2×#500C22semidirect
36. C2×#494C22semidirect
37. C2×#487C22semidirect
38. C2×#487C22semidirect
42. C2×#477C22semidirect
49. C2×#462C22semidirect
56. #1649C22semidirect
57. #1644C22semidirect
58. #1637C22semidirect
59. #1632C22semidirect
60. #1550C22semidirect
61. C2×#333C22semidirect
62. C2×#332C22semidirect
63. #1493C22semidirect
64. C2×Q8⋊C8C22semidirect
65. C2×Q8⋊C8C22semidirect
168. #500C23semidirect
170. #494C23semidirect
173. #487C23semidirect
174. #487C23semidirect
178. #477C23semidirect
187. #462C23semidirect
214. #333C23semidirect
215. #332C23semidirect
216. Q8⋊C8C23semidirect
217. Q8⋊C8C23semidirect
235. C22×D4Q16semidirect
464. C2×D4C2×Q16semidirect
689. C2#18665direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#32492
C2×#31702
C2×#30772
#186652
C2×#16492
C2×#16372
C2×#16442
C2×#16322
C2×#15502
C2×#14932
C22×D4⋊C82