Finite Group Databasebeta

← C2×#28650 #6464052

Subgroups — C2×#28650 #6464052

Order 512.

2348 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#490C211yesmaximal, characteristic
2C22×#446C211yesmaximal, characteristic, complemented
3C2×#2548C211yesmaximal, characteristic
4C2×#2584C211yesmaximal, characteristic, complemented
5#28650C2116yesmaximal, direct product factor
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic, complemented
8C2×#2065C211yesmaximal, characteristic, complemented
9C2×#2065C211yesmaximal, characteristic, complemented
10C2×#1989C211yesmaximal, characteristic, complemented
11C2×#1982C211yesmaximal, characteristic
12C2×#1936C211yesmaximal, characteristic, complemented
13C2×#1936C211yesmaximal, characteristic, complemented
14C2×#1934C211yesmaximal, characteristic, complemented
15C2×#1934C211yesmaximal, characteristic, complemented
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×#127C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#124—21no—
20C22×C4×Q8C2211yescharacteristic
21C2×#490—28no—
22C2×#490C2218yes—
23C2×#446—28no—
24C2×#446C2218yescomplemented
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#97C2211yescharacteristic
39C22×#97C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41#2584—24no—
42#2584C2218yescomplemented
43#2548—24no—
44#2548C2218yes—
45C2×#398—21no—
46C2×#395—21no—
47C2×#394—21no—
48C2×#394—21no—
49C2×#390—21no—
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#389—21no—
55C2×#389—21no—
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#389C2211yescharacteristic
59C2×#389C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C4⋊C4C2211yescharacteristic
62#2067C2218yescomplemented
63#2067C2218yescomplemented
64#2065C2218yescomplemented
65#2065C2218yescomplemented
66#1989C2218yescomplemented
67#1982C2218yes—
68#1936C2218yescomplemented
69#1936C2218yescomplemented
70#1934C2218yescomplemented
71#1934C2218yescomplemented
72#1776C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8D411yescharacteristic
80C23×D4—21no—
81C23×D4—21no—
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×#127—24no—
89C2×#127—22no—
90C2×#127—24no—
91C2×#127C2314yes—
92C2×#125—22no—
93C2×#125—24no—
94C2×#125—24no—
95C2×#125C2314yes—
96C2×#124—28no—
97C2×#124—48no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8D414yes—
101C2×C4×Q8C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
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×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#490—232no—
131#446—232no—
132C2×#107—24no—
133C2×#107C2314yes—
134C2×#106—24no—
135C2×#106C2314yes—
136C2×C4⋊C8—24no—
137C2×C4⋊C8—24no—
138C2×C4⋊C8—24no—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#102—24no—
143C2×#102—24no—
144C2×#102—24no—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102—24no—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#97—24no—
159C2×#97D418yes—
160C2×#97C2314yes—
161C2×#97C2314yes—
162C2×C4×C8—24no—
163C2×C4×C8C2314yes—
164#398—28no—
165#395—28no—
166#394—28no—
167#394—28no—
168#390—28no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390C2314yes—
174#390C2314yes—
175#390C2314yes—
176#390C2314yes—
177#389—28no—
178#389—28no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#389C2314yes—
186#389C2314yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2314yes—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203C25—21no—
204C25—21no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—42no—
208C22×Q8C2×D414yes—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—24no—
212C22×D4—21no—
213C22×D4—21no—
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—44no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—82no—
230C22×D4—82no—
231C22×D4C2412yes—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8C2×D414yes—
257C22×C8C2×D414yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262#127—28no—
263#127—28no—
264#127—48no—
265#125—28no—
266#125—28no—
267#125—48no—
268#124—432no—
269C4×Q8—216no—
270C4×Q8—28no—
271C4×Q8—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
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—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4#4114yes—
307C2×C4⋊C4#4114yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
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—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×C22⋊C4—82no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—44no—
337C2×C42—82no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D414yes—
342C2×C42#4114yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107—28no—
347#106—28no—
348C4⋊C8—28no—
349C4⋊C8—28no—
350C4⋊C8—28no—
351#102—28no—
352#102—28no—
353#102—28no—
354#102—28no—
355#102—28no—
356#102—28no—
357#102—28no—
358#102—28no—
359#97—28no—
360#97C2×D4116yes—
361C4×C8—28no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368C24—22no—
369C24—22no—
370C24—42no—
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—28no—
386C2×Q8—28no—
387C2×Q8—44no—
388C2×Q8—416no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—816no—
422C2×D4—816no—
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—24no—
433C22×C4—24no—
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—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#12114yes—
463C22×C4#12114yes—
464C22×C4AGL(1,ℤ/8)14yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C42−1+412yes—
477C22×C42−1+412yes—
478C22×C4C2511yescharacteristic
479C2×C8—28no—
480C2×C8—28no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8C22×D414yes—
494C2×C8C22×D414yes—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
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—48no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—48no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4C2×#4114yes—
521C4⋊C4C2×#4114yes—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—48no—
537C42—84no—
538C42#12118yes—
539C42C22×D414yes—
540C42C2×#4114yes—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23C2×#12112yes—
570C23#54812yes—
571C23#54812yes—
572C23C2×AGL(1,ℤ/8)12yes—
573C23#57512yes—
574C23#57612yes—
575C23#57712yes—
576C23C23×D411yescharacteristic
577Q8—416no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—816no—
587D4—816no—
588D4—84no—
589D4—84no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
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—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4C2×#12114yes—
636C2×C4C2×#12114yes—
637C2×C4C2×#12112yes—
638C2×C4#53214yes—
639C2×C4#53614yes—
640C2×C4#54812yes—
641C2×C4#54812yes—
642C2×C4#55614yes—
643C2×C4#55614yes—
644C2×C4C2×AGL(1,ℤ/8)14yes—
645C2×C4C23×D411yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C8—24no—
652C8—24no—
653C8—42no—
654C8—42no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—168no—
670C22—168no—
671C22#254514yes—
672C22#305514yes—
673C22#306214yes—
674C22#336614yes—
675C22#336814yes—
676C22#337014yes—
677C22#337114yes—
678C22C22×#12111yescharacteristic
679C22C2×#54811yescharacteristic
680C22C2×#54811yescharacteristic
681C22C22×AGL(1,ℤ/8)11yescharacteristic
682C22C2×#57511yescharacteristic
683C22C2×#57611yescharacteristic
684C22C2×#57711yescharacteristic
685C4—24no—
686C4—24no—
687C4—24no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—44no—
692C4—44no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4#254514yes—
698C4C2×#53211yescharacteristic
699C4C2×#53611yescharacteristic
700C4C2×#55611yescharacteristic
701C4C2×#55611yescharacteristic
702C2—162no—
703C2—162no—
704C2#2865018yesdirect product factor
705C2C2×#254511yescharacteristic
706C2C2×#305511yescharacteristic
707C2C2×#306211yescharacteristic
708C2C2×#336611yescharacteristic
709C2C2×#336811yescharacteristic
710C2C2×#337011yescharacteristic
711C2C2×#337111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#446C2semidirect
4. C2×#2584C2semidirect
5. #28650C2direct (N × H)
6. C2×#2067C2semidirect
7. C2×#2067C2semidirect
8. C2×#2065C2semidirect
9. C2×#2065C2semidirect
10. C2×#1989C2semidirect
12. C2×#1936C2semidirect
13. C2×#1936C2semidirect
14. C2×#1934C2semidirect
15. C2×#1934C2semidirect
16. C2×#1776C2semidirect
24. C2×#446C22semidirect
42. #2584C22semidirect
62. #2067C22semidirect
63. #2067C22semidirect
64. #2065C22semidirect
65. #2065C22semidirect
66. #1989C22semidirect
68. #1936C22semidirect
69. #1936C22semidirect
70. #1934C22semidirect
71. #1934C22semidirect
72. #1776C22semidirect
704. C2#28650direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4902
C22×#4462
C2×#25482
C2×#25842
#286502
C2×#20672
C2×#20652
C2×#19892
C2×#19822
C2×#19362
C2×#19342
C2×#17762