Finite Group Databasebeta

← C2×#26981 #6443636

Subgroups — C2×#26981 #6443636

Order 512.

2440 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×#467C211yesmaximal, characteristic
2C22×#441C211yesmaximal, characteristic, complemented
3C2×#2421C211yesmaximal, characteristic
4C2×#2429C211yesmaximal, characteristic
5#26981C2116yesmaximal, direct product factor
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2035C211yesmaximal, characteristic
8C2×#2016C211yesmaximal, characteristic, complemented
9C2×#2001C211yesmaximal, characteristic
10C2×#1960C211yesmaximal, characteristic, complemented
11C2×#1935C211yesmaximal, characteristic, complemented
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1892C211yesmaximal, characteristic
14C2×#1878C211yesmaximal, characteristic, complemented
15C2×#1863C211yesmaximal, characteristic, complemented
16C2×#1752C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C24×C8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C22⋊Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C2×#467—28no—
27C2×#467C2218yes—
28C2×#441—28no—
29C2×#441C2218yescomplemented
30C22×#107C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41C22×#97C2211yescharacteristic
42C22×#97C2211yescharacteristic
43#2429—24no—
44#2429C2218yes—
45#2421—24no—
46#2421C2218yes—
47C2×#400—21no—
48C2×#398—21no—
49C2×#397—21no—
50C2×#396—21no—
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#389—21no—
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#388—21no—
57C2×#388—21no—
58C2×#387—21no—
59C2×#382C2211yescharacteristic
60C2×#380C2211yescharacteristic
61C2×C4×C22⋊C4C2211yescharacteristic
62#2085C2218yescomplemented
63#2035C2218yes—
64#2016C2218yescomplemented
65#2001C2218yes—
66#1960C2218yescomplemented
67#1935C2218yescomplemented
68#1935C2218yescomplemented
69#1892C2218yes—
70#1878C2218yescomplemented
71#1863C2218yescomplemented
72#1752C2218yes—
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×Q16—24no—
84C22×Q16—24no—
85C22×Q16—24no—
86C22×Q16—24no—
87C22×Q16—24no—
88C22×Q16—42no—
89C22×Q16C2314yes—
90C22×Q16C2314yes—
91C23×C8—24no—
92C23×C8—24no—
93C23×C8—42no—
94C23×C8C2311yescharacteristic
95C23×C8C2314yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8C2314yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—42no—
110C2×C22⋊Q8—42no—
111C2×C22⋊Q8—42no—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2314yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8—42no—
120C2×C4×Q8—42no—
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2314yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4D411yescharacteristic
142C22×C22⋊C4D411yescharacteristic
143C22×C42—21no—
144C22×C42—21no—
145C22×C42C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#467—232no—
148#441—232no—
149C2×#107—24no—
150C2×#107C2314yes—
151C2×#106—24no—
152C2×#106C2314yes—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103—24no—
157C2×#103—42no—
158C2×#103—42no—
159C2×#103—42no—
160C2×#103D418yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×C22⋊C8—24no—
170C2×C22⋊C8C2314yes—
171C2×#97—24no—
172C2×#97—42no—
173C2×#97C2314yes—
174C2×#97C2314yes—
175#400—28no—
176#398—28no—
177#397—28no—
178#396—28no—
179#391—22no—
180#391—22no—
181#391C2314yes—
182#391C2314yes—
183#389—22no—
184#389—22no—
185#389—28no—
186#389C2314yes—
187#389C2314yes—
188#388—28no—
189#388—28no—
190#387—28no—
191#382—22no—
192#382C2314yes—
193#380—22no—
194#380C2314yes—
195C4×C22⋊C4D414yes—
196C4×C22⋊C4C2314yes—
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×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25C2×D411yescharacteristic
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×Q8C2×D414yes—
235C22×Q8#4114yes—
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261C2×Q16—232no—
262C2×Q16—28no—
263C2×Q16—28no—
264C2×Q16—28no—
265C2×Q16—28no—
266C2×Q16—28no—
267C2×Q16—44no—
268C2×Q16—416no—
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×C8—22no—
278C22×C8—28no—
279C22×C8—28no—
280C22×C8—44no—
281C22×C8—416no—
282C22×C8C2×D414yes—
283C22×C8#4114yes—
284C22×C8#4114yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—28no—
290C4⋊Q8—44no—
291C4⋊Q8—48no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—44no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C22⋊Q8—48no—
299C22⋊Q8—48no—
300C22⋊Q8—48no—
301C4×Q8—28no—
302C4×Q8—28no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×Q8—48no—
306C4×Q8—48no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
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—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—42no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4C2×D414yes—
356C2×C22⋊C4C2×D414yes—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—44no—
364C2×C42—44no—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2412yes—
368C2×C42C2412yes—
369#107—28no—
370#106—28no—
371#103—28no—
372#103—28no—
373#103—28no—
374#103—28no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103C2×D4116yes—
379C22⋊C8—28no—
380#97—28no—
381#97—44no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95C2×D414yes—
392#95C2×D414yes—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24#13714yes—
400C24C22×D412yes—
401C24C22×D411yescharacteristic
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—28no—
409C2×Q8—28no—
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—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8C22×D414yes—
440C2×Q8C2×#4114yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
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—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
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—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
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—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4H(5,𝔽2)12yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C4C2511yescharacteristic
514Q16—232no—
515Q16—416no—
516C2×C8—28no—
517C2×C8—28no—
518C2×C8—28no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—416no—
529C2×C8—44no—
530C2×C8C22×D414yes—
531C2×C8C2×#4114yes—
532C2×C8C2×#4114yes—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—48no—
546C4⋊C4—48no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—48no—
551C4⋊C4—48no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4C22×D414yes—
557C22⋊C4—28no—
558C22⋊C4—44no—
559C22⋊C4—48no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—48no—
563C42—24no—
564C42—24no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—48no—
570C42—48no—
571C23—28no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—28no—
576C23—28no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23#53212yes—
585C23D4212yes—
586C23#54612yes—
587C23C2×#13714yes—
588C23C2×#13712yes—
589C23#57512yes—
590C23#57812yes—
591C23#57812yes—
592C23C23×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—24no—
596Q8—24no—
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—48no—
602Q8—48no—
603Q8—42no—
604Q8—42no—
605Q8—42no—
606Q8—42no—
607Q8—84no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—44no—
634C2×C4—48no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—48no—
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—42no—
659C2×C4—42no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4#52514yes—
666C2×C4#53212yes—
667C2×C4#54014yes—
668C2×C4D4212yes—
669C2×C4#54614yes—
670C2×C4#54612yes—
671C2×C4#55314yes—
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—24no—
680C8—24no—
681C8—42no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—42no—
686C22—42no—
687C22—416no—
688C22—42no—
689C22—42no—
690C22#246414yes—
691C22#317714yes—
692C22#318114yes—
693C22#326914yes—
694C22#327014yes—
695C22#331214yes—
696C22#331314yes—
697C22C2×#53211yescharacteristic
698C22C2×D4211yescharacteristic
699C22C2×#54611yescharacteristic
700C22C22×#13711yescharacteristic
701C22C2×#57511yescharacteristic
702C22C2×#57811yescharacteristic
703C22C2×#57811yescharacteristic
704C4—42no—
705C4—44no—
706C4—42no—
707C4—42no—
708C4—44no—
709C4—44no—
710C4—42no—
711C4—42no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4#246414yes—
718C4C2×#52511yescharacteristic
719C4C2×#54011yescharacteristic
720C4C2×#54611yescharacteristic
721C4C2×#55311yescharacteristic
722C2—44no—
723C2#2698118yesdirect product factor
724C2C2×#246411yescharacteristic
725C2C2×#317711yescharacteristic
726C2C2×#318111yescharacteristic
727C2C2×#326911yescharacteristic
728C2C2×#327011yescharacteristic
729C2C2×#331211yescharacteristic
730C2C2×#331311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#441C2semidirect
5. #26981C2direct (N × H)
6. C2×#2085C2semidirect
8. C2×#2016C2semidirect
10. C2×#1960C2semidirect
11. C2×#1935C2semidirect
12. C2×#1935C2semidirect
14. C2×#1878C2semidirect
15. C2×#1863C2semidirect
29. C2×#441C22semidirect
62. #2085C22semidirect
64. #2016C22semidirect
66. #1960C22semidirect
67. #1935C22semidirect
68. #1935C22semidirect
70. #1878C22semidirect
71. #1863C22semidirect
723. C2#26981direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4672
C22×#4412
C2×#24212
C2×#24292
#269812
C2×#20852
C2×#20352
C2×#20162
C2×#20012
C2×#19602
C2×#19352
C2×#18922
C2×#18782
C2×#18632
C2×#17522