Finite Group Databasebeta

← C2×#23686 #6432267

Subgroups — C2×#23686 #6432267

Order 512.

2790 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#442C211yesmaximal, characteristic
2C22×#440C211yesmaximal, characteristic, complemented
3C2×C4×#124C211yesmaximal, characteristic, complemented
4C2×#2293C211yesmaximal, characteristic, complemented
5#23686C2116yesmaximal, direct product factor
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2073C211yesmaximal, characteristic, complemented
8C2×#2066C211yesmaximal, characteristic, complemented
9C2×#2067C211yesmaximal, characteristic, complemented
10C2×#2055C211yesmaximal, characteristic, complemented
11C2×#2054C211yesmaximal, characteristic
12C2×#2042C211yesmaximal, characteristic, complemented
13C2×#2012C211yesmaximal, characteristic
14C2×#1767C211yesmaximal, characteristic, complemented
15C2×#1763C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×#117—21no—
24C2×#442—28no—
25C2×#442C2218yes—
26C2×#440—28no—
27C2×#440C2218yescomplemented
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#97C2211yescharacteristic
41C22×#97C2211yescharacteristic
42C4×#124—24no—
43C4×#124C2218yescomplemented
44C2×#391C2211yescharacteristic
45#2293—24no—
46#2293C2218yescomplemented
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#388—21no—
54C2×#388—21no—
55C2×#388—21no—
56C2×#382C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4—21no—
59C2×C4×C22⋊C4—21no—
60C2×C4×C22⋊C4—21no—
61C2×C43C2211yescharacteristic
62#2085C2218yescomplemented
63#2073C2218yescomplemented
64#2067C2218yescomplemented
65#2066C2218yescomplemented
66#2055C2218yescomplemented
67#2054C2218yes—
68#2042C2218yescomplemented
69#2012C2218yes—
70#1767C2218yescomplemented
71#1763C2218yescomplemented
72#1762C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4—21no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16—24no—
87C22×SD16C2314yes—
88C22×D8—24no—
89C22×D8—24no—
90C22×D8—24no—
91C22×D8C2314yes—
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×#124—24no—
97C2×#124—24no—
98C2×#124—22no—
99C2×#124C2314yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4D418yes—
110C2×C4×D4C2314yes—
111C2×C4×D4C2314yes—
112C2×#117—28no—
113C2×#117—48no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C42—21no—
133C22×C42—21no—
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#442—232no—
142#440—232no—
143C2×#107—24no—
144C2×#107C2314yes—
145C2×#106—24no—
146C2×#106C2314yes—
147C2×C4⋊C8—24no—
148C2×C4⋊C8—24no—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102—24no—
160C2×#102—24no—
161C2×#102—24no—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#97—24no—
168C2×#97—24no—
169C2×#97C2314yes—
170C2×#97C2314yes—
171#391—22no—
172#391C2314yes—
173#390—22no—
174#390—22no—
175#390—22no—
176#390C2314yes—
177#390C2314yes—
178#390C2314yes—
179#389—22no—
180#389—22no—
181#389C2314yes—
182#389C2314yes—
183#388—28no—
184#388—28no—
185#388—28no—
186#388—28no—
187#382—22no—
188#382C2314yes—
189C4×C4⋊C4—22no—
190C4×C4⋊C4C2314yes—
191C4×C22⋊C4—28no—
192C4×C22⋊C4—28no—
193C4×C22⋊C4—28no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C43D414yes—
202C43C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2314yes—
209C25—21no—
210C25—21no—
211C22×Q8—21no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8#4114yes—
218C22×Q8C2412yes—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4C2×D414yes—
232C22×D4#4114yes—
233C22×D4C2412yes—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—28no—
238C23×C4—24no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—232no—
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16—28no—
260C2×D8—232no—
261C2×D8—28no—
262C2×D8—28no—
263C2×D8—28no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8#4114yes—
275C22×C8#4114yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280#124—216no—
281#124—28no—
282#124—28no—
283C4×Q8—28no—
284C4×Q8—28no—
285C4×Q8—48no—
286C4×D4—216no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—28no—
290C4×D4—48no—
291C4×D4C2×D4116yes—
292#117—432no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42C2×D414yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D414yes—
355C2×C42C2×D414yes—
356C2×C42C2×D414yes—
357C2×C42C2×D412yes—
358C2×C42C2×D414yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2412yes—
363#107—28no—
364#106—28no—
365C4⋊C8—28no—
366C4⋊C8—28no—
367#103—28no—
368#103—28no—
369#103—28no—
370#102—28no—
371#102—28no—
372#102—28no—
373#102—28no—
374#102—28no—
375#97—28no—
376#97—28no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384C24—24no—
385C24—24no—
386C24—22no—
387C24—22no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24C22×D411yescharacteristic
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8C2×#4114yes—
414C2×D4—24no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—24no—
419C2×D4—28no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4C22×D414yes—
439C2×D4C2×#4114yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—216no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—48no—
465C22×C4—432no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—48no—
486C22×C4—816no—
487C22×C4#12114yes—
488C22×C4#12114yes—
489C22×C4AGL(1,ℤ/8)14yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D412yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4C2511yescharacteristic
504SD16—232no—
505D8—232no—
506C2×C8—28no—
507C2×C8—28no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—44no—
519C2×C8—44no—
520C2×C8C2×#4114yes—
521C2×C8C2×#4114yes—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—48no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—48no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—48no—
539C4⋊C4C22×D414yes—
540C22⋊C4—216no—
541C22⋊C4—216no—
542C22⋊C4—216no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C42—24no—
551C42—24no—
552C42—28no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—44no—
558C42—44no—
559C42—44no—
560C42—48no—
561C42—48no—
562C42#12118yes—
563C42C22×D414yes—
564C42C2×#4114yes—
565C23—24no—
566C23—24no—
567C23—48no—
568C23—48no—
569C23—48no—
570C23—48no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—48no—
576C23—48no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23C2×#12112yes—
594C23#52512yes—
595C23#53212yes—
596C23C2×AGL(1,ℤ/8)12yes—
597C23#57512yes—
598C23#57512yes—
599C23#57512yes—
600C23C23×D411yescharacteristic
601Q8—24no—
602Q8—24no—
603Q8—48no—
604Q8—42no—
605Q8—42no—
606D4—24no—
607D4—24no—
608D4—24no—
609D4—24no—
610D4—48no—
611D4—48no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—42no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—432no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—48no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—816no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4C2×#12114yes—
664C2×C4C2×#12114yes—
665C2×C4C2×#12112yes—
666C2×C4#52512yes—
667C2×C4#52514yes—
668C2×C4#52514yes—
669C2×C4#52914yes—
670C2×C4#52914yes—
671C2×C4#53212yes—
672C2×C4C2×AGL(1,ℤ/8)14yes—
673C2×C4C23×D411yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C22×#4111yescharacteristic
677C2×C4C22×#4111yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C8—24no—
680C8—24no—
681C8—42no—
682C8—42no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—816no—
696C22—816no—
697C22—82no—
698C22—82no—
699C22#239214yes—
700C22#303614yes—
701C22#304014yes—
702C22#311114yes—
703C22#311514yes—
704C22#318014yes—
705C22#318314yes—
706C22C22×#12111yescharacteristic
707C22C2×#52511yescharacteristic
708C22C2×#53211yescharacteristic
709C22C22×AGL(1,ℤ/8)11yescharacteristic
710C22C2×#57511yescharacteristic
711C22C2×#57511yescharacteristic
712C22C2×#57511yescharacteristic
713C4—24no—
714C4—24no—
715C4—24no—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—44no—
720C4—44no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4#239214yes—
726C4C2×#52511yescharacteristic
727C4C2×#52511yescharacteristic
728C4C2×#52911yescharacteristic
729C4C2×#52911yescharacteristic
730C2—84no—
731C2—84no—
732C2#2368618yesdirect product factor
733C2C2×#239211yescharacteristic
734C2C2×#303611yescharacteristic
735C2C2×#304011yescharacteristic
736C2C2×#311511yescharacteristic
737C2C2×#311111yescharacteristic
738C2C2×#318011yescharacteristic
739C2C2×#318311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#440C2semidirect
3. C2×C4×#124C2semidirect
4. C2×#2293C2semidirect
5. #23686C2direct (N × H)
6. C2×#2085C2semidirect
7. C2×#2073C2semidirect
8. C2×#2066C2semidirect
9. C2×#2067C2semidirect
10. C2×#2055C2semidirect
12. C2×#2042C2semidirect
14. C2×#1767C2semidirect
15. C2×#1763C2semidirect
16. C2×#1762C2semidirect
27. C2×#440C22semidirect
43. C4×#124C22semidirect
46. #2293C22semidirect
62. #2085C22semidirect
63. #2073C22semidirect
64. #2067C22semidirect
65. #2066C22semidirect
66. #2055C22semidirect
68. #2042C22semidirect
70. #1767C22semidirect
71. #1763C22semidirect
72. #1762C22semidirect
732. C2#23686direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4422
C22×#4402
C2×C4×#1242
C2×#22932
#236862
C2×#20852
C2×#20732
C2×#20662
C2×#20672
C2×#20552
C2×#20542
C2×#20422
C2×#20122
C2×#17672
C2×#17632
C2×#17622