Finite Group Databasebeta

← C2×#26256 #6444495

Subgroups — C2×#26256 #6444495

Order 512.

2444 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#497C211yesmaximal, characteristic
2C22×C4×SD16C211yesmaximal, characteristic
3C2×#2447C211yesmaximal, characteristic
4C2×#2446C211yesmaximal, characteristic, complemented
5#26256C2116yesmaximal, direct product factor
6C2×#2056C211yesmaximal, characteristic, complemented
7C2×#2055C211yesmaximal, characteristic
8C2×#2043C211yesmaximal, characteristic, complemented
9C2×#2042C211yesmaximal, characteristic
10C2×#1973C211yesmaximal, characteristic
11C2×#1971C211yesmaximal, characteristic, complemented
12C2×#1969C211yesmaximal, characteristic, complemented
13C2×#1943C211yesmaximal, characteristic, complemented
14C2×#1878C211yesmaximal, characteristic, complemented
15C2×#1877C211yesmaximal, characteristic
16C2×#1771C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C23×SD16C2211yescharacteristic
20C23×SD16C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25C2×#497—28no—
26C2×#497C2218yes—
27C2×C4×SD16—28no—
28C2×C4×SD16C2218yes—
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#97C2211yescharacteristic
41C22×#97C2211yescharacteristic
42C22×C4×C8C2211yescharacteristic
43#2447—24no—
44#2447C2218yes—
45#2446—24no—
46#2446C2218yescomplemented
47C2×#400—21no—
48C2×#398—21no—
49C2×#397—21no—
50C2×#397—21no—
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#390C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#389C2211yescharacteristic
59C2×#387—21no—
60C2×#387—21no—
61C2×#385—21no—
62C2×#381—21no—
63C2×C4×C4⋊C4C2211yescharacteristic
64#2056C2218yescomplemented
65#2055C2218yes—
66#2043C2218yescomplemented
67#2042C2218yes—
68#1973C2218yes—
69#1971C2218yescomplemented
70#1969C2218yescomplemented
71#1943C2218yescomplemented
72#1878C2218yescomplemented
73#1877C2218yes—
74#1771C2218yescomplemented
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C22×Q16—24no—
85C22×Q16—24no—
86C22×Q16—24no—
87C22×Q16—24no—
88C22×Q16—24no—
89C22×Q16—42no—
90C22×Q16C2314yes—
91C22×Q16C2314yes—
92C22×SD16—24no—
93C22×SD16—24no—
94C22×SD16—24no—
95C22×SD16—24no—
96C22×SD16—42no—
97C22×SD16D418yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—42no—
107C2×C4⋊Q8C2314yes—
108C2×#124—24no—
109C2×#124—22no—
110C2×#124—42no—
111C2×#124C2314yes—
112C2×C4×Q8—22no—
113C2×C4×Q8—24no—
114C2×C4×Q8—42no—
115C2×C4×Q8C2314yes—
116C2×C4×D4—24no—
117C2×C4×D4—22no—
118C2×C4×D4—42no—
119C2×C4×D4C2314yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#497—232no—
141C4×SD16—232no—
142C2×#107—24no—
143C2×#107C2314yes—
144C2×#106—24no—
145C2×#106C2314yes—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2314yes—
148C2×#103—24no—
149C2×#103—42no—
150C2×#103—42no—
151C2×#103—42no—
152C2×#103—42no—
153C2×#103—42no—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102—24no—
161C2×#102—42no—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#97—24no—
165C2×#97—24no—
166C2×#97C2314yes—
167C2×#97C2314yes—
168C2×C4×C8—24no—
169C2×C4×C8C2314yes—
170#400—28no—
171#398—28no—
172#397—28no—
173#397—28no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391C2314yes—
180#391C2314yes—
181#391C2314yes—
182#391C2314yes—
183#391C2314yes—
184#390—22no—
185#390C2314yes—
186#389—22no—
187#389—22no—
188#389C2314yes—
189#389C2314yes—
190#387—28no—
191#387—28no—
192#385—28no—
193#381—28no—
194C4×C4⋊C4D414yes—
195C4×C4⋊C4C2314yes—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25—21no—
215C22×Q8—21no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8C2×D414yes—
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4C2×D414yes—
246C22×D4C2412yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—24no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—82no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×Q16—28no—
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—28no—
269C2×Q16—28no—
270C2×Q16—44no—
271C2×Q16—416no—
272C2×Q16—416no—
273C2×SD16—28no—
274C2×SD16—232no—
275C2×SD16—28no—
276C2×SD16—28no—
277C2×SD16—28no—
278C2×SD16—44no—
279C2×SD16—416no—
280C2×SD16C2×D4116yes—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8C2×D414yes—
290C22×C8#4114yes—
291C22×C8#4114yes—
292C22×C8#4114yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C4⋊Q8—28no—
298C4⋊Q8—44no—
299C4⋊Q8—48no—
300#124—28no—
301#124—44no—
302#124—48no—
303C4×Q8—28no—
304C4×Q8—44no—
305C4×Q8—48no—
306C4×D4—28no—
307C4×D4—44no—
308C4×D4—48no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—82no—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—82no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—44no—
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2412yes—
363#107—28no—
364#106—28no—
365C4⋊C8—28no—
366#103—28no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#102—28no—
373#102—44no—
374#97—28no—
375#97—28no—
376C4×C8—28no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D414yes—
390#95C2×D414yes—
391C24—24no—
392C24—22no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24C22×D411yescharacteristic
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8C22×D414yes—
439C2×D4—28no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—42no—
450C2×D4—44no—
451C2×D4C22×D414yes—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—48no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—48no—
499C22×C4—84no—
500C22×C4—816no—
501C22×C4#13714yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4C2511yescharacteristic
516Q16—416no—
517Q16—416no—
518SD16—232no—
519SD16—416no—
520C2×C8—28no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—44no—
530C2×C8—44no—
531C2×C8#123116yes—
532C2×C8C22×D414yes—
533C2×C8C2×#4114yes—
534C2×C8C2×#4114yes—
535C2×C8C2×#4114yes—
536C4⋊C4—28no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—44no—
546C4⋊C4—48no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—84no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C22⋊C4—84no—
560C42—24no—
561C42—24no—
562C42—24no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—48no—
570C23—24no—
571C23—42no—
572C23—42no—
573C23—48no—
574C23—48no—
575C23—48no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23#53212yes—
585C23D4212yes—
586C23#55012yes—
587C23C2×#13712yes—
588C23#57512yes—
589C23#57712yes—
590C23#57812yes—
591C23C23×D411yescharacteristic
592Q8—24no—
593Q8—24no—
594Q8—42no—
595Q8—48no—
596Q8—42no—
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—84no—
606Q8—84no—
607D4—24no—
608D4—24no—
609D4—42no—
610D4—48no—
611D4—42no—
612C2×C4—22no—
613C2×C4—28no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—48no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
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—816no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4C2×#12314yes—
665C2×C4#53212yes—
666C2×C4#54014yes—
667C2×C4D4212yes—
668C2×C4#54614yes—
669C2×C4#55012yes—
670C2×C4#55914yes—
671C2×C4C2×#13714yes—
672C2×C4C23×D411yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×H(5,𝔽2)11yescharacteristic
678C8—24no—
679C8—42no—
680C8—42no—
681C8C2×#12318yes—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—82no—
686C22—82no—
687C22—816no—
688C22—82no—
689C22—82no—
690C22#244514yes—
691C22#317214yes—
692C22#317814yes—
693C22#326614yes—
694C22#326914yes—
695C22#339814yes—
696C22#340114yes—
697C22C2×#53211yescharacteristic
698C22C2×D4211yescharacteristic
699C22C2×#55011yescharacteristic
700C22C22×#13711yescharacteristic
701C22C2×#57511yescharacteristic
702C22C2×#57711yescharacteristic
703C22C2×#57811yescharacteristic
704C4—24no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—44no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4#244514yes—
718C4C22×#12311yescharacteristic
719C4C2×#54011yescharacteristic
720C4C2×#54611yescharacteristic
721C4C2×#55911yescharacteristic
722C2—84no—
723C2#2625618yesdirect product factor
724C2C2×#244511yescharacteristic
725C2C2×#317211yescharacteristic
726C2C2×#317811yescharacteristic
727C2C2×#326611yescharacteristic
728C2C2×#326911yescharacteristic
729C2C2×#339811yescharacteristic
730C2C2×#340111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#2446C2semidirect
5. #26256C2direct (N × H)
6. C2×#2056C2semidirect
8. C2×#2043C2semidirect
11. C2×#1971C2semidirect
12. C2×#1969C2semidirect
13. C2×#1943C2semidirect
14. C2×#1878C2semidirect
16. C2×#1771C2semidirect
46. #2446C22semidirect
64. #2056C22semidirect
66. #2043C22semidirect
69. #1971C22semidirect
70. #1969C22semidirect
71. #1943C22semidirect
72. #1878C22semidirect
74. #1771C22semidirect
723. C2#26256direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4972
C22×C4×SD162
C2×#24472
C2×#24462
#262562
C2×#20562
C2×#20552
C2×#20432
C2×#20422
C2×#19732
C2×#19712
C2×#19692
C2×#19432
C2×#18782
C2×#18772
C2×#17712