Finite Group Databasebeta

← C2×#26841 #6443872

Subgroups — C2×#26841 #6443872

Order 512.

2558 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#490C211yesmaximal, characteristic
2C22×#440C211yesmaximal, characteristic, complemented
3C2×#2453C211yesmaximal, characteristic, complemented
4C2×#2445C211yesmaximal, characteristic
5#26841C2116yesmaximal, direct product factor
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2059C211yesmaximal, characteristic, complemented
8C2×#2055C211yesmaximal, characteristic, complemented
9C2×#2014C211yesmaximal, characteristic
10C2×#1968C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic, complemented
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1934C211yesmaximal, characteristic, complemented
14C2×#1880C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic, complemented
16C2×#1770C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×#123—21no—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C2×#490—28no—
25C2×#490C2218yes—
26C2×#440—28no—
27C2×#440C2218yescomplemented
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#97C2211yescharacteristic
40C22×#97C2211yescharacteristic
41C22×C4×C8C2211yescharacteristic
42#2453—24no—
43#2453C2218yescomplemented
44#2445—24no—
45#2445C2218yes—
46C2×#400—21no—
47C2×#398—21no—
48C2×#394—21no—
49C2×#394—21no—
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#389—21no—
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#389C2211yescharacteristic
59C2×#387—21no—
60C2×#386—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62#2067C2218yescomplemented
63#2059C2218yescomplemented
64#2055C2218yescomplemented
65#2014C2218yes—
66#1968C2218yescomplemented
67#1936C2218yescomplemented
68#1935C2218yescomplemented
69#1934C2218yescomplemented
70#1880C2218yescomplemented
71#1864C2218yescomplemented
72#1770C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4—21no—
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C22×SD16—24no—
86C22×SD16—24no—
87C22×SD16—24no—
88C22×SD16—24no—
89C22×SD16—24no—
90C22×SD16—42no—
91C22×SD16C2314yes—
92C22×SD16C2314yes—
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C2×#127—24no—
98C2×#127—22no—
99C2×#127—42no—
100C2×#127C2314yes—
101C2×#125—24no—
102C2×#125—22no—
103C2×#125—42no—
104C2×#125C2314yes—
105C2×#123—28no—
106C2×#123—48no—
107C2×C4×Q8—22no—
108C2×C4×Q8—24no—
109C2×C4×Q8—42no—
110C2×C4×Q8C2314yes—
111C2×C4×D4—24no—
112C2×C4×D4—22no—
113C2×C4×D4—42no—
114C2×C4×D4C2314yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#490—232no—
140#440—232no—
141C2×#107—24no—
142C2×#107C2314yes—
143C2×#106—42no—
144C2×#106C2314yes—
145C2×C4⋊C8—24no—
146C2×C4⋊C8C2314yes—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102—24no—
152C2×#102—24no—
153C2×#102—24no—
154C2×#102—24no—
155C2×#102—24no—
156C2×#102D418yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#97—24no—
164C2×#97—24no—
165C2×#97C2314yes—
166C2×#97C2314yes—
167C2×C4×C8—24no—
168C2×C4×C8C2314yes—
169#400—28no—
170#398—28no—
171#394—28no—
172#394—28no—
173#391—22no—
174#391—22no—
175#391C2314yes—
176#391C2314yes—
177#390—22no—
178#390—22no—
179#390—22no—
180#390C2314yes—
181#390C2314yes—
182#390C2314yes—
183#389—28no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389C2314yes—
188#389C2314yes—
189#389C2314yes—
190#387—28no—
191#386—28no—
192C4×C4⋊C4D414yes—
193C4×C4⋊C4C2314yes—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95D411yescharacteristic
202C2×#95D411yescharacteristic
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—21no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—44no—
238C22×D4—82no—
239C22×D4—82no—
240C22×D4C2×D414yes—
241C22×D4C2412yes—
242C22×D4C2412yes—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—24no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—82no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×SD16—232no—
261C2×SD16—28no—
262C2×SD16—28no—
263C2×SD16—28no—
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—44no—
267C2×SD16—416no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8C2×D414yes—
278C22×C8#4114yes—
279C22×C8#4114yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284#127—28no—
285#127—44no—
286#127—48no—
287#125—28no—
288#125—44no—
289#125—48no—
290#123—432no—
291C4×Q8—28no—
292C4×Q8—44no—
293C4×Q8—48no—
294C4×D4—28no—
295C4×D4—44no—
296C4×D4—48no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—82no—
345C2×C22⋊C4—82no—
346C2×C22⋊C4—82no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—44no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2412yes—
361#107—28no—
362#106—44no—
363C4⋊C8—28no—
364#103—28no—
365#103—28no—
366#102—28no—
367#102—28no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102C2×D4116yes—
372#97—28no—
373#97—28no—
374C4×C8—28no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95C2×D414yes—
383#95C2×D414yes—
384C24—24no—
385C24—22no—
386C24—22no—
387C24—42no—
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—42no—
412C2×Q8—44no—
413C2×Q8C2×#4114yes—
414C2×D4—28no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
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—48no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—48no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4—816no—
446C2×D4—816no—
447C2×D4C22×D414yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
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—48no—
474C22×C4—48no—
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—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—84no—
492C22×C4—816no—
493C22×C4AGL(1,ℤ/8)14yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C42−1+412yes—
507C22×C4C2511yescharacteristic
508SD16—232no—
509SD16—416no—
510C2×C8—28no—
511C2×C8—28no—
512C2×C8—28no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—44no—
523C2×C8C22×D414yes—
524C2×C8C2×#4114yes—
525C2×C8C2×#4114yes—
526C4⋊C4—24no—
527C4⋊C4—28no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—44no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—48no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—48no—
548C4⋊C4C22×D414yes—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—24no—
557C42—24no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—48no—
566C23—24no—
567C23—42no—
568C23—42no—
569C23—48no—
570C23—48no—
571C23—48no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23#53212yes—
595C23D4212yes—
596C23#54812yes—
597C23C2×AGL(1,ℤ/8)12yes—
598C23#57512yes—
599C23#57712yes—
600C23#57712yes—
601C23C23×D411yescharacteristic
602Q8—24no—
603Q8—24no—
604Q8—42no—
605Q8—42no—
606Q8—48no—
607D4—24no—
608D4—24no—
609D4—42no—
610D4—42no—
611D4—42no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—48no—
616D4—816no—
617D4—816no—
618D4—84no—
619C2×C4—28no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
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—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—48no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—48no—
646C2×C4—44no—
647C2×C4—44no—
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—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—816no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4—84no—
667C2×C4#52514yes—
668C2×C4#53212yes—
669C2×C4#54214yes—
670C2×C4D4212yes—
671C2×C4#54714yes—
672C2×C4#54812yes—
673C2×C4#55114yes—
674C2×C4C2×AGL(1,ℤ/8)14yes—
675C2×C4C23×D411yescharacteristic
676C2×C4C22×#4111yescharacteristic
677C2×C4C22×#4111yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×H(5,𝔽2)11yescharacteristic
680C2×C4C2×2−1+411yescharacteristic
681C8—24no—
682C8—24no—
683C8—24no—
684C8—42no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—816no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—168no—
701C22#246114yes—
702C22#317414yes—
703C22#318014yes—
704C22#327214yes—
705C22#327514yes—
706C22#336814yes—
707C22#337114yes—
708C22C2×#53211yescharacteristic
709C22C2×D4211yescharacteristic
710C22C2×#54811yescharacteristic
711C22C22×AGL(1,ℤ/8)11yescharacteristic
712C22C2×#57511yescharacteristic
713C22C2×#57711yescharacteristic
714C22C2×#57711yescharacteristic
715C4—24no—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—44no—
720C4—42no—
721C4—42no—
722C4—44no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4#246114yes—
728C4C2×#52511yescharacteristic
729C4C2×#54211yescharacteristic
730C4C2×#54711yescharacteristic
731C4C2×#55111yescharacteristic
732C2—84no—
733C2—162no—
734C2#2684118yesdirect product factor
735C2C2×#246111yescharacteristic
736C2C2×#317411yescharacteristic
737C2C2×#318011yescharacteristic
738C2C2×#327211yescharacteristic
739C2C2×#327511yescharacteristic
740C2C2×#336811yescharacteristic
741C2C2×#337111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
2. C22×#440C2semidirect
3. C2×#2453C2semidirect
5. #26841C2direct (N × H)
6. C2×#2067C2semidirect
7. C2×#2059C2semidirect
8. C2×#2055C2semidirect
10. C2×#1968C2semidirect
11. C2×#1936C2semidirect
12. C2×#1935C2semidirect
13. C2×#1934C2semidirect
14. C2×#1880C2semidirect
15. C2×#1864C2semidirect
16. C2×#1770C2semidirect
27. C2×#440C22semidirect
43. #2453C22semidirect
62. #2067C22semidirect
63. #2059C22semidirect
64. #2055C22semidirect
66. #1968C22semidirect
67. #1936C22semidirect
68. #1935C22semidirect
69. #1934C22semidirect
70. #1880C22semidirect
71. #1864C22semidirect
72. #1770C22semidirect
734. C2#26841direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4902
C22×#4402
C2×#24532
C2×#24452
#268412
C2×#20672
C2×#20592
C2×#20552
C2×#20142
C2×#19682
C2×#19362
C2×#19352
C2×#19342
C2×#18802
C2×#18642
C2×#17702