Finite Group Databasebeta

← C2×#25280 #6448008

Subgroups — C2×#25280 #6448008

Order 512.

2446 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#488C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic, complemented
3C2×#2544C211yesmaximal, characteristic
4C2×#2370C211yesmaximal, characteristic, complemented
5#25280C2116yesmaximal, direct product factor
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic, complemented
8C2×#2042C211yesmaximal, characteristic, complemented
9C2×#2041C211yesmaximal, characteristic
10C2×#1983C211yesmaximal, characteristic, complemented
11C2×#1983C211yesmaximal, characteristic, complemented
12C2×#1982C211yesmaximal, characteristic
13C2×#1939C211yesmaximal, characteristic, complemented
14C2×#1928C211yesmaximal, characteristic, complemented
15C2×#1926C211yesmaximal, characteristic, complemented
16C2×C4×#102C211yesmaximal, characteristic, complemented
17C22×#127C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#488—28no—
23C2×#488C2218yescomplemented
24C2×#478—28no—
25C2×#478C2218yescomplemented
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C22×C4×C8C2211yescharacteristic
44#2544—24no—
45#2544C2218yes—
46C2×#394—21no—
47C2×#394—21no—
48#2370—24no—
49#2370C2218yescomplemented
50C2×#391C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#389—21no—
53C2×#389C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#387—21no—
56C2×#387—21no—
57C2×#386—21no—
58C2×#386—21no—
59C2×C4×C4⋊C4—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C4⋊C4C2211yescharacteristic
62C2×C4×C4⋊C4C2211yescharacteristic
63C2×C4×C4⋊C4C2211yescharacteristic
64#2067C2218yescomplemented
65#2066C2218yescomplemented
66#2042C2218yescomplemented
67#2041C2218yes—
68#1983C2218yescomplemented
69#1983C2218yescomplemented
70#1982C2218yes—
71#1939C2218yescomplemented
72#1928C2218yescomplemented
73#1926C2218yescomplemented
74C4×#102C2218yescomplemented
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4—21no—
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×#127—24no—
90C2×#127—22no—
91C2×#127—42no—
92C2×#127C2314yes—
93C2×#125—22no—
94C2×#125—24no—
95C2×#125D418yescomplemented
96C2×#125C2314yes—
97C2×#124—24no—
98C2×#124—22no—
99C2×#124—22no—
100C2×#124—24no—
101C2×#124—24no—
102C2×#124—42no—
103C2×#124C2314yes—
104C2×#124C2314yes—
105C2×C4×D4—22no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#488—232no—
134#478—232no—
135C2×#107—24no—
136C2×#107C2314yes—
137C2×#106—24no—
138C2×#106C2314yes—
139C2×C4⋊C8—24no—
140C2×C4⋊C8—42no—
141C2×C4⋊C8—42no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103—24no—
148C2×#103—42no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102—24no—
154C2×#102—24no—
155C2×#102—24no—
156C2×#102—24no—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102—24no—
160C2×#102—42no—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×C4×C8—24no—
170C2×C4×C8C2314yes—
171#394—28no—
172#394—28no—
173#391—22no—
174#391C2314yes—
175#390—22no—
176#390C2314yes—
177#389—28no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#387—28no—
183#387—28no—
184#386—28no—
185#386—28no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—28no—
190C4×C4⋊C4D414yes—
191C4×C4⋊C4C2314yes—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2314yes—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C25—21no—
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8C2412yes—
218C22×D4—22no—
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—44no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—82no—
238C22×D4—82no—
239C22×D4#4114yes—
240C22×D4C2412yes—
241C22×D4C2412yes—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—24no—
249C23×C4—44no—
250C23×C4—82no—
251C23×C4—82no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#127—28no—
276#127—44no—
277#127—48no—
278#125—216no—
279#125—28no—
280#125C2×D4116yescomplemented
281#124—28no—
282#124—28no—
283#124—28no—
284#124—44no—
285#124—48no—
286#124—48no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—48no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
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—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4#4114yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
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—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—44no—
344C2×C42—82no—
345C2×C42—82no—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D414yes—
350C2×C42#4114yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354#107—28no—
355#106—28no—
356C4⋊C8—28no—
357C4⋊C8—44no—
358C4⋊C8—44no—
359#103—28no—
360#103—28no—
361#103—28no—
362#103—44no—
363#102—28no—
364#102—28no—
365#102—28no—
366#102—28no—
367#102—28no—
368#102—28no—
369#102—28no—
370#102—44no—
371C4×C8—28no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95C2×D414yes—
380#95C2×D414yes—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24C22×D411yescharacteristic
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
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—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—48no—
440C2×D4—84no—
441C2×D4—84no—
442C2×D4—816no—
443C2×D4—816no—
444C2×D4C2×#4114yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
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—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—48no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4—816no—
488C22×C4#12114yes—
489C22×C4#12114yes—
490C22×C4#13514yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D412yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C42−1+412yes—
504C22×C4C2511yescharacteristic
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
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—44no—
518C2×C8—44no—
519C2×C8—44no—
520C2×C8—44no—
521C4⋊C4—216no—
522C4⋊C4—216no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—28no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—48no—
538C4⋊C4—84no—
539C4⋊C4C22×D414yes—
540C4⋊C4C22×D414yes—
541C4⋊C4C2×#4114yes—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C42—24no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—48no—
556C42—84no—
557C42—84no—
558C42#12118yes—
559C42C22×D414yes—
560C42C2×#4114yes—
561C23—24no—
562C23—42no—
563C23—48no—
564C23—42no—
565C23—48no—
566C23—42no—
567C23—48no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23C2×#12112yes—
590C23#54612yes—
591C23#54812yes—
592C23C2×#13512yes—
593C23#57512yes—
594C23#57612yes—
595C23#57712yes—
596C23C23×D411yescharacteristic
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—84no—
602D4—24no—
603D4—24no—
604D4—42no—
605D4—48no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—816no—
612D4—816no—
613D4—84no—
614C2×C4—22no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
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—48no—
636C2×C4—48no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
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—816no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4C2×#12114yes—
661C2×C4C2×#12114yes—
662C2×C4C2×#12112yes—
663C2×C4#52914yes—
664C2×C4#54214yes—
665C2×C4#54612yes—
666C2×C4#54812yes—
667C2×C4#54814yes—
668C2×C4#55014yes—
669C2×C4C2×#13514yes—
670C2×C4C23×D411yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C2×H(5,𝔽2)11yescharacteristic
675C2×C4C2×2−1+411yescharacteristic
676C8—42no—
677C8—42no—
678C8—42no—
679C8—42no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—816no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—168no—
696C22#242614yes—
697C22#303514yes—
698C22#304514yes—
699C22#329714yes—
700C22#332314yes—
701C22#336514yes—
702C22#337014yes—
703C22C22×#12111yescharacteristic
704C22C2×#54611yescharacteristic
705C22C2×#54811yescharacteristic
706C22C22×#13511yescharacteristic
707C22C2×#57511yescharacteristic
708C22C2×#57611yescharacteristic
709C22C2×#57711yescharacteristic
710C4—24no—
711C4—24no—
712C4—24no—
713C4—44no—
714C4—44no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—44no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4#242614yes—
723C4C2×#52911yescharacteristic
724C4C2×#54211yescharacteristic
725C4C2×#54811yescharacteristic
726C4C2×#55011yescharacteristic
727C2—84no—
728C2—162no—
729C2#2528018yesdirect product factor
730C2C2×#242611yescharacteristic
731C2C2×#303511yescharacteristic
732C2C2×#304511yescharacteristic
733C2C2×#329711yescharacteristic
734C2C2×#332311yescharacteristic
735C2C2×#336511yescharacteristic
736C2C2×#337011yescharacteristic
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
1. C22×#488C2semidirect
2. C22×#478C2semidirect
4. C2×#2370C2semidirect
5. #25280C2direct (N × H)
6. C2×#2066C2semidirect
7. C2×#2067C2semidirect
8. C2×#2042C2semidirect
10. C2×#1983C2semidirect
11. C2×#1983C2semidirect
13. C2×#1939C2semidirect
14. C2×#1928C2semidirect
15. C2×#1926C2semidirect
16. C2×C4×#102C2semidirect
23. C2×#488C22semidirect
25. C2×#478C22semidirect
49. #2370C22semidirect
64. #2067C22semidirect
65. #2066C22semidirect
66. #2042C22semidirect
68. #1983C22semidirect
69. #1983C22semidirect
71. #1939C22semidirect
72. #1928C22semidirect
73. #1926C22semidirect
74. C4×#102C22semidirect
95. C2×#125D4semidirect
280. #125C2×D4semidirect
729. C2#25280direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4882
C22×#4782
C2×#25442
C2×#23702
#252802
C2×#20662
C2×#20672
C2×#20422
C2×#20412
C2×#19832
C2×#19822
C2×#19392
C2×#19282
C2×#19262
C2×C4×#1022