Finite Group Databasebeta

← C2×#28877 #6460076

Subgroups — C2×#28877 #6460076

Order 512.

2292 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2511C211yesmaximal, characteristic, complemented
2C2×#2445C211yesmaximal, characteristic
3#28877C2116yesmaximal, direct product factor
4C2×#2084C211yesmaximal, characteristic, complemented
5C2×#2065C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic, complemented
7C2×#2014C211yesmaximal, characteristic
8C2×#1968C211yesmaximal, characteristic, complemented
9C2×#1968C211yesmaximal, characteristic, complemented
10C2×#1957C211yesmaximal, characteristic
11C2×#1934C211yesmaximal, characteristic, complemented
12C2×#1933C211yesmaximal, characteristic, complemented
13C2×#1926C211yesmaximal, characteristic, complemented
14C2×#1877C211yesmaximal, characteristic, complemented
15C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
16C2×C4×#102C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C22×#123—21no—
21C22×C4×D4C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2511—24no—
37#2511C2218yescomplemented
38#2445—24no—
39#2445C2218yes—
40C2×#400—21no—
41C2×#400—21no—
42C2×#394—21no—
43C2×#394—21no—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#387—21no—
53C2×#387—21no—
54C2×#386—21no—
55C2×#384C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2084C2218yescomplemented
59#2065C2218yescomplemented
60#2055C2218yescomplemented
61#2014C2218yes—
62#1968C2218yescomplemented
63#1968C2218yescomplemented
64#1957C2218yes—
65#1934C2218yescomplemented
66#1933C2218yescomplemented
67#1926C2218yescomplemented
68#1877C2218yescomplemented
69C8×C4⋊C4C2218yescomplemented
70C4×#102C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16—42no—
87C22×SD16—42no—
88C22×SD16—42no—
89C22×SD16C2314yes—
90C22×SD16C2314yes—
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⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
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×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C2×#106—42no—
128C2×#106—42no—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2314yes—
133C2×#103—24no—
134C2×#103—42no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102—24no—
141C2×#102—42no—
142C2×#102—42no—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C4×C8—24no—
150C2×C4×C8—42no—
151C2×C4×C8—42no—
152C2×C4×C8C2314yes—
153C2×C4×C8C2314yes—
154C2×C4×C8C2314yes—
155#400—28no—
156#400—28no—
157#394—28no—
158#394—28no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#391C2314yes—
165#390—22no—
166#390—22no—
167#390—22no—
168#390C2314yes—
169#390C2314yes—
170#390C2314yes—
171#389—22no—
172#389—22no—
173#389C2314yes—
174#389C2314yes—
175#387—28no—
176#387—28no—
177#386—28no—
178#384—22no—
179#384C2314yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4D414yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2314yes—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
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—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8C2412yes—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—24no—
214C22×D4—24no—
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—
235C23×C4—21no—
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×SD16—28no—
253C2×SD16—28no—
254C2×SD16—28no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×SD16—416no—
259C2×SD16—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—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4SD1618yescomplemented
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
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—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—82no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#106—44no—
344#106—44no—
345C4⋊C8—28no—
346#103—28no—
347#103—44no—
348#102—28no—
349#102—28no—
350#102—28no—
351#102—28no—
352#102—44no—
353#102—44no—
354C4×C8—28no—
355C4×C8—44no—
356C4×C8—44no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95C2×D414yes—
367#95C2×D414yes—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24C22×D411yescharacteristic
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—28no—
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—48no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—84no—
428C2×D4—84no—
429C2×D4—816no—
430C2×D4—816no—
431C2×D4C2×#4114yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
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—48no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—48no—
472C22×C4—84no—
473C22×C4—816no—
474C22×C4C2×SD1614yes—
475C22×C4C2×SD1614yes—
476C22×C4C2×SD1614yes—
477C22×C4#13514yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4C2511yescharacteristic
492SD16—416no—
493SD16—416no—
494C2×C8—28no—
495C2×C8—28no—
496C2×C8—28no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—44no—
507C2×C8C2×#4114yes—
508C2×C8C2×#4114yes—
509C2×C8C2×#4114yes—
510C4⋊C4—28no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C2×SD16116yescomplemented
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C42—24no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—84no—
548C23—24no—
549C23—42no—
550C23—42no—
551C23—48no—
552C23—48no—
553C23—42no—
554C23—42no—
555C23—48no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23#53212yes—
577C23#54612yes—
578C23#55012yes—
579C23C22×SD1612yes—
580C23C2×#13512yes—
581C23#57612yes—
582C23#57712yes—
583C23C23×D411yescharacteristic
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—84no—
589D4—24no—
590D4—24no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—48no—
598D4—816no—
599D4—816no—
600D4—84no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—816no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4#52914yes—
647C2×C4#53212yes—
648C2×C4#53814yes—
649C2×C4#54612yes—
650C2×C4#55012yes—
651C2×C4#55114yes—
652C2×C4#55314yes—
653C2×C4C22×SD1614yes—
654C2×C4C22×SD1614yes—
655C2×C4C22×SD1614yes—
656C2×C4C2×#13514yes—
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C8—24no—
664C8—24no—
665C8—24no—
666C8—42no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—82no—
671C22—82no—
672C22—816no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—168no—
683C22#255414yes—
684C22#316714yes—
685C22#317914yes—
686C22#329514yes—
687C22#329614yes—
688C22#339214yes—
689C22#339614yes—
690C22C2×#53211yescharacteristic
691C22C2×#54611yescharacteristic
692C22C2×#55011yescharacteristic
693C22C23×SD1611yescharacteristic
694C22C22×#13511yescharacteristic
695C22C2×#57611yescharacteristic
696C22C2×#57711yescharacteristic
697C4—24no—
698C4—24no—
699C4—24no—
700C4—24no—
701C4—42no—
702C4—42no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4#255414yes—
710C4C2×#52911yescharacteristic
711C4C2×#53811yescharacteristic
712C4C2×#55111yescharacteristic
713C4C2×#55311yescharacteristic
714C2—84no—
715C2—162no—
716C2#2887718yesdirect product factor
717C2C2×#255411yescharacteristic
718C2C2×#316711yescharacteristic
719C2C2×#317911yescharacteristic
720C2C2×#329511yescharacteristic
721C2C2×#329611yescharacteristic
722C2C2×#339211yescharacteristic
723C2C2×#339611yescharacteristic
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. C2×#2511C2semidirect
3. #28877C2direct (N × H)
4. C2×#2084C2semidirect
5. C2×#2065C2semidirect
6. C2×#2055C2semidirect
8. C2×#1968C2semidirect
9. C2×#1968C2semidirect
11. C2×#1934C2semidirect
12. C2×#1933C2semidirect
13. C2×#1926C2semidirect
14. C2×#1877C2semidirect
15. C2×C8×C4⋊C4C2semidirect
16. C2×C4×#102C2semidirect
37. #2511C22semidirect
58. #2084C22semidirect
59. #2065C22semidirect
60. #2055C22semidirect
62. #1968C22semidirect
63. #1968C22semidirect
65. #1934C22semidirect
66. #1933C22semidirect
67. #1926C22semidirect
68. #1877C22semidirect
69. C8×C4⋊C4C22semidirect
70. C4×#102C22semidirect
307. C2×C4⋊C4SD16semidirect
530. C4⋊C4C2×SD16semidirect
716. C2#28877direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25112
C2×#24452
#288772
C2×#20842
C2×#20652
C2×#20552
C2×#20142
C2×#19682
C2×#19572
C2×#19342
C2×#19332
C2×#19262
C2×#18772
C2×C8×C4⋊C42
C2×C4×#1022