Finite Group Databasebeta

← C2×#28885 #6460107

Subgroups — C2×#28885 #6460107

Order 512.

2336 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic, complemented
2C2×#2445C211yesmaximal, characteristic, complemented
3#28885C2116yesmaximal, direct product factor
4C2×#2085C211yesmaximal, characteristic, complemented
5C2×#2065C211yesmaximal, characteristic, complemented
6C2×#2054C211yesmaximal, characteristic, complemented
7C2×#2011C211yesmaximal, characteristic
8C2×#1972C211yesmaximal, characteristic, complemented
9C2×#1970C211yesmaximal, characteristic, complemented
10C2×#1961C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic, complemented
12C2×#1934C211yesmaximal, characteristic, complemented
13C2×#1925C211yesmaximal, characteristic, complemented
14C2×#1879C211yesmaximal, characteristic, complemented
15C2×#1770C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C22×#123—21no—
21C22×C4×D4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2513—24no—
37#2513C2218yescomplemented
38#2445—24no—
39#2445C2218yescomplemented
40C2×#397—21no—
41C2×#397—21no—
42C2×#394—21no—
43C2×#394—21no—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#388—21no—
54C2×#386—21no—
55C2×#382C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2085C2218yescomplemented
59#2065C2218yescomplemented
60#2054C2218yescomplemented
61#2011C2218yes—
62#1972C2218yescomplemented
63#1970C2218yescomplemented
64#1961C2218yescomplemented
65#1936C2218yescomplemented
66#1934C2218yescomplemented
67#1925C2218yescomplemented
68#1879C2218yescomplemented
69#1770C2218yescomplemented
70#1762C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C22×D8—24no—
84C22×D8—24no—
85C22×D8—24no—
86C22×D8—42no—
87C22×D8—42no—
88C22×D8—42no—
89C22×D8C2314yes—
90C22×D8C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×#127—24no—
96C2×#127—22no—
97C2×#127—42no—
98C2×#127C2314yes—
99C2×#123—28no—
100C2×#123—48no—
101C2×C4×D4—24no—
102C2×C4×D4—22no—
103C2×C4×D4—42no—
104C2×C4×D4C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
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×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C2×#107—42no—
132C2×#107C2314yes—
133C2×#106—42no—
134C2×#106C2314yes—
135C2×C4⋊C8—24no—
136C2×C4⋊C8C2314yes—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102—24no—
141C2×#102—24no—
142C2×#102—42no—
143C2×#102—42no—
144C2×#102—42no—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#97—42no—
154C2×#97—42no—
155C2×#97C2314yes—
156C2×#97C2314yes—
157C2×C4×C8—24no—
158C2×C4×C8C2314yes—
159#397—28no—
160#397—28no—
161#394—28no—
162#394—28no—
163#390—22no—
164#390—22no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2314yes—
171#390C2314yes—
172#390C2314yes—
173#390C2314yes—
174#390C2314yes—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2314yes—
179#388—28no—
180#388—28no—
181#386—28no—
182#382—22no—
183#382C2314yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4D414yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2314yes—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95D411yescharacteristic
194C2×#95D411yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25—21no—
203C25—21no—
204C22×D4—21no—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4—82no—
231C22×D4—82no—
232C22×D4#4114yes—
233C22×D4C2412yes—
234C22×D4C2412yes—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—82no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×D8—28no—
253C2×D8—28no—
254C2×D8—28no—
255C2×D8—44no—
256C2×D8—44no—
257C2×D8—44no—
258C2×D8—416no—
259C2×D8—416no—
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#4114yes—
270C22×C8#4114yes—
271C22×C8#4114yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276#127—28no—
277#127—44no—
278#127—48no—
279#123—432no—
280C4×D4—28no—
281C4×D4—44no—
282C4×D4—48no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4—82no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—82no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2412yes—
351#107—44no—
352#106—44no—
353C4⋊C8—28no—
354#102—28no—
355#102—28no—
356#102—28no—
357#102—28no—
358#102—28no—
359#102—44no—
360#102—44no—
361#102—44no—
362#97—44no—
363#97—44no—
364C4×C8—28no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95C2×D414yes—
371#95C2×D414yes—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C22×D411yescharacteristic
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—28no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—816no—
442C2×D4—816no—
443C2×D4C2×#4114yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
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—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—42no—
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—84no—
484C22×C4—816no—
485C22×C4AGL(1,ℤ/8)14yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C4C2511yescharacteristic
500D8—416no—
501D8—416no—
502C2×C8—28no—
503C2×C8—28no—
504C2×C8—28no—
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—44no—
515C2×C8C2×#4114yes—
516C2×C8C2×#4114yes—
517C2×C8C2×#4114yes—
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—48no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—84no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—24no—
551C42—24no—
552C42—24no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—84no—
560C23—24no—
561C23—42no—
562C23—42no—
563C23—48no—
564C23—48no—
565C23—42no—
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—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23—84no—
601C23—84no—
602C23#53212yes—
603C23#54612yes—
604C23#55012yes—
605C23C2×AGL(1,ℤ/8)12yes—
606C23#57512yes—
607C23#57612yes—
608C23#57612yes—
609C23C23×D411yescharacteristic
610D4—24no—
611D4—24no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—42no—
618D4—42no—
619D4—42no—
620D4—42no—
621D4—42no—
622D4—48no—
623D4—816no—
624D4—816no—
625D4—84no—
626D4—84no—
627C2×C4—28no—
628C2×C4—22no—
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—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
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—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—816no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4#52914yes—
672C2×C4#53212yes—
673C2×C4#53814yes—
674C2×C4#54612yes—
675C2×C4#55012yes—
676C2×C4#55114yes—
677C2×C4#55314yes—
678C2×C4C2×AGL(1,ℤ/8)14yes—
679C2×C4C22×#4111yescharacteristic
680C2×C4C22×#4111yescharacteristic
681C2×C4C22×#4111yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C2×H(5,𝔽2)11yescharacteristic
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C8—24no—
686C8—24no—
687C8—24no—
688C8—42no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—816no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—168no—
712C22—168no—
713C22#255414yes—
714C22#317614yes—
715C22#318014yes—
716C22#332214yes—
717C22#332514yes—
718C22#339914yes—
719C22#340314yes—
720C22C2×#53211yescharacteristic
721C22C2×#54611yescharacteristic
722C22C2×#55011yescharacteristic
723C22C22×AGL(1,ℤ/8)11yescharacteristic
724C22C2×#57511yescharacteristic
725C22C2×#57611yescharacteristic
726C22C2×#57611yescharacteristic
727C4—24no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—42no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4#255414yes—
739C4C2×#52911yescharacteristic
740C4C2×#53811yescharacteristic
741C4C2×#55111yescharacteristic
742C4C2×#55311yescharacteristic
743C2—84no—
744C2—162no—
745C2—162no—
746C2#2888518yesdirect product factor
747C2C2×#255411yescharacteristic
748C2C2×#317611yescharacteristic
749C2C2×#318011yescharacteristic
750C2C2×#332211yescharacteristic
751C2C2×#332511yescharacteristic
752C2C2×#339911yescharacteristic
753C2C2×#340311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2011C2
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. C2×#2513C2semidirect
2. C2×#2445C2semidirect
3. #28885C2direct (N × H)
4. C2×#2085C2semidirect
5. C2×#2065C2semidirect
6. C2×#2054C2semidirect
8. C2×#1972C2semidirect
9. C2×#1970C2semidirect
10. C2×#1961C2semidirect
11. C2×#1936C2semidirect
12. C2×#1934C2semidirect
13. C2×#1925C2semidirect
14. C2×#1879C2semidirect
15. C2×#1770C2semidirect
16. C2×#1762C2semidirect
37. #2513C22semidirect
39. #2445C22semidirect
58. #2085C22semidirect
59. #2065C22semidirect
60. #2054C22semidirect
62. #1972C22semidirect
63. #1970C22semidirect
64. #1961C22semidirect
65. #1936C22semidirect
66. #1934C22semidirect
67. #1925C22semidirect
68. #1879C22semidirect
69. #1770C22semidirect
70. #1762C22semidirect
746. C2#28885direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25132
C2×#24452
#288852
C2×#20852
C2×#20652
C2×#20542
C2×#20112
C2×#19722
C2×#19702
C2×#19612
C2×#19362
C2×#19342
C2×#19252
C2×#18792
C2×#17702
C2×#17622