Finite Group Databasebeta

← C2×#28893 #6460087

Subgroups — C2×#28893 #6460087

Order 512.

2174 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C211yesmaximal, characteristic, complemented
2C2×#2447C211yesmaximal, characteristic, complemented
3#28893C2116yesmaximal, direct product factor
4C2×#2084C211yesmaximal, characteristic, complemented
5C2×#2065C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic, complemented
7C2×#2012C211yesmaximal, characteristic
8C2×#1972C211yesmaximal, characteristic
9C2×#1968C211yesmaximal, characteristic, complemented
10C2×#1959C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic, complemented
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1879C211yesmaximal, characteristic
15C2×#1771C211yesmaximal, characteristic, complemented
16C2×C4×#102C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#126—21no—
20C22×#124C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2512—24no—
37#2512C2218yescomplemented
38#2447—24no—
39#2447C2218yescomplemented
40C2×#397—21no—
41C2×#397—21no—
42C2×#397—21no—
43C2×#396—21no—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#385—21no—
54C2×#384C2211yescharacteristic
55C2×#381—21no—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2084C2218yescomplemented
59#2065C2218yescomplemented
60#2055C2218yescomplemented
61#2012C2218yes—
62#1972C2218yes—
63#1968C2218yescomplemented
64#1959C2218yescomplemented
65#1936C2218yescomplemented
66#1935C2218yescomplemented
67#1927C2218yescomplemented
68#1879C2218yes—
69#1771C2218yescomplemented
70C4×#102C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×SD16—24no—
82C22×SD16—42no—
83C22×SD16—42no—
84C22×SD16C2314yes—
85C22×D8—24no—
86C22×D8—24no—
87C22×D8—42no—
88C22×D8C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×#126—28no—
94C2×#126—48no—
95C2×#124—24no—
96C2×#124—22no—
97C2×#124—42no—
98C2×#124C2314yes—
99C2×C4×D4—24no—
100C2×C4×D4—22no—
101C2×C4×D4—42no—
102C2×C4×D4C2314yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
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×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C2×#106—42no—
130C2×#106—42no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8—24no—
134C2×C4⋊C8C2314yes—
135C2×#103—24no—
136C2×#103—24no—
137C2×#103—42no—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102—24no—
142C2×#102—24no—
143C2×#102—24no—
144C2×#102—42no—
145C2×#102—42no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#97—24no—
152C2×#97—42no—
153C2×#97C2314yes—
154C2×#97C2314yes—
155C2×C4×C8—42no—
156C2×C4×C8C2314yes—
157#397—28no—
158#397—28no—
159#397—28no—
160#396—28no—
161#391—22no—
162#391—22no—
163#391C2314yes—
164#391C2314yes—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2314yes—
171#390C2314yes—
172#390C2314yes—
173#389—22no—
174#389—22no—
175#389C2314yes—
176#389C2314yes—
177#388—28no—
178#385—28no—
179#384—22no—
180#384C2314yes—
181#381—28no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4D414yes—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
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—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#4114yes—
225C22×D4C2412yes—
226C22×D4C2412yes—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—24no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—82no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×SD16—28no—
245C2×SD16—44no—
246C2×SD16—44no—
247C2×SD16—416no—
248C2×D8—28no—
249C2×D8—28no—
250C2×D8—44no—
251C2×D8—416no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8#4114yes—
262C22×C8#4114yes—
263C22×C8#4114yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268#126—432no—
269#124—28no—
270#124—44no—
271#124—48no—
272C4×D4—28no—
273C4×D4—44no—
274C4×D4—48no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
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—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
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—44no—
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—28no—
348#103—44no—
349#102—28no—
350#102—28no—
351#102—28no—
352#102—44no—
353#102—44no—
354#97—28no—
355#97—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#95C2×D414yes—
365#95C2×D414yes—
366C24—24no—
367C24—22no—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24C22×D411yescharacteristic
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—22no—
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—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4C2×#4114yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
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—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
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—48no—
465C22×C4—84no—
466C22×C4—816no—
467C22×C4AGL(1,ℤ/8)14yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4C2511yescharacteristic
482SD16—416no—
483D8—416no—
484C2×C8—28no—
485C2×C8—28no—
486C2×C8—28no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—44no—
497C2×C8C2×#4114yes—
498C2×C8C2×#4114yes—
499C2×C8C2×#4114yes—
500C4⋊C4—28no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C42—24no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—48no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—48no—
546C23—48no—
547C23—48no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23#53212yes—
571C23#54612yes—
572C23#55012yes—
573C23C2×AGL(1,ℤ/8)12yes—
574C23#57512yes—
575C23#57712yes—
576C23#57712yes—
577C23C23×D411yescharacteristic
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—84no—
583D4—24no—
584D4—24no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—48no—
592D4—84no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
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—816no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4#52914yes—
642C2×C4#53212yes—
643C2×C4#53814yes—
644C2×C4#54612yes—
645C2×C4#55012yes—
646C2×C4#55114yes—
647C2×C4#55314yes—
648C2×C4C2×AGL(1,ℤ/8)14yes—
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C8—24no—
656C8—24no—
657C8—24no—
658C8—42no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—816no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—168no—
675C22#255414yes—
676C22#317114yes—
677C22#317314yes—
678C22#330614yes—
679C22#330714yes—
680C22#339714yes—
681C22#340114yes—
682C22C2×#53211yescharacteristic
683C22C2×#54611yescharacteristic
684C22C2×#55011yescharacteristic
685C22C22×AGL(1,ℤ/8)11yescharacteristic
686C22C2×#57511yescharacteristic
687C22C2×#57711yescharacteristic
688C22C2×#57711yescharacteristic
689C4—24no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4#255414yes—
702C4C2×#52911yescharacteristic
703C4C2×#53811yescharacteristic
704C4C2×#55111yescharacteristic
705C4C2×#55311yescharacteristic
706C2—84no—
707C2—162no—
708C2#2889318yesdirect product factor
709C2C2×#255411yescharacteristic
710C2C2×#317111yescharacteristic
711C2C2×#317311yescharacteristic
712C2C2×#330611yescharacteristic
713C2C2×#330711yescharacteristic
714C2C2×#339711yescharacteristic
715C2C2×#340111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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×#2512C2semidirect
2. C2×#2447C2semidirect
3. #28893C2direct (N × H)
4. C2×#2084C2semidirect
5. C2×#2065C2semidirect
6. C2×#2055C2semidirect
9. C2×#1968C2semidirect
10. C2×#1959C2semidirect
11. C2×#1936C2semidirect
12. C2×#1935C2semidirect
13. C2×#1927C2semidirect
15. C2×#1771C2semidirect
16. C2×C4×#102C2semidirect
37. #2512C22semidirect
39. #2447C22semidirect
58. #2084C22semidirect
59. #2065C22semidirect
60. #2055C22semidirect
63. #1968C22semidirect
64. #1959C22semidirect
65. #1936C22semidirect
66. #1935C22semidirect
67. #1927C22semidirect
69. #1771C22semidirect
70. C4×#102C22semidirect
708. C2#28893direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25122
C2×#24472
#288932
C2×#20842
C2×#20652
C2×#20552
C2×#20122
C2×#19722
C2×#19682
C2×#19592
C2×#19362
C2×#19352
C2×#19272
C2×#18792
C2×#17712
C2×C4×#1022